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Study of Calabi-Yau geometry Kanazawa, Atsushi 2014

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Study of Calabi–Yau GeometrybyAtsushi KanazawaB.Sc., University of Tokyo, 2008M.Sc., University of Tokyo, 2010A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)April 2014c© Atsushi Kanazawa 2014AbstractThis thesis studies various aspects of Calabi–Yau manifolds and related geometry. It isorganized into 6 chapters.Chapter 1 is the introduction of the thesis. It is devoted to background materials onK3 surfaces and Calabi–Yau threefolds. This chapter also serves to set conventions andnotations.Chapter 2 studies the trilinear intersection forms and Chern classes of Calabi–Yauthreefolds. It is concerned with an old question of Wilson [Wil3]. We demonstrate somenumerical relations between the trilinear forms and Chern classes.Chapter 3 provides the classification of Calabi–Yau threefolds with infinite fundamentalgroup, based on Oguiso and Sakurai’s work [OS]. Such Calabi–Yau threefolds are classifiedinto two types: type A and type K.Chapter 4 investigates Calabi–Yau threefolds of type K from the viewpoint of mirrorsymmetry, namely Yukawa couplings and Strominger–Yau–Zaslow conjecture. We obtainseveral results parallel to what is known for Borcea–Voisin threefolds: Voisin’s work onYukawa couplings [Voi], and Gross and Wilson’s work on special Lagrangian fibrations[GW].Chapter 5 studies some non-commutative projective Calabi–Yau schemes. The aimof this chapter is twofold: to construct the first examples of non-commutative projectiveCalabi–Yau schemes, in the sense of Artin and Zhang [AZ1], and to introduce a virtualcounting theory of stable modules on them.Chapter 6 is the conclusion of this thesis. We recapitulate the results obtained in thisthesis and also discuss future research directions.iiPrefaceChapter 2 is a version of the joint work [KW] with Pelham Wilson. Chapters 3 and 4are based on the collaboration [HK1, HK2] with Kenji Hashimoto. Chapters 5 is myindependent work.The publication relevant to this thesis is [KW]: A. Kanazawa and P. H. M. Wilson,Trilinear forms and Chern classes of Calabi–Yau threefolds, Osaka J. Math. Vol 51. No.1.203-213, 2014.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Lattices and Calabi–Yau Surfaces . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Lattice Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 K3 Surfaces and Enriques Surfaces . . . . . . . . . . . . . . . . . . 51.2 Calabi–Yau Threefolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Trilinear Forms and Chern Classes of Calabi–Yau Threefolds . . . . . . 92.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Bound for c2(X) ∪H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Bound for c3(X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 Residual Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Calabi–Yau Threefolds of Type K: Classification . . . . . . . . . . . . . . 213.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.1 Work of Oguiso and Sakurai . . . . . . . . . . . . . . . . . . . . . . 243.2.2 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2.3 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2.4 Moduli Spaces of Complex Structures . . . . . . . . . . . . . . . . . 37ivTable of Contents3.3 Key Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.3.1 Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.3.2 Proof of Key Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 413.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.5 Calabi–Yau Threefolds of Type A . . . . . . . . . . . . . . . . . . . . . . . 494 Calabi–Yau Threefolds of Type K: Mirror Symmetry . . . . . . . . . . . 534.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.2 Calabi–Yau Threefolds of Type K . . . . . . . . . . . . . . . . . . . . . . . 544.2.1 Some Topological Computation . . . . . . . . . . . . . . . . . . . . 564.2.2 Lattices H2(S,Z)G and H2(S,Z)HC2 . . . . . . . . . . . . . . . . . . 584.3 Yukawa Couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.3.1 Moduli Spaces and Mirror Maps . . . . . . . . . . . . . . . . . . . . 614.3.2 Brauer Group and A-Model Moduli Space . . . . . . . . . . . . . . 634.3.3 A- and B-Yukawa Couplings . . . . . . . . . . . . . . . . . . . . . . 644.3.4 Asymptotic Behavior of Yukawa Couplings . . . . . . . . . . . . . . 664.3.5 B-Model Computation . . . . . . . . . . . . . . . . . . . . . . . . . 674.4 Special Lagrangian Fibrations . . . . . . . . . . . . . . . . . . . . . . . . . 704.4.1 K3 Surfaces as HyperKa¨hler Manifolds . . . . . . . . . . . . . . . . 724.4.2 Calabi–Yau Threefolds of Type K . . . . . . . . . . . . . . . . . . . 724.5 Appendix (Borcea–Voisin Threefolds) . . . . . . . . . . . . . . . . . . . . . 755 Non-Commutative Projective Calabi–Yau Schemes . . . . . . . . . . . . 775.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.2 Non-Commutative Calabi–Yau Projective Schemes . . . . . . . . . . . . . . 795.2.1 Non-commutative Projective Schemes . . . . . . . . . . . . . . . . . 795.2.2 Calabi–Yau Condition . . . . . . . . . . . . . . . . . . . . . . . . . . 805.2.3 Proof of gl.dim(tails(An)) = n− 2 . . . . . . . . . . . . . . . . . . . 845.3 Hilbert Schemes of Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.4 Moduli Spaces via Differential Graded Lie Algebra . . . . . . . . . . . . . . 915.4.1 Differential Graded Lie Algebra . . . . . . . . . . . . . . . . . . . . 915.4.2 Deformation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.4.3 Geometric Invariant Theory . . . . . . . . . . . . . . . . . . . . . . 975.5 Symmetric Obstruction Theory . . . . . . . . . . . . . . . . . . . . . . . . . 995.5.1 Odd Symplectic Structure . . . . . . . . . . . . . . . . . . . . . . . 995.5.2 Calabi–Yau 3 Higher Tangent Complex . . . . . . . . . . . . . . . . 1016 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.1 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.2 Future Research Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . 106vTable of ContentsBibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109viList of Tables1.1 Discriminant groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Number of fix points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.1 Triplets (S,L, G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 Exponent (a, b), (a′, b′), (e, f) . . . . . . . . . . . . . . . . . . . . . . . . . 303.3 (2, 2, 2)-complete intersection . . . . . . . . . . . . . . . . . . . . . . . . . . 323.4 Dimensions of Γ, V and W . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.5 Moduli spaceMGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.6 rankΛH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.7 Lattice over Z2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.8 Case H = C2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.9 Case H = C3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.10 Invariant lattice H2(S,Z)G . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.11 Homology group H1(X,Z) ∼= (Z/2Z)n . . . . . . . . . . . . . . . . . . . . . 463.12 2-torsion points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.1 Trilinear intersection forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.2 Symplectic invariant lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.3 Lattices H2(S,Z)G and H2(S,Z)HC2 . . . . . . . . . . . . . . . . . . . . . . . 594.4 Complex structure, holomorphic 2-form and Ka¨hler form . . . . . . . . . . . 72viiList of Figures3.1 Tiling of the hyperbolic plane by geodesic triangles . . . . . . . . . . . . . . 484.1 A-model moduli space for Br(X) ∼= Z4 . . . . . . . . . . . . . . . . . . . . . 64viiiAcknowledgementsFirst and foremost, I would like to express my deepest gratitude to my family for theiraffection and care. They have always believed in me and encouraged me to pursue mygoals. Their guidance has made all the difference in my life and I am very grateful for alltheir support.I would like to thank my advisor Kai Behrend for his patience, support and guidance.I have had the privilege to learn much from him for three years. I am grateful to mycommittee members Jim Bryan and Zinovy Reichstein for their inspiration and guidance.They have had a great influence on my view of mathematics. Special thanks go to ShinobuHosono, who first introduced me to this rich subject, Calabi–Yau geometry, and keptencouraging me for years. I also wish to give thanks to Charles Doran and Keiji Oguisofor useful discussions and reference letters in my job application. I feel very lucky to meetall these senior mathematicians with various mathematical backgrounds.My thanks also go to the staff and faculty members at the UBC mathematics depart-ment for creating a comfortable environment in which this thesis was written. Amongothers, Marlowe Dirkson, Lee Yupitun, and Jessica Trat have always been so helpful. Iam grateful to Alejandro Adem, Bill Casselman, Jingyi Chen, Nassif Ghoussoub, JuliaGordon, Kalle Karu, Young-Heon Kim, Dale Rolfsen, and Lior Silberman for teaching mebeautiful mathematics, both in and out of class. I enjoyed stimulating discussions withgraduate students and postdoctoral fellows at UBC. Jose Gonzalez, Martijn Kool, EdwardRichmond, and Jonathan Wise have been good examples of senior mathematicians. I amvery appreciative to my office mates: Carmen Bruni, Vincent Chan, Robert Klinzmann,Maxim Stykow, and Thomas Wong, without whom this thesis would have been completedmuch earlier, but with much less fun.I am really grateful to my collaborators Kenji Hashimoto and Pelham Wilson for veryenjoyable and productive joint research experiences, and for allowing me to include someparts of our joint works in this thesis. I would also like to express gratitude to MakotoMiura, who has been a source of my inspiration. I benefited greatly from many discussionswith him and a large part of my research is influenced by his insight.ixAcknowledgementsThere are many friends outside of the mathematics department contributing to mymeaningful life in Vancouver. My time in Vancouver would have been very boring had itnot been for St. John’s College. It was a great pleasure to interact with and learn frommany brilliant people at the college. Among many others, I have to thank Mimi Brown,Kamila Kolpashnikova, Tianhan Liu, Szu Shen, Tatchai Titichetrakun, and Sheng-Jun Xufor sharing wonderful memories with me. I am also very proud of being a part of theLightburn family. Vancouver will always be my second home town.Several institutions made it possible for me to travel in my Ph.D. years. I enjoyedmy one-month stays at the Institut Joseph Fourier in Grenoble and at the MathematicalScience Research Institute in Berkeley, three-week stay at the University of Tokyo, andtwo-month stay at the Fields Institute in Toronto. Some parts of this thesis were carriedout at these institutes and I warmly thank them for their hospitality. I am particularlygrateful to Noriko Yui for her invitation to the Fields Institute in summer 2013.I am partially supported by the Four-Year Doctoral Fellowship, Itoko Muraoka Fel-lowship, and Reginald and Annie Van Fellowship. I am very honoured to hold thesescholarships.xDedicationTo Junji and Haru Kanazawa, Izumi and Ikiko Kazami.xiChapter 1IntroductionThe present thesis studies various aspects of Calabi–Yau manifolds and related geome-try. A particular emphasis is on Calabi–Yau threefolds and mirror symmetry. Calabi–Yauthreefolds are 3-dimensional analogues of elliptic curves and K3 surfaces, which have playedan important role in many fields of mathematics. In contrast to elliptic curves and K3 sur-faces, not much is known about Calabi–Yau theefolds. For example, it is an open problemwhether or not the number of topological types of Calabi–Yau threefolds is bounded. Onemotivation to study Calabi–Yau threefolds comes from superstring theory, which conjec-tures that our spacetime takes the form of M1,3 ×X, where M1,3 is the Minkowski spaceand X is a Calabi–Yau threefold. A detailed study of superstring theory led to the idea ofmirror symmetry.Conjecture 1.0.1 (Mirror Symmetry). For any Calabi–Yau threefold X, there exists aCalabi–Yau threefold Y such that the complex geometry of X is equivalent to the symplecticgeometry of Y , and vice versa.The above equivalence has been formulated and confirmed in many examples. Wecan think of mirror symmetry as a generalization of the classical Fourier transform (T-dulaity)1. Mirror symmetry relates various mathematical objects on distinct Calabi–Yauthreefolds in highly non-trivial ways. The process of building a mathematical foundationof mirror symmetry has given impetus to new fields in mathematics, such as Gromov–Witten theory and Fukaya category. Moreover, in the development of mirror symmetry, ithas become more apparent that Calabi–Yau threefolds enjoy very rich properties. Indeed,they are related to various fields of mathematics and can be studied with a wide range oftechniques. For example, in this thesis we use algebraic geometry, topology, representationtheory, lattice theory, differential geometry, Hodge theory, modular forms, derived categoryand non-commutative geometry.1.1 Lattices and Calabi–Yau SurfacesWe begin with a brief summary of the basics of lattices, K3 surfaces and Enriques surfacesthat we need throughout this thesis. Standard references are [BHPV, CS, Nik2].1 The Strominger–Yau–Zaslow conjecture [SYZ] claims that fiberwise Fourier transforms along specialLagrangian fibrations should geometrically explain mirror symmetry.11.1. Lattices and Calabi–Yau Surfaces1.1.1 Lattice TheoryA lattice is a free Z-module L of finite rank together with a symmetric bilinear form〈∗, ∗∗〉 : L × L → Z. With a slight abuse of notation, we denote a lattice simply by L.With respect to a choice of basis, the symmetric bilinear form is represented by a Grammatrix. The discriminant disc(L) of a lattice L is the determinant of the Gram matrixof L. We denote by O(L) the group of automorphisms of L. We define L(n) to be thelattice obtained by multiplying the bilinear form L by an integer n. We denote by 〈a〉the lattice of rank 1 generated by x with x2 := 〈x, x〉 = a. A lattice L is called even ifx2 ∈ 2Z for all x ∈ L. L is non-degenerate if disc(L) 6= 0 and unimodular if disc(L) = ±1.If L is a non-degenerate lattice, the signature of L is the pair (t+, t−) where t+ and t−respectively denote the dimensions of the positive and negative eigenspaces of L⊗ R. Wedefine signL := t+ − t−.A sublattice M of a lattice L is a submodule of L with the bilinear form of L restrictedto M . A sublattice M of L is called primitive if L/M is torsion free. For a sublattice Mof L, we denote the orthogonal complement of M in L by M⊥L (or simply M⊥):M⊥L := {x ∈ L | 〈x, y〉 = 0 (∀y ∈M)}.We assume that an action of a group G on a lattice L preserves the bilinear form unlessotherwise stated. For a lattice L with an action of G, we define the invariant part LG andthe coinvariant part LG of L byLG := {x ∈ L | g · x = x (∀g ∈ G)}, LG := (LG)⊥L .We simply denote L〈g〉 and L〈g〉 by Lg and Lg respectively for g ∈ G. If another group Hacts on L, we denote LG ∩ LH by LGH .The hyperbolic lattice U is the rank 2 lattice whose Gram matrix is given byU :=[0 11 0].The corresponding basis of U is called the standard basis and often denoted by e, f . Uis an indefinite, unimodular even lattice. Let Am, Dn, El, (m ≥ 1, n ≥ 4, l = 6, 7, 8)be the lattices defined by the corresponding Cartan matrices. Among others, the latticeE8 plays an important role in the study of K3 surfaces. E8 is the unique positive definite,21.1. Lattices and Calabi–Yau SurfacesL A(L) | disc(L)| |R|An Z/(n+ 1)Z n+ 1 n(n+ 1)Dn{Z/2Z× Z/2Z n evenZ/4Z n odd4 2n(n+ 1)E6 Z/3Z 3 72E7 Z/2Z 2 126E8 0 1 240 Table 1.1: Discriminant groupseven unimodular lattice of rank 8 up to isomorphism. The Gram matrix is given byE8 :=−2 0 1 0 0 0 0 00 −2 0 1 0 0 0 01 0 −2 1 0 0 0 00 1 1 −2 1 0 0 00 0 0 1 −2 1 0 00 0 0 0 1 −2 1 00 0 0 0 0 1 −2 10 0 0 0 0 0 1 −2.An even unimodular lattice of signature (3, 19) is isomorphic to Λ := U⊕3 ⊕ E8(−1)⊕2,which is called the K3 lattice.For indefinite unimodular lattices, the classification was given by Milnor [Mil]. Forn,m ≥ 1, there exists a unique odd unimodular lattice In,m := 〈1〉⊕n⊕〈−1〉⊕m of signature(n,m), up to isomorphism. The classification of indefinite even unimodular lattices is alsovery simple. Every indefinite even unimodular lattice can be realized as an orthogonalsum of copies of U and E8(±1) in an essentially unique way, the only relation beingE8 ⊕ E8(−1) ∼= U⊕8.Let L be a non-degenerate even lattice. The bilinear form of L determined a canonicalembedding L ↪→ L∨ := Hom(L,Z). The quotient group A(L) := L∨/L is a finite abeliangroup of order | disc(L)| and called the discriminant group of L. Table (1.1) shows somebasic examples, where |R| denotes the number of roots in L. The bilinear form on Lnaturally extends to the one on the dual L∨ and defines a quadratic mapq(L) : A(L) −→ Q/2Z, x+ L 7→ 〈x, x〉+ 2Z.We call the pair (A(L), q(L)) the discriminant form of L.31.1. Lattices and Calabi–Yau SurfacesExample 1.1.1 (L = An). Let e1, . . . en be a basis of An with e2i = −2 and 〈ei, ej〉 = δi,j−1for i < j. A generator of A(An) ∼= Z/(n + 1)Z is given by e := (∑ni=1 ei)/(n + 1) withq(An)(e) = −n/(n+ 1).Two even lattices L and L′ have isomorphic discriminant form if and only if they arestably equivalent, that is, L ⊕ K ∼= L′ ⊕ K ′ for some even unimodular lattices K andK ′. Since the rank of an even unimodular is divisible by 8, sign q(L) := signL mod 8 iswell-defined. Let M ↪→ L be a primitive embedding of non-degenerate even lattices andsuppose that L is unimodular, then there is a natural isomorphism(A(M), q(M)) ∼= (A(M⊥),−q(M⊥)).The genus of L is defined as the set of isomorphism classes of lattices L′ such that thesignature of L′ is the same as that of L and (A(L), q(L)) ∼= (A(L′), q(L′)).Theorem 1.1.2 ([Nik2, O’Mea]). Let L be a non-degenerate even lattice with rankL ≥ 3.If L ∼= U(n) ⊕ L′ for a positive integer n and a lattice L′, then the genus of L consists ofonly one class.Theorem 1.1.3 (Nikulin [Nik2]). Let L be a non-degenerate indefinite even lattice withrankL ≥ l(A(L))+ 2, where l(A(L)) is the minimum number of generators of A(L). Thenthe genus gL of L contains only one class.Let L be a lattice and M a module such that L ⊂M ⊂ L∨. We say that M equippedwith the induced bilinear form 〈∗, ∗∗〉 is an overlattice of L if 〈∗, ∗∗〉 takes integer valueson M . Any lattice which includes L as a sublattice of finite index is considered as anoverlattice of L.Proposition 1.1.4 ([Nik2]). Let L be a non-degenerate even lattice and M a submoduleof L∨ such that L ⊂M . Then M is an even overlattice of L if and only if the image of Min A(L) is an isotropic subgroup, that is, the restriction of q(L) to M/L is zero. Moreover,there is a natural one-to-one correspondence between the set of even overlattices of L andthe set of isotropic subgroups of A(L).Proposition 1.1.5 ([Nik2]). Let K and L be non-degenerate even lattices. Then thereexists a primitive embedding of K into an even unimodular lattice Γ such that K⊥ ∼= L,if and only if (A(K), q(K)) ∼= (A(L),−q(L)). More precisely, any such Γ is of the formΓλ ⊂ K∨ ⊕ L∨ for some isomorphismλ : (A(K), q(K))→ (A(L),−q(L)),where Γλ is the lattice corresponding to the isotropic subgroup{(x, λ(x)) ∈ A(K)⊕A(L) | x ∈ A(K)} ⊂ A(K)⊕A(L).41.1. Lattices and Calabi–Yau SurfacesLemma 1.1.6. Let L be a non-degenerate lattice and ι ∈ O(L) an involution. ThenL/(Lι ⊕ Lι) ∼= (Z/2Z)n for some n ≤ min{rankLι, rankLι}.Proof. For any x ∈ L, we have a decomposition x = x+ + x− with x+ ∈ Lι ⊗ Q andx− ∈ Lι⊗Q. We have 2x+ = x+ ι(x) ∈ L. We define φ(x mod Lι⊕Lι) = 2x+ mod 2Lι.We can easily see that φ : L/(Lι⊕Lι)→ Lι/2Lι is a well-defined injection. Hence we haveL/(Lι ⊕ Lι) ∼= (Z/2Z)n with n ≤ rankLι. Similarly, we have n ≤ rankLι.1.1.2 K3 Surfaces and Enriques SurfacesDefinition 1.1.7. A K3 surface S is a simply-connected compact complex surface S withtrivial canonical bundle KS = 0.It is a deep theorem due to Siu that any K3 surface is a Ka¨hler surface [Siu].Let S be a K3 surface. By Noether’s formula, b2(S) = 22, and since S is simply-connected, the second cohomology group H2(S,Z) is a free Z-module of rank 22. ByHodge index theorem and Wu’s formula, H2(S,Z) with its intersection form is an evenunimodular lattice of signature (3, 19). These uniquely characterize the lattice H2(S,Z)and we conclude that it is isomorphic to the K3 lattice Λ = E8(−1)⊕2 ⊕ U⊕3. Since S isKa¨hler, H2(S,Z) is endowed with a weight-two Hodge structureH2(S,C) = H2(S,Z)⊗ C = H2,0(S)⊕H1,1(S)⊕H0,2(S).The space H2,0(S) ∼= C is generated by the class of a nowhere vanishing holomorphic 2-form ωS , which we denote by the same ωS . The weight-two Hodge structure on the latticeH2(S,Z) is determined by the line CωS ⊂ H2(S,C). The algebraic lattice NS(S) and thetranscendental lattice T (S) of S are defined byNS(S) := {x ∈ H2(S,Z)∣∣ 〈x, ωS〉 = 0}, T (S) := NS(S)⊥H2(S,Z).Here we extend the bilinear form 〈∗, ∗∗〉 on H2(S,Z) to that on H2(S,Z)⊗C linearly. Theexponential exact sequence inducess an exact sequenceH1(S,OS) −→ H1(S,O×S ) ∼= Pic(S)c1−→ H2(S,Z) −→ H2(S,OS),from which we see that c1 yields an isomorphism between Pic(S) and NS(S).The open subset KS ⊂ H1,1(S,R) := H2(S,R) ∩H1,1(S) of classes of Ka¨hler forms isthe Ka¨hler cone of S. Set∆+ := {x ∈ NS(S) | x2 = −2, x effective},Hδ := {x ∈ H2(S,Z) | 〈x, δ〉 = 0} (δ ∈ ∆+).51.1. Lattices and Calabi–Yau SurfacesBy the Hodge index theorem, the set {x ∈ H1,1(S) | x2 > 0} consists of two disjointconnected cones. Let CS be the connected cone that contains the Ka¨hler cone KS . We callthe connected components of CS \⋃δ∈∆+ Hδ the chambers of CS . Then the Ka¨hler coneKS is the chamber given by{x ∈ CS | 〈x, δ〉 > 0 ∀δ ∈ ∆+}.The study of automorphism of K3 surfaces reduces to lattice theory due to the followingtheorems.Theorem 1.1.8 (Global Torelli Theorem [BHPV]). Let S and T be K3 surfaces. Letφ : H2(S,Z)→ H2(T,Z) be an isomorphism of lattices satisfying the following conditions.1. (φ⊗ C)(H2,0(S)) = H2,0(T ).2. There exists an element κ ∈ KS such that (φ⊗ R)(κ) ∈ KT .Then there exists a unique isomorphism f : T → S such that f∗ = φ.Theorem 1.1.9 (Surjectivity of the period map [BHPV]). Assume that vectors ω ∈ Λ⊗Cand κ ∈ Λ⊗ R satisfy the following conditions:1. 〈ω, ω〉 = 0, 〈ω, ω〉 > 0, 〈κ, κ〉 > 0 and 〈κ, ω〉 = 0.2. 〈κ, x〉 6= 0 for any x ∈ (ω)⊥Λ such that 〈x, x〉 = −2.Then there exist a K3 surface S and an isomorphism α : H2(S,Z)→ Λ of lattices such thatCω = (α⊗ C)(H2,0(S)) and κ ∈ (α⊗ R)(KS).An action of a group G on a K3 surface S induces a (left) G-action on H2(S,Z) byg · x := (g−1)∗x, g ∈ G, x ∈ H2(S,Z).The following lemma is useful to study finite group actions on K3 surfaces.Lemma 1.1.10 (Oguiso–Sakurai [OS, Lemma 1.7]). Let S be a K3 surface with an actionof a finite group G and let x be an element in NS(S)G ⊗ R with x2 > 0. Suppose that〈x, δ〉 6= 0 for any δ ∈ NS(S) with δ2 = −2. Then there exists γ ∈ O(H2(S,Z)) such thatγ(H2,0(S)) = H2,0(S), γ(x) ∈ KS, and γ commutes with G.Let g be an automorphism of a K3 surface S. We denote the fixed locus of g by Sg. Theautomorphism g is said to be symplectic if g∗ωS = ωS , and anti-symplectic if g∗ωS = −ωS .If g is an anti-symplectic involution, then g is locally linearized as (z1, z2) 7→ (z1,−z2) atany fixed point, thus Sg is the disjoint union of smooth curves, which is possibly empty.61.1. Lattices and Calabi–Yau Surfacesord(g) 2 3 4 5 6 7 8|Sg| 8 6 4 4 2 3 2Table 1.2: Number of fix pointsTheorem 1.1.11 ([Nik1]). Let g be a symplectic automorphism of S of finite order, thenord(g) ≤ 8 and the number of fixed points depends only on ord(g) as given in Table (1.2).Definition 1.1.12. An Enriques surface is the quotient surface S/〈ι〉 of a K3 surface Sby a fixed point free involution ι. A fixed point free involution ι of a K3 surface S is calledan Enriques involution.Theorem 1.1.13 ([Nik3]). Let ι be an involution of a K3 surface S. If ι is an Enriquesinvolution, then ι is anti-symplectic andH2(S,Z)ι ∼= U(2)⊕ E8(−2), H2(S,Z)ι ∼= U ⊕ U(2)⊕ E8(−2).Conversely, if H2(S,Z)ι ∼= U(2) ⊕ E8(−2) for an involution ι, then ι is an Enriquesinvolution.Example 1.1.14. Let S be a smooth tridegree (2, 2, 2)-hypersurface in P1×P1×P1. ThenS has an anti-symplectic involution given by ι : (x, y, z) 7→ (−x,−y,−z), where x, y, zare inhomogeneous coordinate of P1 × P1 × P1. The quotient surface T := S/〈ι〉 is anEnriques surface. The K3 surface S is realized as a surface of degree 12 in P7 via the Segreembedding.An example of a K3 surface with an Enriques involution we keep in mind in this thesisis the following Horikawa model.Example 1.1.15 (Horikawa model [BHPV, Section V.23]). Consider the double coveringpi : S → P1 × P1 branching along a smooth curve B of bidegree (4, 4). Then S is a K3surface. We denote by θ the covering involution on S. Assume that B is invariant underthe involution ι of P1×P1 given by (x, y) 7→ (−x,−y), where x and y are the inhomogeneouscoordinates of P1 × P1. The involution ι lifts to a symplectic involution of the K3 surfaceS. Then θ ◦ ι is an involution of S without fixed points unless B passes through one ofthe four fixed points of ι on P1 × P1. The quotient surface T = S/〈θ ◦ ι〉 is therefore anEnriques surface.S/〈θ〉id// S/〈θ◦ι〉P1 × P1 TProposition 1.1.16 ([BHPV, Proposition XIII.18.1]). Any generic K3 surface with anEnriques involution is realized as a Horikawa model defined above.71.2. Calabi–Yau Threefolds1.2 Calabi–Yau ThreefoldsDefinition 1.2.1. A Calabi–Yau threefold X is a compact Ka¨hler threefold with trivialcanonical bundle KX = 0 such that H1(X,OX) = 0.It is worth remarking that we do not assume that X is simply-connected. A rea-son for this is that simply-connected Calabi–Yau threefolds are not closed under mirrorsymmetry, that is, a mirror partner of a simply-connected Calabi–Yau threefold may notbe simply-connected (look at [GP] for example). With a view toward mirror symmetry,our definition seems more natural than strict Calabi–Yau threefolds, for which we requiresimply-connectedness. Since X is a Ka¨hler threefold, the cohomology groups admits theHodge decompositions:Hk(X,C) =⊕p+q=kHp,q(X) with Hp,q(X) = Hq,p(X).The only non-trivial part of the decomposition is H1,1(X) and H2,1(X). The former vectorspace H1,1(X) contains the Ka¨hler cone as an open cone and it represents the deformationof the Ka¨hler structure of X. The latter vector space H2,1(X) is isomorphic to H1(X,TX)by the Calabi–Yau condition KX = 0 and represents the first order deformation of thecomplex structure of X. In fact, the Bogomolov–Tian–Todorov theorem asserts that themoduli space of Calabi–Yau threefolds is unobstructed and therefore H1(X,TX) reallyrepresents the local complex moduli space around X. Mirror symmetry is a duality ofthe symplectic (Ka¨hler) geometry and the complex geometry of two distinct Calabi–Yauthreefolds X and Y , and swaps H1,1(X) and H1,1(Y ) in a non-trivial way.In contrast to elliptic curves and K3 surfaces, the topological structure of Calabi–Yautheefolds is not unique and the classification is far from understood. One useful theorem,which we will heavily rely on this thesis is the following:Theorem 1.2.2 ([Bea1]). Let X be a compact Ka¨bler manifold with trivial canonical bundleKX . Then there exists a finite e´tale covering Y of X which is a product of Ka¨bler manifoldsof the following form:Y = T ×M1 × · · ·Mk ×N1 × · · ·Nl,where T is a torus, Mi’s are simply-connected Calabi–Yau manifold and Nj’s are hy-perKa¨bler manifolds.Here a Calabi–Yau manifold M is a compact Ka¨hler manifold with trivial canonicalbundle KM = 0 such that Hp(M,OM ) = 0 for 1 < p < dimM . A hyperKa¨hler manifoldN is a simply-connected compact Ka¨hler manifold such that H0(N,Ω2N ) is spanned by anowhere non-degenerate 2-form. Theorem 1.2.2 in particular implies that a Calabi–Yauthreefold X with infinite fundamental group admits an e´tale covering either by an abelianthreefold or by the product of a K3 surface and an elliptic curve.8Chapter 2Trilinear Forms and Chern Classesof Calabi–Yau Threefolds2.1 IntroductionThis chapter is concerned with the interplay of the symmetric trilinear form µ on the secondcohomology group H2(X,Z) and the Chern classes c2(X), c3(X) of a Calabi–Yau threefoldX. It is an open problem whether or not the number of topological types of Calabi–Yauthreefolds is bounded and the original motivation of this work was to investigate topologicaltypes of Calabi–Yau threefolds via the trilinear form µ on H2(X,Z). The role that thetrilinear form µ plays in the geography of real sixfolds is indeed prominent as Wall provedthe following celebrated theorem by using surgery methods and homotopy informationassociated with these surgeries.Theorem 2.1.1 (Wall [Wal]). Diffeomorphism classes of simply-connected, spin, oriented,closed sixfolds X with torsion-free cohomology correspond bijectively to isomorphism classesof systems of invariants consisting of1. free Abelian groups H2(X,Z) and H3(X,Z),2. a symmetric trilinear from µ : H2(X,Z)⊗3 → H6(X,Z) ∼= Z defined byµ(x, y, z) := x ∪ y ∪ z,3. a linear map p1 : H2(X,Z)→ H6(X,Z) ∼= Z defined byp1(x) := p1(X) ∪ x,where p1(X) ∈ H4(X,Z) is the first Pontrjagin class of X,subject to: for any x, y ∈ H = H2(X,Z),µ(x, x, y) + µ(x, y, y) ≡ 0 (mod 2),4µ(x, x, x)− p1(x) ≡ 0 (mod 24).The isomorphism H6(X,Z) ∼= Z above is given by pairing the cohomology class with thefundamental class [X] with natural orientation.92.1. IntroductionAt present the classification of trilinear forms, which is as difficult as that of diffeomor-phism classes of real sixfolds, is unknown. In the light of the essential role of the K3 latticein the study of K3 surfaces, we would like to propose the following question:Question 2.1.2. What kind of trilinear forms µ occur on Calabi–Yau threefolds?The quantized version of the trilinear forms, known as Gromov–Witten invariants orA-model Yukawa couplings, are also of interest to both mathematicians and physicists.One advantage of working with complex threefolds is that we can reduce our questions tothe theory of complex surfaces by considering linear systems of divisors. Furthermore, forCalabi–Yau threefolds X, the second Chern class c2(X) and the Ka¨hler cone KX turn outto encode important information about µ (look at [Wil2, Wil4] for details). The purpose ofour study is to take the first step towards an investigation on how the Calabi–Yau structureaffects the trilinear form µ and the Chern classes of the underlying manifold.It is worth mentioning some relevant work from elsewhere. Let (X,H) be a polarizedCalabi–Yau threefold. A bound for the value c2(X) ∪H in terms of the triple intersectionH3 is well-known (see for example [Wil3]) and hence there are only finitely many possibleHilbert polynomialsχ(X,OX(nH)) =H36n3 + c2(X) ∪H12nfor such (X,H). It is shown by Oguiso and Peternell [OP] that we can always pass froman ample divisor H on a Calabi–Yau threefold to a very ample one 10H. Put it anotherway, an ample line bundle is essentially the same as a very ample line bundle for a Calabi–Yau threefold. Then, by the standard Hilbert scheme theory, we see that the Calabi–Yauthreefold X belongs to a finite number of families. This implies that once we fix a positiveinteger n ∈ N, there are only finitely many diffeomorphism classes of polarized Calabi–Yauthreefolds (X,H) with H3 = n, and in particular only finitely many possibilities for theChern classes c2(X) and c3(X) of X. Explicit bounds on the Euler characteristic c3(X) interms of H3 for certain types of Calabi–Yau threefolds are given in [CK1]; the idea of thischapter is to record the following simple explicit result which holds in general, and whichmay be useful for both mathematicians and physicists.Theorem 2.1.3. Let (X,H) be a very amply polarized Calabi–Yau threefold, i.e. x = His a very ample divisor on X. Then the following inequality holds:−36µ(x, x, x)− 80 ≤ c3(X)2= h1,1(X)− h2,1(X) ≤ 6µ(x, x, x) + 40.Moreover, the above inequality can be sharpened by replacing the left hand side by −80,−180 and right hand side by 28, 54 when µ(x, x, x) = 1, 3 respectively.102.2. Bound for c2(X) ∪HIn the last section, we study the cubic form µ(x, x, x) : H2(X,Z) → Z for a Ka¨hlerthreefold X, assuming that µ(x, x, x) has a linear factor over R. Some properties of thelinear form and the residual quadratic form on H2(X,R) are obtained; possible signaturesof the residual quadratic form are determined under a certain condition (for example, whenX is a Calabi–Yau threefold).2.2 Bound for c2(X) ∪HIn this section, we collect some properties of the trilinear form and the second Chern classesof a Calabi–Yau threefold. We will always work over the field of complex numbers C.Let X be a smooth Ka¨hler threefold. Throughout this thesis, we write ci(X) := ci(TX)the i-th Chern class of the tangent bundle TX. Ka¨hler classes constitute an open coneKX ⊂ H1,1(X,C) ∩H2(X,R),called the Ka¨hler cone. The closure KX then consists of nef classes and hence is called thenef cone. The second Chern class c2(X) ∈ H4(X,Z) defines a linear function on H2(X,R).Under the assumption that X is minimal (for instance a Calabi–Yau threefold), results ofMiyaoka [Miy] imply that for any nef class x ∈ KX , we have c2(X) ∪ x ≥ 0.Let X be a smooth complex threefold. We define a symmetric trilinear form µ :H2(X,Z)⊗3 → H6(X,Z) ∼= Z by setting µ(x, y, z) := x ∪ y ∪ z for x, y, z ∈ H2(X,Z). Bysmall abuse of notation we also use µ for its scalar extension. For a Calabi–Yau threefoldX, the exponential exact sequence gives an identificationPic(X) = H1(X,O×X) ∼= H2(X,Z).The divisor class [D] is then identified with the first Chern class c1(OX(D)) of the associatedline bundle OX(D). In the following we freely use this identification.Theorem 2.2.1 (Hirzebruch–Riemann–Roch). For a Calabi–Yau threefold X, we haveχ(X,OX(D)) =16µ(x, x, x) + 112c2(X) ∪ xfor any x = D ∈ H2(X,Z). Therefore2µ(x, x, x) + c2(X) ∪ x ≡ 0 (mod 12).112.2. Bound for c2(X) ∪HIn particular, c2(X) ∪ x is an even integer for any x ∈ H2(X,Z). In the case where Xis simply-connected and the cohomology groups are torsion-free, it also follows from thefact p1(X) = −2c2(X) and Wall’s theorem (Theorem 2.1.1). The role played by p1(X) inhis theorem is now replaced by c2(X) for Calabi–Yau threefolds.For a compact complex surface S, the geometric genus pg(S) is defined by pg(S) :=dimCH0(S,Ω2S). The basic strategy we take in the following is to reduce the question onCalabi–Yau threefolds to compact complex surface theory by considering linear systems ofdivisors.Proposition 2.2.2. Let X be a Calabi–Yau threefold.1. For any ample x = H ∈ KX ∩H2(X,Z) with |H| free and dimC |H| ≥ 2, the followinginequalities hold.12c2(X) ∪ x ≤ 2µ(x, x, x) + Cwhere C = 18 when µ(x, x, x) even and C = 15 otherwise.2. If furthermore the canonical map Φ|KH | : H → P|KH | (which is given by the restrictionof the map Φ|H| to H) is birational onto its image, the following inequality holds.12c2(X) ∪ x ≤ µ(x, x, x) + 203. If furthermore the image of the canonical map in (2) is generically an intersection ofquadrics, the following inequality holds.c2(X) ∪ x ≤ µ(x, x, x) + 48Proof. (1) By Bertini’s theorem, a general member of the complete linear system |H|is irreducible and gives us a smooth compact complex surface S ⊂ X. Applying theHirzebruch–Riemann–Roch theorem and the Kodaira vanishing theorem to the ample linebundle OX(H), we can readily show that the geometric genus is given bypg(S) =16µ(x, x, x) + 112c2(X) ∪ x− 1.Since KS is ample, the surface S is a minimal surface of general type. Then the Noether’sinequality12K2S ≥ pg(S)− 2yields the desired two equalities depending on the parity of K2S = µ(x, x, x).122.2. Bound for c2(X) ∪H(2) The proof is almost identical to the first case. Since the surface S obtained aboveis a minimal canonical surface, i.e. the canonical map Φ|KS | : S → P|KS | is birational ontoits image, the Castelnuovo inequality for minimal canonical surfacesK2S ≥ 3pg(S)− 7yields the inequality.(3) We say that an irreducible variety S ⊂ Ppg−1 is generically an intersection ofquadrics if S is one component of the intersection of all quadrics through S. In this case,M. Reid [Rei] improved the above inequality toK2S ≥ 4pg(S) + q(S)− 12.The irregularity q(S) := dimH1(S,OS) = 0 in our case.If x ∈ KX is very ample, the conditions in Proposition 2.2.2 (1) and (2) are automat-ically satisfied. The first two inequalities are optimal in the sense that equalities hold forthe complete intersection Calabi–Yau threefolds P(14,4) ∩ (8) and P4 ∩ (5) respectively.Recall that the ∆-genus ∆(X,H) of a polarized variety (X,H) is defined as∆(X,H) := dimX + degX − h0(X,H)It is worth noting that the polarized Calabi–Yau threefolds (X,H) with ∆-genus ∆(X,H) ≤2 are classified by Oguiso [Ogu1] and it is observed in [Wil3] that the inequalityc2(X) ∪H ≤ 10H3holds for those with ∆(X,H) > 2. Schimmrigk’s experimental observation [Sch] howeverconjectures the existence of a better linear upper bound of c2(X) for Calabi–Yau hyper-surfaces in weighted projective spaces.Proposition 2.2.3. The surface S in the proof of Proposition 2.2.2 is a minimal surfaceof general type with non-positive second Segre class s2(S). Moreover, s2(S) is negative ifand only if c2(X) is not identically zero.Proof. Let i : S ↪→ X be the inclusion and we identifyH4(S,Z) ∼= Z. A simple computationshows c1(S) = −i∗(x) and c2(S) = µ(x, x, x) + c2(X) ∪ x. Since x ∈ KX ,s2(S) = c1(S)2 − c2(S) = −c2(X) ∪ x ≤ 0by the result of Y. Miyaoka [Miy]. The second claim follows from the fact that KX ⊂H2(X,R) is an open cone.If X is a Calabi–Yau threefold and the linear form c2(X) is identically zero, it is wellknown that X is the quotient of an abelian threefold by a finite group acting freely on it(see [OS, HK1] and Chapter 3.5).132.3. Bound for c3(X)2.3 Bound for c3(X)In this section, we apply to smooth projective threefolds the Fulton–Lazarsfeld theory fornef vector bundles developed by Demailly, Peternell and Schneider [DPS]. This gives usseveral inequalities among Chern classes and cup products of certain cohomology classes.When X is a Calabi–Yau threefold, these inequalities simplify and provide us with effectivebounds for the Chern classes.Definition 2.3.1. A vector bundle E on a complex manifold X is called nef if the Serreline bundle OP(E)(1) on the projectivized bundle P(E) is nef.Theorem 2.3.2 (Demailly–Peternell–Schneider [DPS]). Let E be a nef vector bundle overa complex manifold X equipped with a Ka¨hler class ωX ∈ KX . Then for any Schur poly-nomial Pλ of degree 2r and any complex submanifold Y of dimension d, we have∫YPλ(c(E)) ∧ ωd−rX ≥ 0.Here we let deg ci(E) = 2i for 0 ≤ i ≤ rankE and the Schur polynomial Pλ(c(E)) ofdegree 2r is defined byPλ(c(E)) := det(cλi−i+j(E))for each partition λ := (λ1, λ2, . . . ) a r of a non-negative integer r ≤ dimY with λk ≥ λk+1for all k ∈ N.Example 2.3.3 ([Laz, Example 8.3.4]). Let X be a complex threefold and E a vectorbundle of rankE = 3, thenP(1)(c(E)) = c1(E), P(2)(c(E)) = c2(E), P(1,1)(c(E)) = c1(E)2 − c2(E)P(3)(c(E)) = c3(E), P(2,1)(c(E)) = c1(E) ∪ c2(E)− c3(E),P(1,1,1)(c(E)) = c1(E)3 − 2c1(E) ∪ c2(E) + c3(E).Proposition 2.3.4. Let X be a smooth projective threefold, x, y ∈ KX ∩ H2(X,Z) andassume x is very ample, then the following inequalities hold.1. 8µ(x, x, x) + 2c2(X) ∪ x ≥ 4µ(c1(X), x, x) + c3(X)2. 64µ(x, x, x) + 4µ(c1(X), c1(X), x) + 4c2(X) ∪ x+ c3(X)≥ 32µ(c1(X), x, x) + c1(X) ∪ c2(X)3. 80µ(x, x, x) + 10µ(c1(X), c1(X), x) + 2c1(X) ∪ c2(X)≥ 40µ(c1(X), x, x) + µ(c1(X), c1(X), c1(X)) + 10c2(X) ∪ x+ c3(X)142.3. Bound for c3(X)4. 12µ(x, x, y) + c2(X) ∪ y ≥ 4µ(c1(X), x, y)5. 24µ(x, x, y) + µ(c1(X), c1(X), y) ≥ 8µ(c1(X), x, y) + c2(X) ∪ y6. 6µ(x, y, y) ≥ µ(c1(X), y, y)Proof. The very ample divisor x = H gives us an embedding Φ|H| : X → P(V ), whereV = H0(X,OX(H)). Using the Euler sequence and the Koszul complex, we obtain thefollowing exact sequence of sheaves0 −→ Ωk+1P(V ) −→k+1∧V ⊗OP(V )((−k − 1)H) −→ ΩkP(V ) −→ 0for each 1 ≤ k ≤ dimC V − 1. We thus see that ΩP(V )(2H) is a quotient of O⊕(dimC V2 )P(V ) . Thevector bundle ΩX(2H) is then generated by global sections because it is a quotient of theglobally generated vector bundle ΩP(V )|X(2H). We hence conclude that ΩX(2H) is a nefvector bundle (see [Laz, Proposition 6.1.2]). Applying Theorem 2.3.2 (or rather the inequal-ities derived using the above example) to our nef vector bundle ΩX(2H), straightforwardcomputation shows the desired inequalities.The above result (with appropriate modification) certainly carries over to complexmanifolds of dimension other than 3.Corollary 2.3.5. Let X be a Calabi–Yau threefold, x, y ∈ KX ∩H2(X,Z) and assume xis very ample, then the following inequalities hold.1. 8µ(x, x, x) + 2c2(X) ∪ x ≥ c3(X)2. 64µ(x, x, x) + 4c2(X) ∪ x+ c3(X) ≥ 03. 80µ(x, x, x) ≥ 10c2(X) ∪ x+ c3(X)4. 24µ(x, x, y) ≥ c2(X) ∪ yIn recent literature there has been some interest in finding practical bounds for topolog-ical invariants of Calabi–Yau threefolds. As is mentioned in the introduction, the standardHilbert scheme theory assures that possible Chern classes of a polarized Calabi–Yau three-fold (X,H) are in principle bounded once we fix a triple intersection number H3 = n ∈ N,but now that we have effective bounds for the Chern classes (with a bit of extra data for thesecond Chern class c2(X)) as follows. Recall first that it is shown by Oguiso and Peternell[OP] that we can always pass from an ample divisor H on a Calabi–Yau threefold to a veryample one 10H. More precisely:152.4. Residual Quadratic FormsTheorem 2.3.6 (Oguiso–Peternell [OP]). Let (X,H) be a polarized Calabi–Yau threefoldwith at worst Q-factorial terminal singularities. Then the following hold:1. |mH| gives a birational map for m ≥ 5,2. |mH| is free for m ≥ 5,3. mL is simply-generated2, in particular very ample for m ≥ 10.Then the last inequality in Corollary 2.3.5 says that once we know the trilinear formµ on the ample cone KX there are only finitely many possibilities for the linear functionc2(X) : H2(X,Z)→ Z. We shall now give a simple explicit formula to give a range of theEuler characteristic c3(X) of a Calabi–Yau threefold X.Theorem 2.3.7. Let (X,H) be a very amply polarized Calabi–Yau threefold, i.e. x = His a very ample divisor on X. Then the following inequality holds:−36µ(x, x, x)− 80 ≤ c3(X)2= h1,1(X)− h2,1(X) ≤ 6µ(x, x, x) + 40.Moreover, the above inequality can be sharpened by replacing the left hand side by −80,−180 and right hand side by 28, 54 when µ(x, x, x) = 1, 3 respectively.Proof. This is readily proved by combining Proposition 2.2.2 (1), (2) and Corollary 2.3.5(1), (2), (4).The smallest and largest known Euler characteristics c3(X) of a Calabi–Yau threefoldX are −960 and 960 respectively. Our formula may replace the question of finding a rangeof c3(X) by that of estimating the value µ(x, x, x) for an ample class x ∈ KX ∩H2(X,Z).2.4 Residual Quadratic FormsIn this section we further study the cubic form µ(x, x, x) : H2(X,Z) → Z for a Ka¨hlerthreefold X, assuming that µ(x, x, x) has a linear factor over R. We will see that thelinear factor and the residual quadratic form are not independent. Possible signatures ofthe residual quadratic form are also determined under a certain condition. If the secondBetti number b2(X) > 3, the residual quadratic form may endow the second cohomologyH2(X,Z) mod torsion with a lattice structure.2 A line bundle H on a normal variety X is said to be simply-generated if C-graded algebra⊕∞k=0H0(X,OX(kH)) is generated by the linear piece H0(X,OX(H)). It is well-known that an amplesimply-generated line bundle is very ample.162.4. Residual Quadratic FormsWe start with fixing our notation. Let ξ : V → R be a real quadratic form. Oncewe fix a basis of the R-vector space V , ξ may be represented as ξ(x) = xtAξx for somesymmetric matrix Aξ. The signature of a quadratic form ξ is a triple (s+, s0, s−) where s0is the number of zero eigenvalues of Aξ and s+ (s−) is the number of positive (negative)eigenvalues of Aξ. Aξ also defines a linear map Aξ : V → V ∨ (or a symmetric bilinear formAξ : V ⊗2 → R). The quadratic form ξ is called (non-)degenerate if dimRKer(Aξ) > 0 (= 0).We say that ξ is definite if it is non-degenerate and either s+ or s− is zero, and indefiniteotherwise.Let X be a Ka¨hler threefold and assume that its cubic form µ(x, x, x) factors asµ(x, x, x) = ν(x)ξ(x),where ν is a linear and ξ is quadratic map H2(X,R) → R respectively. We can alwayschoose the linear form ν so that it is positive on the Ka¨hler cone KX ⊂ H2(X,R). It isproven (see the proof of Lemma 4.3 in [Wil1]) that there exists a non-zero point on thequadric hypersurfaceQξ := {x ∈ H2(X,R) | ξ(x) = 0}and hence ξ is indefinite provided that the irregularity q(X) = 0 and the second Bettinumber b2(X) > 3.Proposition 2.4.1. Let X be a Ka¨hler threefold. Assume that the trilinear form µ(x, x, x)decomposes as ν(x)ξ(x) over R (if the quadratic form is not a product of linear forms, thenwe may work over Q by the Galois theory) and the linear form ν is positive on the Ka¨hlercone KX . Then the following hold.1. dimRKer(Aξ) ≤ 1. If ξ is a degenerate quadratic form, its restriction ξ|Hν to thehyperplaneHν := {x ∈ H2(X,R) | ν(x) = 0}is non-degenerate.2. If the irregularity q(X) = 0 (for example a Calabi–Yau threefold), then the signatureof ξ is either(2, 0, b2(X)− 2), (1, 1, b2(X)− 2) or (1, 0, b2(X)− 1).3. The above three signatures are realized by some Calabi–Yau threefolds with b2(X) = 2.Proof. (1) Let ωX ∈ KX be a Ka¨hler class. The Hard Lefschetz theorem states thatthe map H2(X,R) → H4(X,R) defined by α 7→ ωX ∪ α is an isomorphism. Hence thecubic form µ(x, x, x) depends on exactly b2(X) variables. Then the quadratic form ξ must172.4. Residual Quadratic Formsdepend on at least b2(X)− 1 variables and thus we have dimRKer(Aξ) ≤ 1. Assume nextthat the quadratic form ξ is degenerate. Then the linear form ν is not the zero form onKer(Aξ) (otherwise µ(x, x, x) depends on less than b2(X) variables). The restriction ξ|Hνis non-degenerate becauseH2(X,R) = Hν ⊕Ker(Aξ)as a R-vector space.(2) Let L1 ∈ KX ∩H2(X,R) be an ample class such that µ(L1, L1, L1) = 1. Since theKa¨hler cone KX ⊂ H2(X,R) is an open cone, X is projective by the Kodaira embeddingtheorem. Then the Hodge index theorem states that the symmetric bilinear formbµ,L1 := µ(L1, ∗, ∗∗) : H2(X,R)⊗2 ∼= (NS(X)⊗ R)⊗2 → Rhas signature (1, 0, b2(X) − 1), where NS(X) is the Neron–Severi group of X. Note thatdimR(L⊥1 ∩Hν) ≥ b2(X)− 2, where L⊥1 denotes the orthogonal space to L1 with respect tothe non-degenerate bilinear form bµ,L1 . We then have two cases; the first is when dimR(L⊥1 ∩Hν) = b2(X)− 1 (i.e. L⊥1 = Hν). In this case we can write down a basis L2, . . . , Lb2(X) forthe subspace Hν which diagonalizes the quadratic form bµ,L1 |Hν , and hence (noting thatL1 6∈ Hν) the Gramian matrix of bµ,L1 with respect to the basis L1, . . . , Lb2(X) of H2(X,R)isAbµ,L1 := (bµ,L1(Li, Lj)) = diag(1,−1, . . . ,−1).If dimR(L⊥1 ∩ Hν) = b2(X) − 2, then we can write down a basis L2, . . . , Lb2(X)−1 for thesubspace L⊥1 ∩Hν which diagonalizes the quadratic form bµ,L1 |L⊥1 ∩Hν , and then extend itto a basis L2, . . . , Lb2(X) of Hν . Thus in both cases L1, . . . , Lb2(X) is a basis for H2(X,R);the corresponding matrix Abµ,L1 will not be diagonal in this second case, but the first(b2(X)− 1)-principal minor is, with one +1 and b2(X)− 2 entries −1 on the diagonal.Let us define a new basis {Mi}b2(X)i=1 ofH2(X,R) by settingMi = Li for 1 ≤ i ≤ b2(X)−1andMb2(X) = Lb2(X) +b2(X)∑i=2bµ,L1(Li, Lb2(X))Li ∈ Hν .Let x =∑b2(X)i=1 aiMi. Then the hyperplane Hν is defined by the equation a1 = 0 and theKa¨hler cone KX lies on the side where a1 > 0 by the assumption on ν. Therefore we haveµ(x, x, x) = a1(a21 −b2(X)−1∑i=2a2i + Ca1ab2(X) +Da2b2(X))182.4. Residual Quadratic Formsfor some (explicit) constants C,D ∈ R. Since the quadratic form is positive on the theKa¨hler cone KX , there must be at least one positive eigenvalue and hence possible signa-tures are (2, 0, b2(X)− 2), (1, 1, b2(X)− 2) and (1, 0, b2(X)− 1).(3) Consider a Calabi–Yau threefold XII7 (1, 1, 1, 2, 2)2−186 [HLY, p.575] given as a reso-lution of a degree 7 hypersurface in the weighted projective space P(1,1,1,2,2). Its cubic formis given bya1(14a21 + 21a1a2 + 9a22),whose quadratic form has signature (2, 0, 0). The cubic form of a hypersurface Calabi–Yauthreefold (P3 × P1) ∩ (4, 2) is2a31 + 12a21a2,whose quadratic form has signature either (1, 0, 1) or (1, 1, 0), depending on its decompo-sition.The restriction ξ|Hν may be degenerate if ξ is non-degenerate. The cubic form of theabove Calabi–Yau threefold (P3 × P1) ∩ (4, 2) gives an example of such phenomenon. Letν(a) = 2a1 and ξ(a) = a1(a1 + 6a2). Then ξ is hyperbolic and non-degenerate, but itsrestriction to Hν is trivial.Let X be a Ka¨hler threefold. If b2(X) > 3, the cubic form µ cannot consist of threelinear factors over R and hence if µ contains a linear factor it must be rational (see alsothe comment after Lemma 4.2 [Wil1]). Hence an appropriate scalar multiple of ξ endowsthe second cohomology H2(X,Z) mod torsion with a lattice structure.Example 2.4.2 (Enriques Calabi–Yau threefold [HK1]). Let X be a K3 surface with anEnriques involution ιS. Let E be an elliptic curve with the negation ιE. Then we can definea new involution ι of S × E by ι := (ιS , ιE). The free quotientX := (S × E)/〈ι〉is a Calabi–Yau threefold with b2(X) = 11. The cubic form µ(x, x, x) of X has a linearfactor (which, we assume, is positive on the Ka¨hler cone KX) and the residual quadraticform ξ has signature (1, 1, 9). More precisely, the lattice structure on H2(X,Z) mod torsionassociated with appropriate ξ is given byU ⊕ E8(−1)⊕ 〈0〉.Proposition 2.4.3. Let G be a finite group acting on a Ka¨hler threefold X and φ :G → GL(H2(X,Z)) the induced representation. Assume that the trilinear form decom-poses µ(x, x, x) = ν(x)ξ(x) as above. Then the image of φ : G→ GL(H2(X,Z)) lies in theorthogonal group O(ξ) associated with the quadratic form ξ.192.4. Residual Quadratic FormsProof. Since the cubic form µ : H2(X,R) → R is invariant under G, it is enough to showthat the linear form ν is invariant under G. There exists x ∈ KX such that Rx is a trivialrepresentation of G (by averaging a Ka¨hler class over G) and then the representation φ isa direct sum of two subrepresentations Rx ⊕Hν . Since ν is a linear form, this shows theinvariance of ν under G.This proposition may be useful to study group actions on the cohomology groupH2(X,Z).Although the case where the trilinear form µ has a linear factor is rather special, it isintimately related to the famous open problem: whether or not Calabi–Yau threefolds withρ = 3 admit infinite automorphism group (see also the remark after Proposition 3.4.6).Proposition 2.4.4 (Lazic´, Oguiso and Peternell [LOP]). Let X be a Calabi–Yau threefoldwith ρ(X) = 3. Then the automorphism group Aut(X) is either finite or it is an almostabelian group of rank 1, i.e. it is isomorphic to Z up to finite kernel and cokerel.Moreover, they reduced the problem to the case where µ has a linear factor [LOP].20Chapter 3Calabi–Yau Threefolds of Type K:Classification3.1 IntroductionThe present chapter is concerned with the Calabi–Yau threefolds with infinite fundamentalgroup. Let X be a Calabi–Yau threefold with infinite fundamental group. Then theBogomolov decomposition theorem (Theorem 1.2.2) implies that X admits an e´tale Galoiscovering either by an abelian threefold or by the product of a K3 surface and an ellipticcurve. We callX of type A in the former case and of type K in the latter case. Among manycandidates for such coverings, we can always find a unique smallest one, up to isomorphismas a covering [Bea2, Proposition 3]. We call the smallest covering the minimal splittingcovering of X. The main result of this chapter is the following:Theorem 3.1.1 (Theorem 3.2.1). There exist exactly eight Calabi–Yau threefolds of typeK, up to deformation equivalence. The equivalence class is uniquely determined by theGalois group G of the minimal splitting covering. Moreover, the Galois group is isomorphicto one of the following combinations of cyclic and dihedral groupsC2, C2 × C2, C2 × C2 × C2, D6, D8, D10, D12, or C2 ×D8.Most Calabi–Yau threefolds we know have finite fundamental groups: for example,complete intersection Calabi–Yau threefolds in toric varieties or homogeneous spaces, and(resolutions of singularities of) finite quotients thereof. Calabi–Yau threefolds with infinitefundamental group were only partially explored before the pioneering work of Oguiso andSakurai [OS]. In their paper, they made a list of possible Galois groups for type K but itwas not settled whether they really occur or not. In this chapter, we complement their workby providing the full classification (Theorem 3.1.1) and also give an explicit presentationfor the deformation classes of the eight Calabi–Yau threefolds of type K.The results described in this chapter represent the first step in our program which isaimed at more detailed understanding of Calabi–Yau threefolds of type K. Calabi–Yauthreefolds of type K are relatively simple yet rich enough to display the essential complexi-ties, and we expect that they will provide good testing-ground for general theories and con-213.1. Introductionjectures. Indeed, the simplest example, known as the Enriques Calabi–Yau threefold (or theFHSV-model [FHSV]), has been one of the most tractable compact Calabi–Yau threefoldsboth in string theory and mathematics (see for example [36, FHSV, Asp, KM2, MP, PP]).A particularly nice property of Calabi–Yau threefolds of type K is their fibration struc-ture; they all admit a K3 fibration, an Abelian surface fibration, and an elliptic fibration(Proposition 3.4.3). This rich structure suggests that they play an important role in dual-ities among various string theories. In the next chapter, we will discuss mirror symmetryof Calabi–Yau threefolds of type K.We will also provide the full classification of Calabi–Yau threefolds of type A, againbased on Oguiso and Sakurai’s work [OS]. In contrast to Calabi–Yau threefolds of typeK, Calabi–Yau threefolds of type A are classified not by the Galois groups of the minimalsplitting coverings, but by the minimal totally splitting coverings, where abelian threefoldsthat cover Calabi–Yau threefolds of type A split into the product of three elliptic curves(Theorem 3.5.3). Together with the classification of Calabi–Yau threefolds of type K, wefinally complete the full classification of Calabi–Yau threefolds with infinite fundamentalgroup:Theorem 3.1.2 (Theorem 3.5.5). There exist exactly fourteen deformation classes ofCalabi–Yau threefolds with infinite fundamental group. More precisely, six of them areof type A, and eight of them are of type K.It is remarkable that we can study Calabi–Yau threefolds very concretely by simplyassuming that their fundamental groups are infinite. We hope that our results unveil aninteresting class of Calabi–Yau threefolds and shed some light on the further investigationof general compact Calabi–Yau threefolds.The layout of this chapter is as follows. Section 2 begins with a review of the funda-mental work of Oguiso and Sakurai [OS], which essentially reduces the study of Calabi–Yauthreefolds of type K to that of K3 surfaces equipped with Calabi–Yau actions (Definition3.2.6). We will then provide the full classification of Calabi–Yau threefolds of type K,presenting all the deformation equivalence classes. Section 3 is devoted to the proof of akey technical result, Lemma 3.2.14 (Key Lemma), which plays a crucial role in Section 3.Section 4 addresses some basic properties of Calabi–Yau threefolds of type K. In Section 5,we will improve Oguiso and Sakurai’s work on Calabi–Yau threefolds of type A. This sec-tion finally completes the classification of Calabi–Yau threefolds with infinite fundamentalgroup.223.2. Classification3.2 ClassificationIn this chapter, we consider the Calabi–Yau threefolds with infinite fundamental group.The Bogomolov decomposition theorem [Bea1] implies that such a Calabi–Yau threefoldX admits an e´tale Galois covering either by an abelian threefold or by the product of aK3 surface and an elliptic curve. We call X of type A in the former case and of typeK in the latter case. Among many candidates of such coverings, we can always find aunique smallest one, up to isomorphism as a covering [Bea2]. We call the smallest coveringthe minimal splitting covering of X. The goal of this section is to provide the followingclassification theorem of Calabi–Yau threefolds of type K:Theorem 3.2.1. There exist exactly eight Calabi–Yau threefolds of type K, up to defor-mation equivalence. The equivalence class is uniquely determined by the Galois group Gof the minimal splitting covering. Moreover, the Galois group is isomorphic to one of thefollowing combinations of cyclic and dihedral groupsC2, C2 × C2, C2 × C2 × C2, D6, D8, D10, D12, or C2 ×D8.We will also give an explicit presentation for the eight Calabi–Yau threefolds. For thereader’s convenience, here we outline the proof of Theorem 3.2.1. Firstly, by the work ofOguiso and Sakurai [OS], the classification of Calabi–Yau threefolds of type K essentiallyreduces to that of K3 surfaces S equipped with a Calabi–Yau action (Definition 3.2.6) ofa finite group of the form G = H o 〈ι〉. Here the action of H on S is symplectic and ι isan Enriques involution of S. A sketch of the proof of the classification of such K3 surfacesis the following:1. We make a list of examples of K3 surfaces S with a Calabi–Yau G-action. They aregiven as double coverings of P1 × P1 (H-equivariant Horikawa models).2. It is proven that there exists an element v ∈ NS(S)G such that v2 = 4 (Key Lemma).3. It is shown that S has a projective model of degree 4 and admits a G-equivariantdouble covering of a quadric hypersurface in P3, which is isomorphic to P1 × P1 if itis smooth. Therefore, S is generically realized as an H-equivariant Horikawa model.4. We classify the deformation equivalence classes of S on a case-by-case basis and alsoexclude an unrealizable Galois group.It is worth noting that a realization of a K3 surface S with a Calabi–Yau G-action as aHorikawa model is in general not unique (Propositions 3.2.11 and 3.2.13).233.2. Classification3.2.1 Work of Oguiso and SakuraiWe begin with a brief review of Oguiso and Sakurai’s work [OS]. Let X be a Calabi–Yau threefold of type K. Then the minimal splitting covering pi : S × E → X is obtainedby imposing the condition that the Galois group of the covering pi does not contain anyelements of the form (idS , non-zero translation of E).Definition 3.2.2. We call a finite group G a Calabi–Yau group if there exist a K3 surfaceS, an elliptic curve E and a faithful G-action on S ×E such that the following conditionshold.1. G contains no elements of the form (idS ,non-zero translation of E).2. The G-action on H3,0(S × E) ∼= C is trivial.3. The G-action is free, that is, (S × E)g = ∅ for all g ∈ G, g 6= 1.4. G does not preserve any holomorphic 1-form, that is, H0(S × E,ΩS×E)G = 0.We call S × E a target threefold of G.The Galois group G of the minimal splitting covering S × E → X of a Calabi–Yauthreefold X of type K is a Calabi–Yau group. Conversely, if G is a Calabi–Yau group witha target space S ×E of G, then the quotient (S ×E)/G is a Calabi–Yau threefold of typeK.Let G be a Calabi–Yau group and S × E a target threefold of G. Thanks to a resultof Beauville [? ], we have a canonical isomorphism Aut(S × E) ∼= Aut(S)× Aut(E). Theimages of G ⊂ Aut(S×E) under the two projections to Aut(S) and Aut(E) are denoted byGS and GE respectively. It can be proven that GS ∼= G ∼= GE via the natural projections:Aut(S)⋃Aut(S × E)p1oo⋃p2// Aut(E)⋃GS Gp1|G∼=oop2|G∼=// GE .We denote by gS and gE the elements in GS and GE respectively corresponding to g ∈ G,that is, p1(g) = gS , p2(g) = gE .Proposition 3.2.3 (Oguiso–Sakurai [OS, Lemma 2.28]). Let G be a Calabi–Yau groupand S × E a target threefold of G. Define H := Ker(G → GL(H2,0(S))) and take anyι ∈ G \H. Then the following hold.243.2. Classification1. ord(ι) = 2 and G = H o 〈ι〉, where the semi-direct product structure is given byιhι = h−1 for all h ∈ H.2. gS is an Enriques involution for any g ∈ G \H.3. ιE = − idE and HE = 〈ta〉×〈tb〉 ∼= Cn×Cm under an appropriate origin of E, whereta and tb are translations of order n and m respectively such that n|m. Moreover wehave (n,m) ∈ {(1, k) (1 ≤ k ≤ 6), (2, 2), (2, 4), (3, 3)}.Although the case (n,m) = (2, 4) is eliminated from the list of possible Calabi–Yaugroups in [OS], there is an error in the proof of Lemma 2.29 in [OS], which is used to provethe proposition above3. In fact, there exists a Calabi–Yau group of the form (C2×C4)oC2 ∼=C2 × D8 (Proposition 3.2.11). For the sake of completeness, here we settle the proof ofLemma 2.29 in [OS]. We do not repeat the whole argument but give a proof of the non-trivial part: (n,m) 6= (1, 7), (1, 8), (2, 6), (4, 4). The reader can skip this part, assumingProposition 3.2.3.Proof of (n,m) 6= (1, 7), (1, 8), (2, 6), (4, 4).1. (1, 7) : Let HS = 〈g〉 ∼= C7. Since ιgι = g−1, 〈ι〉 ∼= C2 acts on Sg, which hascardinality 3 and thus has a fixed point. But this contradicts with the fact thatSf = ∅ for any f ∈ GS \HS .2. (1, 8) : Let HS = 〈g〉 ∼= C8. Note that 〈g, ι〉/〈g2〉 ∼= C2×C2 and acts on Sg2 \Sg whichhas cardinality 4− 2 = 2. Then this action induces a homomorphism φ : C2 × C2 →S2. Since Sf = ∅ for all f ∈ GS \ HS , Ker(φ)(6= 1) ⊂ 〈g〉/〈g2〉 ∼= C2 and thusKer(φ) ∼= 〈g〉/〈g2〉. This contradicts with our subtracting Sg from Sg2 .3. (2, 6) : Let HS = 〈g〉 × 〈h〉 ∼= C2 × C6. |Sh| = 2 and thus there is a homomorphismφ : 〈g, ι〉 ∼= C2 × C2 → S2. Since Sf = ∅ for all f ∈ GS \ HS , Ker(φ) = 〈g〉 ∼= C2.Let p ∈ Sg be one of the fixed points. Then we have a faithful representationHS = C2 × C6 → SL(TpS) ∼= SL(2,C). This contradicts with the classification offinite subgroups in SL(2,C).4. (4, 4) : Let HS = 〈g〉×〈h〉 ∼= C4×C4. Note that 〈g, h, ι〉/〈g2, h2〉 ∼= C2×C2×C2 andacts on Sh2 . Note also that |Sh2\Sh| = 8−4 = 4. Then this induces a homomorphismφ : C2×C2×C2 → S4. Since S4 does not contain C2×C2×C2, Ker(φ) ⊂ 〈g, h〉/〈g2, h2〉is not trivial. Let α be a lift of a non-trivial element of Ker(φ) and take a fixed pointp ∈ Sh2 \ Sh. Then we have a natural injection 〈α, h2〉 → SL(TpS) ∼= SL(2,C). Inaddition using h /∈ Ker(φ), we obtain 〈α, h2〉 ∼= C4 × C2, which contradicts with theclassification of finite subgroups in SL(2,C).3 The error in [OS, Lemma 2.28] is that, with their notation, the group 〈α, h2S〉 is not necessarilyisomorphic to either C2 × C2 or C4, but may be isomorphic to C2.253.2. ClassificationWe now state a main result of [OS] with a slight correction.Theorem 3.2.4 (Oguiso–Sakurai [OS, Theorem 2.23]). Let X be a Calabi–Yau threefoldof type K. Let S×E → X be the minimal splitting covering and G its Galois group. Thenthe following hold.1. G is isomorphic to one of the following:C2, C2 × C2, C2 × C2 × C2, D6, D8, D10, D12, C2 ×D8, or (C3 × C3)o C2.2. In each case the Picard number ρ(X) of X is uniquely determined by G and is cal-culated as ρ(X) = 11, 7, 5, 5, 4, 3, 3, 3, 3 respectively.3. The cases G ∼= C2, C2 × C2, C2 × C2 × C2 really occur.It has not been settled yet whether or not there exist Calabi–Yau threefolds of typeK with Galois group G ∼= D2n (3 ≤ n ≤ 6), C2 × D8 or (C3 × C3) o C2. Note that theexample of a Calabi–Yau threefold of type K with G ∼= D8 presented in Proposition 2.33in [OS] is incorrect4. We will settle this classification problem of Calabi–Yau threefolds oftype K and also give an explicit presentation of the deformation classes.Example 3.2.5 (Enriques Calabi–Yau threefold). Let S be a K3 surface with an Enriquesinvolution ι and E an elliptic curve with the negation −1E. The free quotientX := (S × E)/〈(ι,−1E)〉is the simplest Calabi–Yau threefold of type K, known as the Enriques Calabi–Yau threefold.3.2.2 ConstructionThe goal of this section is to make a list of concrete examples of Calabi–Yau threefoldsof type K. We will later show that the list covers all the generic Calabi–Yau threefolds oftype K. We begin with the definition of Calabi–Yau actions, which is based on Proposition3.2.3.Definition 3.2.6. Let G be a finite group. We say that an action of G on a K3 surface Sis a Calabi–Yau action if the following hold.1. G = Ho〈ι〉 for some H ∼= Cn×Cm with (n,m) ∈ {(1, k) (1 ≤ k ≤ 6), (2, 2), (2, 4), (3, 3)},and ι with ord(ι) = 2. The semi-direct product structure is given by ιhι = h−1 for allh ∈ H.4 The error in [OS, Proposition 2.33] is that, with their notation, Sab 6= ∅.263.2. Classification2. H acts on S symplectically and any g ∈ G \H acts on S as an Enriques involution.Recall that any generic K3 surface with the simplest Calabi–Yau action, namely anEnriques involution, is realized as a Horikawa model (Proposition 1.1.16). We will seethat any K3 surface equipped with a Calabi–Yau G-action is realized as an H-equivariantHorikawa model (Proposition 3.2.11).We can construct a Calabi–Yau G-action on a K3 surface as follows. Let u, x, y, z, wbe affine coordinates of C× C2 × C2. Define L byL :=(C× (C2 \ {0})× (C2 \ {0}))/(C×)2,where the action of (µ1, µ2) ∈ (C×)2 is given by(u, x, y, z, w) 7→ (µ21µ22u, µ1x, µ1y, µ2z, µ2w).The projection C× C2 × C2 → C2 × C2 descends to the mappi : L→ Z :=((C2 \ {0})× (C2 \ {0}))/(C×)2 ∼= P1 × P1.Note that L is naturally identified with the total space of OZ(2, 2). Let F = F (x, y, z, w) ∈H0(OZ(4, 4)) be a homogeneous polynomial of bidegree (4, 4). Assume that the curveB ⊂ Z defined by F = 0 has at most ADE-singularities. We define S0 byS0 := {u2 = F} ⊂ L.In other words, S0 is a double covering of Z branching along B. The minimal resolutionS of S0 is a K3 surface (see the proof of Lemma 3.2.9 below). The group Γ := GL(2,C)×GL(2,C) acts on L by, for γ = M1 ×M2 ∈ Γ,γ(u, x, y, z, w) = (u, x′, y′, z′, w′),[x′y′]= M1[xy],[z′w′]= M2[zw].If F is invariant under the action of γ ∈ Γ, then γ naturally acts on S0 as well. We denoteby γ+ the induced action of γ on S. The covering transformation of S0 → Z, which isdefined by (u, x, y, z, w) 7→ (−u, x, y, z, w), induces the involution j := (√−1 I2 × I2)+ onS. Note that there are two lifts of the action of γ on Z: γ+ and γ− := γ+j. The K3surface S with the polarization L := pi∗OZ(1, 1) is a polarized K3 surface of degree 4. LetAut(S,L) denote the automorphism group of the polarized K3 surface (S,L).Definition 3.2.7. We denote by (S,L, G) a triplet consisting of a polarized K3 surface(S,L) defined above and a finite subgroup G ⊂ Aut(S,L).273.2. ClassificationWe define Aut(S,L)+ to be the subgroup of Aut(S,L) preserving each ruling Z → P1.Then we haveAut(S,L)+ = {γ ∈ Γ∣∣ γ∗F = F}/{λ1I2 × λ2I2∣∣ λ21λ22 = 1}. (3.2.1)Remark 3.2.8. Since the Picard group of Z is isomorphic to Z⊕2 (hence torsion free), aline bundle M on Z such that M⊗2 ∼= OZ(4, 4) is isomorphic to OZ(2, 2). Therefore, adouble covering of Z branching along B is unique and isomorphic to S0.Lemma 3.2.9. Let ωS be a nowhere vanishing holomorphic 2-form on S. If F is invariantunder the action of γ = M1 ×M2 ∈ Γ, we have(γ±)∗ωS = ± det(M1) det(M2)ωS .Proof. Since the residue of (xdy − ydx) ∧ (zdw − wdz)/u gives a nowhere vanishing holo-morphic 2-form on S (for example, see [Muk, Lemma 2.1]), the equality in the assumptionholds.Lemma 3.2.10. Let φ : Y → Y0 be the minimal resolution of a surface Y0 with at mostADE-singularities. Then an automorphism g of Y0 has a fixed point if and only if theinduced action of g on Y has a fixed point.Proof. The assertion follows from the fact that any automorphism of a connected ADE-configuration has a fixed point.Proposition 3.2.11. Let (S,L, G) be a triplet defined in Definition 3.2.7. Assume thatthe action of G on S is a Calabi–Yau action. Then such triplets (S,L, G) are classified intothe types in Table (3.1) up to isomorphism. Here two triplets (S,L, G) and (S′,L′, G′) areisomorphic if there exists an isomorphism f : S → S′ such that f∗L′ = L and f−1 ◦G′ ◦f =G. A triplet (S,L, G) for each type is defined as follows.1. F is invariant under the action of M(a)×M(b) for all (a, b) ∈ Ξ and ι1, whereM(a) :=[exp(2piia) 00 exp(−2piia)], ι1 :=[0 11 0]×[0 11 0].2. H := 〈(M(a)×M(b))+∣∣ (a, b) ∈ Ξ〉.3. G := H o 〈ι〉, ι := ι−1 =(√−1[0 11 0]×[0 11 0])+.4. for any g ∈ G \H, the action of g on B has no fixed point.283.2. ClassificationH Ξ basis of H0(OZ(4, 4))GC1 ∅ xiy4−izjw4−j + x4−iyiz4−jwjC2 {(1/4, 1/4)} xiy4−izjw4−j + x4−iyiz4−jwj (i ≡ j mod 2)C2 {(1/4, 0)} xiy4−izjw4−j + x4−iyiz4−jwj (i ≡ 0 mod 2)C3 {(1/3, 1/3)} xiy4−izjw4−j + x4−iyiz4−jwj (i+ j ≡ 1 mod 3)C4 {(1/8, 1/8)} x4z4 + y4w4, x4w4 + y4z4, x3yzw3 + xy3z3w, x2y2z2w2C4 {(1/8, 1/4)} x4z3w + y4zw3, x4zw3 + y4z3w, x2y2z4 + x2y2w4, x2y2z2w2C5 {(1/5, 2/5)} x4zw3 + y4z3w, x3yz4 + xy3w4, x2y2z2w2C6 {(1/12, 1/6)} x4z4 + y4w4, x4zw3 + y4z3w, x2y2z2w2C2 × C2 {(1/4, 0), (0, 1/4)} xiy4−izjw4−j + x4−iyiz4−jwj (i ≡ j ≡ 0 mod 2)C2 × C4 {(1/8, 1/8), (0, 1/4)} x4z4 + y4w4, x4w4 + y4z4, x2y2z2w2Table 3.1: Triplets (S,L, G)Furthermore, for a generic F ∈ H0(OZ(4, 4))G, the surface S0 is smooth and the condition(4) is satisfied, and thus the action of G on S0(= S) is a Calabi–Yau action.Remark 3.2.12. In Proposition 3.2.11, the group G ⊂ Aut(S,L) for each type acts onH0(OZ(4, 4)) in a natural way by using the generator matrices. Hence we can define theG-invariant space H0(OZ(4, 4))G. We will use a similar convention in Proposition 3.2.13.Proof. Let (S,L, G) be a triplet such that the action of G = H o 〈ι〉 on S is a Calabi–Yauaction. By Definition 3.2.6, H is isomorphic to one in the table or C3×C3. For g ∈ G \H,the action of g preserves each ruling Z → P1; otherwise, Zg is 1-dimensional and Sg 6= ∅.Since G is generated by G \H, we may assume that any element in G is of the form γ±for γ ∈ Γ with γ∗F = F by (3.2.1). By Lemma 3.2.10, the condition (4) is satisfied.We begin with the case H = C1. Since Sι = ∅, it follows that Zι is (at most) 0-dimensional. Hence we may assume thatι =(λ[0 11 0]×[0 11 0])+after changing the coordinates of Z. Since the action of ι on S is anti-symplectic byTheorem 1.1.13, we have λ = ±√−1 by Lemma 3.2.9, thus ι = ι−1 .Let us next consider the case H = Cn (1 ≤ n ≤ 6). Let σ be a generator of H.By the argument above and the relation ισι = σ−1, we may assume that ι = ι−1 andσ = (λM(a) ×M(b))+ for some a, b ∈ Q after changing the coordinates of Z. Since theaction of σ on S is symplectic, we have λ = ±1 by Lemma 3.2.9, thus σ = (M(a)×M(b))+.293.2. Classification(a, b) (a′, b′) (e, f)(i) (1/4, 0) (0, 1/4) (0, 0)(ii) (1/4, 0) (0, 1/4) (0, 1)(iii) (1/8, 1/8) (0, 1/4) (0, 0)(iv) (1/8, 1/4) (0, 1/4) (0, 0)(v) (1/3, 1/3) (0, 1/3) (0, 0)Table 3.2: Exponent (a, b), (a′, b′), (e, f)If ka 6∈ 14Z and kb ∈12Z for some k ∈ Z, then F is divisible by x2y2. Hence this case isexcluded. We can see that (a, b), (b, a), (−a, b), (a+1/2, b), and (ka, kb) with GCD(k, n) = 1give isomorphic triplets. Therefore, we may assume that (a, b) is one of the following:(1/4, 1/4), (1/4, 0), (1/3,1/3), (1/8, 1/8), (1/8, 1/4),(1/5, 1/5), (1/5, 2/5), (1/12, 1/12), (1/12, 1/6).Here we have n = min{k ∈ Z>0 | ka ∈ 12Z, kb ∈12Z}. Suppose that (a, b) = (1/5, 1/5) or(1/12, 1/12). Then F is a linear combination ofx4w4 + y4z4, x3yzw3 + xy3z3w, x2y2z2w2,and hence B has a singular point of multiplicity 4 at, for instance, [1 : 0] × [1 : 0]. Thiscontradicts to the assumption that B admits only ADE-singularities. Therefore the cases(a, b) = (1/5, 1/5) and (1/12, 1/12) are excluded.Lastly, let us consider the cases H = C2 × C2, C2 × C4, and C3 × C3. Let σ, τ begenerators of H such that ord(σ) is divisible by ord(τ). By a similar argument for H = Cn,n = ord(σ), we may assume thatσ = (M(a)×M(b))+, τ =(λM(a′)[0 11 0]e×M(b′)[0 11 0]f)+, ι = ι−1 ,where e, f ∈ {0, 1}, λ2(−1)e+f = 1 and a, b, a′, b′ ∈ Q. We can chose (a, b) as is given inthe table for H = Cn. Moreover, we may assume that a′ = 0 after replacing τ by σkτ forsome k ∈ Z. By the argument for H = C1, it follows that Zhι is 0-dimensional for h ∈ H.This implies that e = 0, and f = 0 if (a, b) 6= (1/4, 0). We may assume that one of thecases in Table (3.2) occurs. We can check that the cases (i) and (ii) for H = C2×C2 giveisomorphic triplets. Since F is divisible by x2y2 in the cases (iv) and (v), these cases areexcluded.303.2. ClassificationConversely, we check that the action of G on S is a Calabi–Yau action for (S,L, G) ofeach type. By the argument above, the action of H on S is symplectic. Let g ∈ G \ H.Then Bg = ∅ and Zg is 0-dimensional, thus Sg is either empty or 0-dimensional. Recallthat the fixed locus of an anti-symplectic involution of a K3 surface is 1-dimensional if itis not empty. This implies that Sg = ∅. Therefore, the action of G on S is a Calabi–Yauaction.To show the smoothness of S0 for a generic F , it suffices, by Bertini’s theorem, to showthat B is smooth on the base locus of the linear system defined by H0(OZ(4, 4))G. We cancheck this directly. We also find that the action of any g ∈ G \H on B has no fixed pointfor a generic F .For H = C2 or C4, we obtain two families of K3 surfaces with a Calabi–Yau action ofG = H o 〈ι〉 by Proposition 3.2.11. As we will see in Proposition 3.2.13 below, if we forgetthe polarizations, they are essentially and generically the same families of K3 surfaces witha Calabi–Yau G-action. LetQ = Q(s1, t1, s2, t2, s3, t3) ∈ H0(OP(2, 2, 2)), P := P1 × P1 × P1,be a homogeneous polynomial of tridegree (2, 2, 2), where [si : ti] is homogeneous coordi-nates of the i-th P1. Assume that the surfaceS0 := {Q = 0} ⊂ Phas at most ADE-singularities. Then the minimal resolution S of S0 is a K3 surface. Letpi : P→ P1 denote the i-th projection. With this notation, we have the following.Proposition 3.2.13. Let G ⊂ PSL(2,C)3 be the group defined by Ξ′ in Table (3.3), in asimilar way to Proposition 3.2.11. More precisely, H is generated by M(a)×M(b)×M(c)for (a, b, c) ∈ Ξ′, and G := H o 〈ι〉, whereι =[0 11 0]×[0 11 0]×[0 11 0].For a generic Q ∈ H0(OP(2, 2, 2))G, the surface S0 is smooth and the action of G onS(= S0) is a Calabi–Yau action. Moreover, a generic triplet for H = Cn (1 ≤ n ≤ 4) orC2 × C2 constructed in Proposition 3.2.11 is of the form (S,L, G), whereL = ((pα × pβ)∗OP1×P1(1, 1))|Swith α, β ∈ {1, 2, 3}, α 6= β.313.2. ClassificationH Ξ′ basis of H0(OP(2, 2, 2))GC1 ∅ si1t2−i1 sj2t2−j2 sk3t2−k3 + s2−i1 ti1s2−j2 tj2s2−k3 tk3C2 {(1/4, 1/4, 0)}si1t2−i1 sj2t2−j2 sk3t2−k3 + s2−i1 ti1s2−j2 tj2s2−k3 tk3(i+ j ≡ 0 mod 2)C3 {(1/3, 1/3, 1/3)}si1t2−i1 sj2t2−j2 sk3t2−k3 + s2−i1 ti1s2−j2 tj2s2−k3 tk3(i+ j + k ≡ 0 mod 3)C4 {(1/8, 1/8, 1/4)}si1t2−i1 sj2t2−j2 sk3t2−k3 + s2−i1 ti1s2−j2 tj2s2−k3 tk3(i+ j + 2k ≡ 0 mod 4)C2 × C2 {(1/4, 0, 1/4), (0, 1/4, 1/4)}si1t2−i1 sj2t2−j2 sk3t2−k3 + s2−i1 ti1s2−j2 tj2s2−k3 tk3(i+ k ≡ j + k ≡ 0 mod 2)Table 3.3: (2, 2, 2)-complete intersectionProof. In each case, we can check, for a generic Q ∈ H0(OP(2, 2, 2))G, that S0 is smoothand that the action of any g ∈ G\H on S(= S0) has no fixed point by a direct computation(see the proof of Proposition 3.2.11). Since the action of H on S is symplectic (Lemma3.2.9), the action of G on S is a Calabi–Yau action. This proves the first assertion.Let us next show the second assertion. We assume that (α, β) = (1, 2) as the othercases are similar. Define the mapφ : V := H0(OP(2, 2, 2))G →W := H0(OP(4, 4, 0))GbyQ = As23 +Bs3t3 + Ct23 7→ F = det[A B/2B/2 C].The branching locus B of the double covering p1 × p2 : S0 → P1 × P1 is defined by F = 0.The map φ gives a correspondence between (2, 2, 2)-hypersurfaces in P with a Calabi–YauG-action and double coverings of P1×P1 branching along a (4, 4)-curve with a Calabi–YauG-action. Hence it suffices to show that φ is dominant. We show this by comparing thedimensions of φ−1(F ), V and W . By the argument above and generic smoothness ([Har,Corollary III.10.7]), we may assume that S0 and φ−1(F ) is smooth (hence S = S0) bytaking a generic Q ∈ V . Let ∆ be a contractible open neighborhood of Q in φ−1(F ). Weconstruct a natural family of embeddings S → P parametrized by ∆ as follows. Since thebranching locus of the double covering(p1 × p2)× id∆ : S → (P1 × P1)×∆, S := {(x,Q′)∣∣ Q′(x) = 0} ⊂ P×∆,323.2. ClassificationH Ξ dimΓ dimV dimWC1 ∅ 1 14 13C2 {(1/4, 1/4)} 1 8 7C2 {(1/4, 0)} 0 8 8C3 {(1/3, 1/3)} 0 5 5C4 {(1/8, 1/8)} 0 4 4C4 {(1/8, 1/4)} 0 4 4C2 × C2 {(1/4, 0), (0, 1/4)} 0 5 5Table 3.4: Dimensions of Γ, V and Wis B ×∆, we have a natural commutative diagramS ×∆ f∼=//(p1×p2)×id∆''OOOOOOOOOOOS(p1×p2)×id∆(P1 × P1)×∆,wherefQ′ := f(∗, Q′) : S → {Q′ = 0}, Q′ ∈ ∆,is a G-equivariant isomorphism and fQ = idS . Since ∆ is connected and the Picard groupof S is discrete, the map fQ′ is represented by γ ∈ GL(2,C)3 such that γ commutes withG in PSL(2,C)3. Furthermore, since (p1 × p2) ◦ fQ′ = p1 × p2, we may assume thatγ = I2 × I2 ×M . Then φ(γ∗Q) = det(M)2F . Therefore, we have ∆ ⊂ Γ∗Q, where Γ is asubgroup of GL(2,C)3 defined byΓ := {γ = I2 × I2 ×M∣∣ γ commutes with G in PSL(2,C)3, det(M) = ±1}.Since dim∆ ≤ dimΓ, φ is dominant if dimΓ ≤ dimV − dimW . This can be checkeddirectly, as indicated in Table (3.4).3.2.3 UniquenessIn this section, we will prove the uniqueness theorem of Calabi–Yau actions (Theorem3.2.19). We will also show the non-existence of a K3 surface with a Calabi–Yau G-actionfor G ∼= (C3 × C3) o C2 (Theorem 3.2.20). Throughout this section, we fix a semi-directproduct decomposition G = H o 〈ι〉 of a Calabi–Yau group G as in Proposition 3.2.3. Thekey to proving the uniqueness is the following lemma, whose proof will be given in Section3.3.333.2. ClassificationLemma 3.2.14 (Key Lemma). Let S be a K3 surface with a Calabi–Yau G-action. Thenthere exists an element v ∈ NS(S)G such that v2 = 4.First, we consider the (coarse) moduli space of K3 surfaces S with a Calabi–Yau G-action. Let ΨG denote the set of actions ψ : G → O(Λ) of G on Λ = U⊕3 ⊕ E8(−1)⊕2such that there exist a K3 surface S with a Calabi–Yau G-action and a G-equivariantisomorphism H2(S,Z) → Λψ. Here we denote Λ with a G-action ψ by Λψ. The groupO(Λ) acts on ΨG by conjugation:(γ · ψ)(g) = γψ(g)γ−1, γ ∈ O(Λ), ψ ∈ ΨG, g ∈ G.Define the period domain D˜G byD˜G :=⊔ψ∈Ψ′GD˜G,ψ, D˜G,ψ := {Cω ∈ P((Λψ)Hι ⊗ C)∣∣ 〈ω, ω〉 = 0, 〈ω, ω〉 > 0},where Ψ′G is a complete representative system of the quotient ΨΛ/O(Λ). For any K3surface S with a Calabi–Yau G-action, there exist a unique ψ ∈ Ψ′G and a G-equivariantisomorphism α : H2(S,Z)→ Λψ. Under the period map, S with the G-action correspondsto the period point(α⊗ C)(H2,0(S)) ∈ D˜G,ψ ⊂ D˜G.Lemma 3.2.15. Let S be a K3 surface with a G-action. If the induced action ψ : G→ O(Λ)(which is defined modulo the conjugate action of O(Λ)) is an element in ΨG, the G-actionon S is a Calabi–Yau action.Proof. In general, a symplectic automorphism g of a K3 surface S of finite order is charac-terized as an automorphism such that H2(S,Z)g is negative definite [Nik1, Theorem 3.1].Also, an Enriques involution is characterized by Lemma 1.1.13.Proposition 3.2.16. For the moduli space MG of K3 surfaces S with a Calabi–Yau G-action, the period map defined above induces an isomorphismτ :MG →⊔ψ∈Ψ′G(DG,ψ/O(Λ, ψ)).HereDG,ψ :={Cω ∈ D˜G,ψ∣∣ 〈ω, δ〉 6= 0 (∀δ ∈ ∆ψ)},∆ψ :={δ ∈ (Λψ)G∣∣ δ2 = −2},O(Λ, ψ) :={γ ∈ O(Λ)∣∣ γψ(g) = ψ(g)γ (∀g ∈ G)}.343.2. ClassificationProof. Let Cω = (α ⊗ C)(H2,0(S)) ∈ D˜G,ψ be the period point of a K3 surface S with aCalabi–Yau G-action. We can check that Cω modulo the action of O(Λ, ψ) is independentof the choice of α. Since G is finite, there exists a G-invariant Ka¨hler class κS of S. Wehave κ := (α⊗R)(κS) ∈ (Λψ)G⊗R. For any δ ∈ ∆ψ, we have 〈κ, δ〉 = 0, and thus 〈ω, δ〉 6= 0(see Section ??). Therefore we see that Cω ∈ DG,ψ. Assume that a K3 surface S′ witha Calabi–Yau G-action is mapped to the same point as S by τ . Then there exists a G-equivariant isomorphism φ : H2(S,Z)→ H2(S′,Z) such that (φ⊗C)(H2,0(S)) = H2,0(S′).By Lemma 1.1.10, we may assume that (φ ⊗ R)(κS) is a Ka¨hler class of S′. By Theorem1.1.8, φ induces a G-equivariant isomorphism between S and S′. Therefore, τ is injective.Let Cω1 ∈ DG,ψ. By the definition of DG,ψ, for any δ ∈ Λψ with δ2 = −2 and 〈ω1, δ〉 = 0,we have 〈δ, (Λψ)G〉 ) {0}. Hence there exists κ1 ∈ (Λψ)G ⊗ R such that ω1 and κ1 satisfythe conditions (1) and (2) in Theorem 1.1.9. Therefore, by Theorem 1.1.9, there exist a K3surface S1 and an isomorphism α1 : H2(S1,Z)→ Λψ such that (α−11 ⊗C)(Cω1) = H2,0(S1)and (α−11 ⊗R)(κ1) is a Ka¨hler class of S1. By Theorem 1.1.8, the G-action on Λψ inducesa G-action on S1 such that α1 is G-equivariant, which is a Calabi–Yau action by Lemma3.2.15. This implies the surjectivity of τ .Next let us consider projective models of the K3 surfaces with a Calabi–Yau action.Lemma 3.2.17. Let S be a K3 surface with a Calabi–Yau G-action. Assume that thereexists an element v ∈ NS(S)G such that v2 = 4. Then there exists a G-invariant linebundle L on S satisfying the following conditions.1. L2 = 4 and h0(L) = 4.2. The linear system |L| defined by L is base-point free and defines a map φL : S → P3.3. dimφL(S) = 2.4. The degree deg φL of the map φL : S → φL(S) is 2, and φL(S) is isomorphic to eitherP1 × P1 or a cone (i.e. a nodal quadric surface).Proof. Note that the closure KS of KS is the nef cone of S. We may assume that v is nefby Lemma 1.1.10. Let L be a line bundle on S representing v. By [SD, Sections 4 and 8],we have h0(L) = 4 and either of the following occurs.(a) L is base-point free, dimφL(S) = 2, and deg φL = 1 or 2. Any connected componentof φ−1L (p) for any p is either a point or an ADE-configuration.(b) L ∼= OS(3E + Γ) and |L| = {D1 +D2 +D3 + Γ∣∣ Di ∼ E}, where E and Γ ∼= P1 areirreducible divisors such that E2 = 0, Γ2 = −2 and 〈E,Γ〉 = 1.353.2. ClassificationIn Case (b), the base locus Γ ∼= P1 of |L| is stable under the action of ι and thus ι has afixed point in Γ, which is a contradiction. Hence Case (a) occurs. Since the fixed locusof any (projective) involution of P3 is at least 1-dimensional, there exists a fixed point pof the action of ι on φL(S). If deg φL = 1, then Sι 6= ∅ by Lemma 3.2.10, which is acontradiction. Hence deg φL = 2, and φL(S) is an irreducible quadric surface in P3, whichis either P1 × P1 or a cone.Proposition 3.2.18. For a generic point in MG, the corresponding K3 surface S witha Calabi–Yau action χ : G → Aut(S) has a projective model as in Proposition 3.2.11.More precisely, S and χ are realized as the double covering of P1 × P1 branching along a(4, 4)-curve and a projective G-action on S.Proof. Let S be a K3 surface with a Calabi–Yau G-action. By Lemmas 3.2.14 and 3.2.17,there exists a G-invariant line bundle L satisfying the conditions (1)–(4) in Lemma 3.2.17.Let φL = u ◦ θ be the Stein factorization of φL. Then θ(S) is a normal surface possiblywith ADE-singularities, and u is a finite map of degree 2. Assume that φL(S) is a conewith the singular point p. By Lemma 3.2.10, u−1(p) consists of two points p1, p2, whichare interchanged by any g ∈ G \ H. Hence each θ−1(pi) is a (−2)-curve Ci on S andC1 − C2 ∈ H2(S,Z)Hι with (C1 − C2)2 = −4. In particular, the Picard number of S isgreater than the generic Picard number = 22 − rankH2(S,Z)Hι . Therefore, if the periodpoint of (S, χ) is contained in⊔ψ∈Ψ′G{Cω ∈ DG,ψ∣∣ 〈ω, δ〉 6= 0 (∀δ ∈ ∆′ψ)}, ∆′ψ := {δ ∈ (Λψ)Hι∣∣ δ2 = −4},then φL(S) ∼= P1 × P1 and the branching curve B of u has at most ADE-singularities.Moreover, since the canonical bundle of S is trivial, the bidegree of B is (4, 4).Let S be a K3 surface with a Calabi–Yau action χ : G → Aut(S). We will prove(Theorem 3.2.19) that the pair (S, χ(G)) is unique up to equivariant deformation. A pair(S, χ) represents a K3 surface S with a Calabi–Yau G-action with a fixed group G, while apair (S, χ(G)) represents a K3 surface S with a subgroup of Aut(S) which gives a Calabi–Yau G-action. The difference is whether or not we keep track of a way to identify G with asubgroup of Aut(S). Let AutH(G) denote the subgroup of Aut(G) consisting of elementswhich preserve H. The group AutH(G) acts on the moduli spaceMG of pairs (S, χ) fromthe right by(S, χ) · σ = (S, χ ◦ σ), σ ∈ AutH(G).Note that Aut(G) does not necessarily act on MG because the action of H on S is sym-plectic by definition. The orbit of (S, χ) under the action of AutH(G) is identified with(S, χ(G)), andMG/AutH(G) is considered as the moduli space of pairs (S, χ(G)).363.2. ClassificationTheorem 3.2.19. Let H = Cn (1 ≤ n ≤ 6), C2 × C2 or C2 × C4. Then there exists aunique subgroup of O(Λ) induced by a Calabi–Yau G-action up to conjugation in O(Λ),that is,ΨG = O(Λ) · ψ ·AutH(G), ψ ∈ ΨG. (3.2.2)Moreover, we haveMG/AutH(G) ∼= DG,ψ/Γψ, Γψ := {γ ∈ O(Λ)∣∣ γψ(G)γ−1 = ψ(G)}. (3.2.3)In particular, the moduli space MG/AutH(G) of pairs (S, χ(G)) as above is irreducible,and a pair (S, χ(G)) exists uniquely up to equivariant deformation.Proof. By Proposition 3.2.18, a generic pair (S, χ(G)) has a projective model as in Propo-sition 3.2.11. Hence the existence of Calabi–Yau G-actions follows from Proposition 3.2.11.Also, the connectedness ofMG/AutH(G) follows from Propositions 3.2.11 and 3.2.13. Thestabilizer subgroup Σ of DG,ψ/O(Λ, ψ) in AutH(G) is given byΣ = {σ ∈ AutH(G)∣∣ ψ ◦ σ = γ · ψ (∃γ ∈ O(Λ))}.Since Σ is naturally isomorphic to Γψ/O(Λ, ψ), we can check (3.2.2) and (3.2.3) by Propo-sition 3.2.16.Theorem 3.2.20. For G ∼= (C3 × C3) o C2, there does not exists a K3 surface with aCalabi–Yau G-action, that is,MG = ∅.Proof. As in the proof of Theorem 3.2.19, a generic pair (S, χ(G)) admits a projectivemodel as in Proposition 3.2.11. However, there is no such a projective model.3.2.4 Moduli Spaces of Complex StructuresIn this section, we describe the complex moduli spaces of Calabi–Yau threefolds of typeK. We in particular show the irreducibility of the moduli space of Calabi–Yau threefoldsof type K with a prescribed Galois group G. Throughout this section, we fix a semi-directproduct decomposition G = H o 〈ι〉 of a Calabi–Yau group G as in Proposition 3.2.3. InSection 3.2.3, we studied the moduli space of K3 surfaces S with a Calabi–Yau G-action,which is denoted byMGS instead ofMG in this section.Let us consider the moduli problem of elliptic curves with a G-action prescribed inProposition 3.2.3, which we also call a Calabi–Yau G-action. LetMGE denote the modulispace of elliptic curves with a Calabi–Yau G-action. An element inMGE is the isomorphismclass of an elliptic curve with a faithful translation action ofH. A faithful translation actionof C2 × C2 on an elliptic curve E is given by a level 2 structure on E. ThereforeMGE forH = C2 × C2 is identified with the (non-compact) modular curve Y (2) := H/Γ(2). In the373.2. ClassificationG C2 × C2 × C2 Cn o C2 C2 ×D8MGE Y (2) Y1(n) Y (2 | 4)Table 3.5: Moduli spaceMGEsame manner, MGE for H = Cn is identified with the modular curve Y1(n) := H/Γ1(n).For H = C2×C4, we want the moduli space of elliptic curves with linearly independent 2-and 4-torsion points. It is not difficult to see that it is identified with the modular curveY (2 | 4) := H/Γ(2 | 4), whereΓ(2 | 4) :={[a bc d]∈ SL(2,Z)∣∣∣∣∣a− 1 ≡ c ≡ 0 mod 2, b ≡ d− 1 ≡ 0 mod 4}.We summarize the argument above in the following lemma.Lemma 3.2.21. Let 1 ≤ n ≤ 6. The moduli spaceMGE of elliptic curves with a Calabi–YauG-action is irreducible and given by Table (3.5).Theorem 3.2.22. Let AutH(G) denote the subgroup of Aut(G) consisting of elementswhich preserve H. The quotient space(MGS ×MGE)/AutH(G)is the coarse moduli space of Calabi–Yau threefolds of type K whose minimal splitting cov-ering has the Galois group isomorphic to G. The moduli space is in particular irreducible.Proof. Two Calabi–Yau threefolds X and Y of type K are isomorphic if and only if thecorresponding minimal splitting coverings SX ×EX and SY ×EY are isomorphic as Galoiscoverings. Suppose that the Galois group is isomorphic to G. The condition is equivalent tothe existence of an isomorphism f : SX×EX → SY ×EY and an automorphism φ ∈ Aut(G)such that the following diagram commutes:Gφ//	GAut(SX × EX) // Aut(SY × EY ),that is, f(g · x) = φ(g) · f(x) for any g ∈ G and any x ∈ SX × EX . Note that we haveφ ∈ AutH(G) because we fix a subgroup H as in Proposition 3.2.3. Since a Calabi–YauG-action on SX × EX induces that on each SX and EX , SX × EX is represented by a383.3. Key Lemmapoint inMGS ×MGE . The quotient space (MGS ×MGE)/AutH(G) is then the coarse modulispace of the isomorphism classes of Calabi–Yau threefolds of type K with Galois groupisomorphic to G. The moduli space is irreducible because the action of AutH(G) on theset of connected components ofMGS is transitive andMGE is irreducible by Theorem 3.2.19and Lemma 3.2.21.Combining Proposition 3.2.11, Theorems 3.2.19, 3.2.20 and 3.2.22, we complete theproof of the main theorem (Theorem 3.2.1) of the present section.3.3 Key LemmaLet S be a K3 surface with a Calabi–Yau G-action. We fix a semi-direct product decom-position G = H o 〈ι〉 as in Proposition 3.2.3. This section is devoted to the proof of theKey Lemma:Key Lemma (Lemma 3.2.14). There exists an element v ∈ NS(S)G such that v2 = PreparationLemma 3.3.1. Set r := rankH2(S,Z)H . We then haverankH2(S,Z)G = r2− 1, rankH2(S,Z)Hι =r2+ 1.Proof. Let X denote a Calabi–Yau threefold (S × E)/G of type K. Since a holomorphic2-form ωS on S is contained in H2(S,C)Hι , we see that H2(S,C)G ⊂ H1,1(S). We thereforehaveH1,1(X) ∼= H1,1(S × E,C)G ∼= (H2(S,C)G ⊗H0(E,C))⊕ (H0(S,C)⊗H2(E,C)).as C-vector spaces and conclude that h1,1(X) = rankH2(S,Z)G + 1. On the other handwe have canonical isomorphisms of C-vector spaces:H2,1(X) ∼= H2,1(S × E,C)G ∼= (H1,1(S)Hι ⊗H1,0(E))⊕ (H2,0(S)⊗H0,1(E)).By the decompositionH2(S,C)Hι ∼= H2,0(S)⊕H1,1(S)Hι ⊕H0,2(S),we conclude that h2,1(X) = rankH2(S,Z)Hι − 1. Since the Euler characteristic e(X) =e(S × E)/|G| = 0, we have h1,1(X) = h2,1(X). Then the claim readily follows.Recall that the action of H on S is symplectic. Hence the quotient surface S/H has atmost ADE-singularities and the minimal resolution ˜S of S/H is again a K3 surface. Let ι˜denote the involution of ˜S induced by ι.393.3. Key LemmaLemma 3.3.2. The involution of S/H induced by ι has no fixed point. In particular, ι˜ isan Enriques involution of ˜S.Proof. If the involution of S/H induced by ι has a fixed point, then the action of hι on Shas a fixed point for some h ∈ H, which is a contradiction.Each irreducible curve Mi which contracts under the resolution ˜S → S/H is a (−2)-curve. We denote by M the negative definite lattice generated by {Mi}i and set K :=M⊥H2(˜S,Z).Lemma 3.3.3. If H = Cn (3 ≤ n ≤ 6), then there is no nef class v ∈ K ι˜ such that v2 = 4.Proof. We assume that a nef class v ∈ K ι˜ satisfies v2 = 4 and derive a contradiction. ByLemma 3.3.2, the (induced) action of ι on S/H has no fixed point. By the same argumentas in the proof of Proposition 3.2.17, the class v induces a morphism ˜f : ˜S → P3 such that˜f(˜S) is a quadric surface and the degree of ˜f is 2. Since we have v⊥M by the assumption,the morphism ˜f induces a morphism f : S/H → P3. By the proof of Lemma 3.2.18, wemay assume that f(S/H) ∼= P1 × P1 by taking a generic S. The action of ι on S/H isof the form στ , where σ is induced by a symplectic involution of ˜S and τ is the coveringtransformation of f . Let τ ∈ Aut(S) be a lift of τ . Note that τ normalizes H. Since finduces a generically one-to-one morphism S/〈H, τ〉 → P1 × P1, it follows that S/〈H, τ〉 issmooth and that the action of τ fixes each singular point of S/H. Hence the actions of agenerator of H and τ are represented by the matrices[ζn 00 ζ−1n]and[0 11 0]respectively,in local coordinates around a point in SH , where ζn := exp(2pii/n). Therefore we haveτhτ = h−1 for any h ∈ H (?).We checked that τ fixes each point in Sing(S/H). Hence the action of σ has 8 fixedpoints qi 6∈ Sing(S/H) (1 ≤ i ≤ 8) by Theorem 1.1.11. Let Qi ⊂ S denote the inverse imageof qi, which consists of |H| points. Take a point p ∈ Qi. Since H acts on Qi transitively,we can take a lift σ ∈ Aut(S) of σ such that σ · p = p. The action of σ around p is locallyidentified with that of σ around qi. Therefore ord(σ) = 2. Since σ τ ∈ Hι, the condition(?) implies that σ commutes with H. Hence the action of σ on each Qi is trivial or free.If n = 3, 5 or 6, this contradicts to the fact |Sσ| = 8. Let us next consider the case n = 4.Let h ∈ H be a generator of H. By a similar argument, for each Qi, we can check that theaction of either σ or σh2 on Qi is trivial. Therefore we have ∪8i=1Qi = Sσ ∪ Sσh2 . On theother hand, Theorem 1.1.11 implies that |∪8i=1Qi| = 8·|H| = 32 and |Sσ∪Sσh2 | = 2·8 = 16.This is a contradiction.403.3. Key LemmaH C1 C2 C3 C4 C5 C6 C2 × C2 C2 × C4 C3 × C3rankΛH 22 14 10 8 6 6 10 6 6Table 3.6: rankΛH3.3.2 Proof of Key LemmaIn the following, we write LR := L ⊗Z R for a lattice L and a Z-module R. The bilinearform on L naturally extends to that on LR which takes values in R. We denote by Zpthe p-adic integers. Lattices over Zp, and their discriminant groups and forms are definedin a similar way to lattices (over Z). Note that a lattice over Z2 is not necessarily even.Assume that L is non-degenerate and even. Then A(LZp) and q(LZp) are the p-parts ofA(L) and q(L) respectively (see [Nik2] for details). In particular, if | disc(L)| is a power ofp, then we have (A(L), q(L)) ∼= (A(LZp), q(LZp)).Some remarks are in order before the proof. We fix an identification H2(S,Z) = Λ :=U⊕3⊕E8(−1)⊕2. Since H2,0(S) is contained in (ΛHι )C, we have NS(S)G = ΛG. By [Nik2],the H-invariant lattice ΛH is non-degenerate, and the rank of ΛH , which depends onlyon the group H, is given in Table (3.6). Since ΛG(1/2) is contained in Λι(1/2), which isisomorphic to U ⊕ E8(−1) by Theorem 1.1.13, it follows that ΛG(1/2) is even. Similarly,K ι˜(1/2) is even by Lemma 3.3.2. Since G is finite, there exists a G-invariant Ka¨hler classof S. Therefore ΛG has signature (1, rankΛG − 1). SetS′ := S \ {p ∈ S∣∣ h · p = p (∃h ∈ H, h 6= 1)},and let pi : S′ → ˜S be the natural map. Since S \ S′ is a finite set, the pushforward pi∗ andPoincare´ duality induce a natural mapf : Λ = H2(S,Z) ∼= H2(S,Z)→ H2(˜S,Z) ∼= H2(˜S,Z).For any x, y ∈ ΛH , we have 〈f(x), f(y)〉 = |H|〈x, y〉. The map f decomposes asf : Λ→ (ΛH)∨ → H2(˜S,Z),where the first map is the restriction of the first projection of the decomposition ΛQ =(ΛH)Q ⊕ (ΛH)Q and the second map is the natural injection. Since ΛH/(ΛG ⊕ ΛHι ) ∼=(Z/2Z)⊕l for some l by Lemma 1.1.6, we have 2(ΛG)∨ ⊂ (ΛH)∨. Hence we find thatf(2(ΛG)∨) ⊂ f((ΛG)Q ∩ (ΛH)∨) ⊂ K ι˜.Set L := ΛG(1/2). Then we have 2(ΛG)∨ ∼= 2L∨(1/2) ∼= L∨(2). Thus we haveL∨(|H|) ∼= f(2(ΛG)∨)(1/2) ⊂ K ι˜(1/2).Therefore L satisfies the following conditions.413.3. Key LemmaN 〈2ka〉 U(2k) =[0 2k2k 0]V (2k) =[2k+1 2k2k 2k+1]A(N) Z/2kZ (Z/2kZ)⊕2 (Z/2kZ)⊕2q(N) 〈a/2k〉 u(2k) :=[0 1/2k1/2k 0]v(2k) :=[1/2k−1 1/2k1/2k 1/2k−1]sign q(N) a+ k(a2 − 1)/2 0 4kTable 3.7: Lattice over Z21. L and L∨(|H|) are even.2. If H = Cn (3 ≤ n ≤ 6), then v2 6= 2/n for any v ∈ L∨.Here (2) is a conclusion of Lemmas 1.1.10 and 3.3.3. These conditions are derived fromgeometry of K3 surfaces. On the other hand, the argument below is essentially latticetheoretic.Proof of Key Lemma. If ΛG contains U(2), we see that the assertion of the Key Lemmaholds.Case H = C1. We have ΛG ∼= U(2)⊕ E8(−2) by Theorem 1.1.13.Case H = C2. This case has been studied by Ito and Ohashi (No. 13 in their paper[IO]). They showed that ΛG ∼= U(2) ⊕ D4(−2). Here we give a proof of this fact for thesake of completeness. By Lemma 3.3.1, the signature of ΛG is (1, 5). For each prime p, thelattice LZp over the local ring Zp admits an orthogonal decomposition LZp ∼=⊕i≥0 L(p)i (pi),where L(p)i ’s are unimodular lattices (see [Nik2] for details). By (1), we have L(p)i = 0 forany p 6= 2 and any i ≥ 1. Thus | disc(L)| is a power of 2. Again, by (1), L(2)0 and L(2)1 areeven, and we have L(2)i = 0 for any i ≥ 2. Let V denote the lattice over Z2 defined by thematrix[2 11 2]. In general, a lattice over Z2 is expressed as an orthogonal sum of the latticesin Table (3.7) [Nik2, Propositions 1.8.1 and 1.11.2]. Here k ≥ 0 and a = ±1,±3. (Notethat 〈±1/2〉 ∼= 〈∓3/2〉.) Since L(2)0 and L(2)1 are even, LZ2 has an orthogonal decompositionLZ2 ∼= U⊕ν ⊕ V ⊕µ ⊕ U(2)⊕ν′ ⊕ V (2)⊕µ′ .Then we haveA(L) ∼= (Z/2Z)⊕2(ν′+µ′), sign q(L) ≡ 4µ′ mod 8.Since we have Λι(1/2) ∼= U ⊕ E8(−1) and ΛG = (Λι)H , it follows that ν ′ + µ′ ≤ 2 byProposition 1.1.5 and Lemma 1.1.6. The fact that signΛG ≡ sign q(L) mod 8 implies that423.3. Key LemmaA(L) q(L) L(a) (Z/2Z)⊕2 v(2) U ⊕D4(−1)(b) (Z/2Z)⊕4 u(2)⊕ v(2) U(2)⊕D4(−1)Table 3.8: Case H = C2A(L) q(L) L(a) Z/3Z 〈−2/3〉 U ⊕A2(−1)(b) (Z/3Z)⊕3 〈2/3〉⊕3 U(3)⊕A2(−1)Table 3.9: Case H = C3µ′ = 1. Hence either of the two cases in Table (3.8) occurs. Here, in each case, L is uniquelydetermined by q(L) by Theorem 1.1.2. In Case (b), we have q(ΛιH(1/2)) ∼= u(2)⊕ v(2) byProposition 1.1.5. Since q(ΛιH(1/2)) takes values in Z/2Z, it follows that ΛιH(1/4) is aneven unimodular lattice of rank 4, which contradicts to the fact that any even unimodularlattice has rank divisible by 8. Hence Case (a) occurs: ΛG = L(2) ∼= U(2)⊕D4(−2).Case H = C3. The signature of ΛG is (1, 3). By a similar argument, the condition (1)implies that A(L) ∼= (Z/3Z)⊕l for some 0 ≤ l ≤ 4. Due to the relations 〈2/3〉⊕2 ∼= 〈−2/3〉⊕2and sign〈±2/3〉 ≡ ±2 mod 8 (see [Nik2] for details), we conclude thatq(L) ∼= 〈2/3〉⊕l−1 ⊕ 〈±2/3〉, sign q(L) ≡ 2(l − 1)± 2 mod 8.We can check that either of the two cases in Table (3.9) occurs. Case (b) cannot occur by(2). Therefore Case (a) occurs: ΛG ∼= U(2)⊕A2(−2).Case H = C4. The signature of ΛG is (1, 2). Similarly, we find that | disc(L)| is apower of 2. Moreover, L(2)0 and L(2)2 are even, and we have L(2)i = 0 for any i ≥ 3. IfL(2)0 = L(2)2 = 0, then L(1/2) ∼= U ⊕ 〈−1〉 by the uniqueness of indefinite odd unimodularlattices. Otherwise, q(L) ∼= 〈−1/2〉 or u(4) ⊕ 〈−1/2〉 because of the relation 〈a/2k〉 ⊕v(2k+1) ∼= 〈5a/2k〉 ⊕ u(2k+1) for any a with a ≡ 1 mod 2 (see [Nik2] for more details).Therefore we conclude that L ∼= U(2k) ⊕ 〈−2〉 for k = 0, 1 or 2. By (2), it follows thatk 6= 1, 2. Thus ΛG ∼= U(2)⊕ 〈−4〉.Case H = C5. In a similar way, we can check that L is an indefinite even lattice ofrank 2 such that A(L) ∼= (Z/5Z)⊕l for some 0 ≤ l ≤ 2. By [CS, Table 15.2], we see thatL ∼= U,[2 11 −2]or U(5).433.3. Key LemmaThe second and third cases cannot occur by (2). Hence we conclude that ΛG ∼= U(2).Case H = C6. The signature of ΛG is (1, 1). We make use of the argument in CaseH = C3. Let h be a generator of H. We define N := Λ〈h2,ι〉(1/2) ∼= U ⊕ A2(−1). Thenh acts on N as an involution and we have Nh = L. By Lemma 1.1.6, we can check thatA(Nh) and A(Nh) are of the form (Z/2Z)⊕l⊕(Z/3Z)⊕m for some 0 ≤ l ≤ 2 and 0 ≤ m ≤ 1.Therefore, according to [CS, Tables 15.1 and 15.2], we haveNh ∼= U, U(2) or ±[2 00 −6]; Nh ∼= A2(−1),[−2 00 −2],[−2 00 −6]or A2(−2).By (1), we have Nh ∼= U or U(2). Note that Nh⊕Nh is a sublattice of N of finite index andthat Nh and Nh are primitive sublattices of N . Hence we have Nh ∼= U and Nh ∼= A2(−1)(see Proposition 1.1.4). Thus ΛG = Nh(2) ∼= U(2).Case H = C2 × C2. The signature of ΛG is (1, 3). An almost identical argument tothat in Case H = C4 shows that L ∼= U(2k) ⊕ 〈−2〉⊕2 for k = 0, 1 or 2. In order to showk 6= 2, we make use of the argument in Case H = C2. Let H ′ ∼= C2 be a subgroup ofH. Then we know that N := Λ〈H′,ι〉(1/2) ∼= U ⊕ D4(−1). Since H/H ′ acts on N as aninvolution and we have NH/H′ = L, it follows that | disc(L)| divides 24 by Lemma 1.1.6.Therefore we have ΛG ∼= U(2k+1) ⊕ 〈−4〉⊕2 for k = 0 or 1, and the assertion of the KeyLemma holds.Case H = C2 × C4. The signature of ΛG is (1, 1). Let H ′ ∼= C4 be a subgroup of H.We then have N := Λ〈H′,ι〉(1/2) ∼= U ⊕ 〈−2〉 by the argument in Case H = C4. Similarly,we find that | disc(NH/H′)| and |disc(NH/H′)| divide 22. By [CS, Table 15.2], we haveNH/H′ ∼= U, U(2) or 〈2〉 ⊕ 〈−2〉; NH/H′ ∼= 〈−2〉 or 〈−4〉.Therefore, by Proposition 1.1.4, we have NH/H′ ∼= U or 〈2〉 ⊕ 〈−2〉, and NH/H′ ∼= 〈−2〉.Thus ΛG ∼= U(2) or 〈4〉 ⊕ 〈−4〉. Hence the assertion of the Key Lemma holds.Case H = C3 × C3. The signature of ΛG is (1, 1). We make use of the argument inCase H = C3. Let H ′,H ′′ ∼= C3 be subgroups of H such that H = H ′ × H ′′. ThenN := Λ〈H′,ι〉(1/2) ∼= U ⊕ A2(−1). By a similar argument to the proof of Lemma 1.1.6, wehave Λ/(ΛH′′ ⊕ ΛH′′) ∼= (Z/3Z)⊕l for some l. Hence N/(L ⊕ L⊥N ) ∼= (Z/3Z)⊕m for some0 ≤ m ≤ 2, and |disc(L)| and | disc(L⊥N )| divide 33. Therefore, by [CS, Tables 15.1 and15.2], we haveL ∼= U, U(3) or ±[2 33 0]; L⊥N ∼= A2(−1), A2(−3) or[2 11 14].443.4. PropertiesG H2(S,Z)GC2 U(2)⊕ E8(−2)C2 × C2 U(2)⊕D4(−2)C2 × C2 × C2 U(2)⊕ 〈−4〉⊕2D6 U(2)⊕A2(−2)D8 U(2)⊕ 〈−4〉D10 U(2)D12 U(2)C2 ×D8 U(2)Table 3.10: Invariant lattice H2(S,Z)GBy Proposition 1.1.4, we can check that L ∼= U or U(3), and that L⊥N ∼= A2(−1). Assumethat L ∼= U(3). Note that N ′ := L⊥N = Λ〈H′,ι〉H′′ (1/2). Hence, by interchanging H′ and H ′′,we haveN ′′ := Λ〈H′′,ι〉H′ (1/2) ∼= A2(−1), L⊕N′ ⊕N ′′ ⊂ Λι(1/2).Since we have N/(L⊕N ′) ∼= Z/3Z, there exist elements v ∈ L∨ and w ∈ (N ′)∨ such thatv2 = 2/3, (w′)2 = −2/3 and v + w′ ∈ N . Similarly, there exists an element w′′ ∈ (N ′′)∨such that (w′′)2 = −2/3 and v + w′′ ∈ Λ〈H′′,ι〉(1/2). This implies that w′ − w′′ ∈ Λι(1/2)and that (w′ − w′′)2 = −4/3. Since Λι(1/2) is integral, this is a contradiction. Thereforewe conclude that ΛG ∼= U(2).Proposition 3.3.4. The G-invariant lattice H2(S,Z)G is given by Table (3.10) 5.Proof. It is worth noting that H2(S,Z)G does not depend on the choice of S. By theproof of the Key Lemma, it suffices to show the assertion for G = C2 × C2 and C2 ×D8.Note that a generic K3 surface S with a Calabi–Yau G-action is realized as a Horikawamodel (Proposition 3.2.11) and thus H2(S,Z)G contains U(2), which is the pullback of theNe´ron–Severi lattice of P1×P1. For G = C2×C2, we have H2(S,Z)G ∼= U(2k+1)⊕〈−4〉⊕2for k = 0 or 1 by the proof of the Key Lemma. Hence we have H2(S,Z)G ∼= U(2)⊕〈−4〉⊕2.Similarly, for G = C2 ×D8, we have H2(S,Z)G ∼= U(2) or 〈4〉 ⊕ 〈−4〉. We thus concludethat H2(S,Z)G ∼= U(2).3.4 PropertiesIn this section, we will investigate some basic properties of Calabi–Yau threefolds of type K.The explicit description obtained in the preceding section plays a central role in our study.5In the proof of Proposition 3.2.11, we already checked that Case H = C3 × C3 does not occur.453.4. PropertiesG C2 C2 × C2 C2 × C2 × C2 D6 D8 D10 D12 C2 ×D8n 3 4 5 3 4 3 4 5Table 3.11: Homology group H1(X,Z) ∼= (Z/2Z)nThroughout this section, X is a Calabi–Yau threefold of type K and pi : S ×E → X is theminimal splitting covering with Galois group G. We also fix a semi-direct decompositionG = H o 〈ι〉.There exist G-equivariant Ricci-flat Ka¨hler metrics gS and gE on S and E respectively[Yau]. Then the product metric gS × gE on S × E descends to a Ricci-flat Ka¨hler metricg′ and g on the quotients (S × E)/H and X respectively. Let T := S/〈ι〉 be the Enriquessurface with the metric gT induced by gS . We denote by Holh(Y ) the holonomy group ofa manifold Y with respect to a metric h (we do not refer to a based point).Proposition 3.4.1. 1. HolgT (T ) ∼= {A ∈ U(2) | detA = ±1} ⊂ U(2).2. Holg(X) ∼= S(U(2)× C2) ⊂ SU(3).Proof. Since the holonomy group HolgT (T ) cannot be SU(2), it must be a C2-extension ofHolgS (S) ∼= SU(2) in U(2). Such an extension is unique and this proves the first assertion.In order to prove the second assertion, we first consider the quotient (S × E)/H, whichadmits a smooth isotrivial K3 fibration (S ×E)/H → E/H. Since the action of H on S issymplectic, we see that Holg′((S×E)/H) ∼= SU(2). Therefore the holonomy group Holg(X)is an extension of SU(2) in SU(3) of index at most 2. Since X contains an Enriques surface,we conclude that Holg(X) ∼= S(U(2)× C2) ⊂ SU(3).Proposition 3.4.2. The following hold.1. pi1(X) = (Z× Z)oG, where the G-action on Z× Z is identified with that on pi1(E).2. H1(X,Z) ∼= (Z/2Z)n, where the exponent n is given by Table (3.11).Proof. The first assertion readily follows from the exact sequence 0 → pi1(S × E) →pi1(X) → G → 0, and the Calabi–Yau G-action on E. The second follows from thefact that H1(X,Z) ∼= pi1(X)Ab, or the Cartan–Leray spectral sequence associated to thee´tale map S × E → X.Proposition 3.4.3. X admits both K3 and Abelian surface fibrations over P1.Proof. By construction, the map X = (S×E)/G→ E/G ∼= P1 is an isotrivial K3 fibrationwith four Enriques fibers. By Lemma 3.2.17, there exists a G-equivariant morphism S → Z,463.4. Propertieswhere Z is isomorphic to either (a) P1×P1 or (b) a cone. In Case (a), let pi : Z → P1 denotethe first (or the second) projection. In Case (b), the resolution of Z is the Hirzebruchsurface F2, and the ruling F2 → P1 descends to pi : Z → P1. In either case, the mapS → Z pi→ P1 =: C is an elliptic fibration. Hence the map S × E → S → C is an abelianfibration. For h ∈ H, the action of h on E is a translation. Since Sh is a finite set byTheorem 1.1.11, if the action of h on C is trivial, then h acts on each smooth fiber ofS → C as a translation. On the other hand, since Sι = ∅ and the action of ι on S is anti-symplectic, ι acts on C non-trivially. Therefore, the map X = (S × E)/G→ C/G ∼= P1 isan abelian fibration.Proposition 3.4.4. There exists no isolated (smooth) rational curve on X. Here we saya curve is isolated if it is not a member of any non-trivial family.Proof. Suppose that there exists an isolated rational curve C ⊂ X. Since pi is e´tale, thepullback pi−1(C) consists of |G| isolated rational curves. On the other hand, there is noisolated rational curve on the product S × E as any morphism P1 → E is constant andany smooth rational curve on any K3 surface has self-intersection number −2. This leadsus to a contradiction.All rational curves show up in families (parametrized by the elliptic curve E). It isshown that they do not contribute to Gromov–Witten invariants but the higher genusquantum corrections are present at least for the Enriques Calabi–Yau threefold [MP].Proposition 3.4.5. Aut(X) = Bir(X).Proof. Any birational morphism between minimal models is decomposed into finitely manyflops up to automorphisms [Kaw]. Hence it is enough to prove that there exists no flopof X. In the case of threefolds, the exceptional locus of any flopping contraction must bea tree of isolated rational curves [KM1, Theorems 1.3 and 3.7]. The previous propositiontherefore shows that there exists no flop of X.Proposition 3.4.6. The following hold.1. If G ∼= D10, D12 or C2 ×D8, we have |Aut(X)| <∞ .2. If G ∼= C2, C2×C2, C2×C2×C2, D6 or D8, and X is generic in the moduli space,we have |Aut(X)| =∞.Proof. It is not difficult to see that there is a canonical one-to-one correspondence betweenAut(X) and N/G, where N denotes the normalizer of G in Aut(S×E). Since Aut(S×E) ∼=Aut(S)×Aut(E), we further have a canonical inclusion N ⊃ Aut(S)G×Aut(E)G of finiteindex. Here Aut(S)G stands for the subgroup of Aut(S) whose elements commute withthe action of G, and similar for Aut(E)G. In order to show the assertion, it is enough to473.4. Propertiese1 e3e2Figure 3.1: Tiling of the hyperbolic plane by geodesic trianglesdetermine whether Aut(S)G is finite or not. Note that Aut(S)G acts on ΛG.Assume that ΛG is isomorphic to U(2) with standard basis e, f . Then we find thatAut(S)G preserves the polarization ±(e+ f), which gives generically a map S → P1 × P1as in Proposition 3.2.11. It follows that Aut(S)G is finite by the well-known fact that theautomorphism group of any polarized K3 surface is finite. This proves the assertion (1)because of Proposition 3.3.4.We will next prove the assertion (2). By Proposition 3.2.13, the K3 surface S is realizedas a (2, 2, 2)-hypersurface in P := P1 × P1 × P1. We will show that the three involutionsof S associated to the three double covering structures S → P1 × P1 generate the infinitegroup C2 ∗C2 ∗C2. Let ei be the class of an elliptic fiber of the i-th projection pi : S → P1.Consider the coneσ := R>0e1 + R>0e2 + R>0e3 ⊂ KS ⊂ H2(S,R).Then the reflection with respect to the face σij := R>0ei +R>0ej (i 6= j) of σ correspondsto the covering transformation ιij of the double covering pi × pj : S → P1 × P1. Note thatP := {v ∈ Re1 + Re2 + Re3∣∣ v2 > 0}/R×is considered as a hyperbolic plane in a natural way. The image σ of σ in P is a geodesictriangle each of whose angles is zero. In this situation, the involution ι¯ij of P induced by ιijis a reflection along the geodesic σij and they generate the infinite group Γ ∼= C2 ∗C2 ∗C2by the standard hyperbolic geometry (Figure 3.1). Since Γ commutes with G, we concludethat Aut(S)G includes Γ. Hence Aut(S)G is infinite.Remark 3.4.7. Proposition 3.4.6 (1) also follows from the result of [LOP].483.5. Calabi–Yau Threefolds of Type ARemark 3.4.8. In Proposition 3.4.6 (2), the genericity assumption is essential at least inthe case G = C2. In fact, if G = C2, it follows that Aut(X) is infinite if and only if theautomorphism group of the Enriques surface S/〈ι〉 is infinite. Although the automorphismgroup of a generic Enriques surface is infinite, there exist Enriques surfaces with finiteautomorphism group (see [Kon1] for the classification of such Enriques surfaces).It is known that the automorphism group of a Calabi–Yau threefold with ρ = 1, 2 isfinite [Ogu2]. On the other hand, it is expected that there is a Calabi–Yau threefold withinfinite automorphism group for each ρ ≥ 4 (see for example [Bor, GM, OT]). Proposition3.4.6 provides a supporting evidence for this folklore conjecture, giving examples for smalland new ρ. It is an open problem whether or not a Calabi–Yau threefold with ρ = 3 admitsinfinite automorphism group [LOP].3.5 Calabi–Yau Threefolds of Type AIn this final section, we slightly change the topic and probe Calabi–Yau threefolds of typeA. Recall that a Calabi–Yau threefold is called of type A if it is an e´tale quotient of anabelian threefold. They are characterized by vanishing of all Chern classes:Theorem 3.5.1 (Kobayashi [Kob]). Let X be a Calabi–Yau threefold. Then the secondChern class c2(X) is identically zero if and only if X is of type A.By refining Oguiso and Sakurai’s fundamental work [OS] on Calabi–Yau threefolds oftype A, we will finally settle the full classification of Calabi–Yau threefolds with infinitefundamental group (Theorem 3.5.5).Let A := Cd/Λ be a d-dimensional complex torus. There is a natural semi-directdecomposition Aut(A) = A o AutLie(A), where the first factor is the translation group ofA and AutLie(A) consists of elements that fix the origin of A. We call the second factor ofg ∈ Aut(A) the Lie part of g and denote it by g0. The fundamental result in the theory ofCalabi–Yau threefolds of type A is the following.Theorem 3.5.2 (Oguiso–Sakurai [OS, Theorem 0.1]). Let X be a Calabi–Yau threefold oftype A. Then the following hold.1. X = A/G, where A is an abelian threefold and G is a finite group acting freely on Ain such a way that either of the following is satisfied:(a) G = 〈a〉 × 〈b〉 ∼= C2 × C2 anda0 =1 0 00 −1 00 0 −1 , b0 =−1 0 00 1 00 0 −1 ,493.5. Calabi–Yau Threefolds of Type AT1 T2 T3 T40 〈(0, 1/2, 1/2)A′〉 〈(1/2, 1/2, 0)A′ , (1/2, 0, 1/2)A′〉 〈(1/2, 1/2, 1/2)A′〉Table 3.12: 2-torsion points(b) G = 〈a, b | a4 = b2 = abab = 1〉 ∼= D8 anda0 =1 0 00 0 −10 1 0 , b0 =−1 0 00 1 00 0 −1 ,where a0 and b0 are the Lie part of a and b respectively and the matrix representationis the one given by an appropriate realization of A as C3/Λ.2. In the first case, ρ(X) = 3 and in the second case ρ(X) = 2.3. Both cases really occur.Theorem 3.5.2 provides a classification of the Lie part of the Galois groups of the min-imal splitting coverings, where the Galois groups do not contain any translation element.We will see that, in contrast to Calabi–Yau threefolds of type K, Calabi–Yau threefolds oftype A are not classified by the Galois groups of the minimal splitting coverings. That is, achoice of Galois group does not determine the deformation family of a Calabi–Yau threefoldof type A. We can improve Theorem 3.5.2 by allowing non-minimal splitting coverings asfollows.Proposition 3.5.3. Let X be a Calabi–Yau threefold of type A. Then X is isomorphic tothe e´tale quotient A/G of an abelian threefold A by an action of a finite group G, where Aand G are given by the following.1. A = A′/T , where A′ is the direct product of three elliptic curves E1, E2 and E3:A′ := E1 ×E2 × E3, Ei := C/(Z⊕ Zτi), τi ∈ Hand T is one of the subgroups of A′ in Table (3.12), which consists of 2-torsion pointsof A′. Here (z1, z2, z3)A′ denotes the image of (z1, z2, z3) ∈ C3 in A′.2. G ∼= C2 × C2 or D8.(a) If G = 〈a〉 × 〈b〉 ∼= C2 × C2, then G is generated bya : (z1, z2, z3)A 7→ (z1 + τ1/2,−z2,−z3)A,b : (z1, z2, z3)A 7→ (−z1, z2 + τ2/2,−z3 + τ3/2)A.503.5. Calabi–Yau Threefolds of Type A(b) If G = 〈a, b | a4 = b2 = abab = 1〉 ∼= D8, then τ2 = τ3 =: τ , T = T2 or T3, andG is generated bya : (z1, z2, z3)A 7→ (z1 + τ1/4,−z3, z2)A,b : (z1, z2, z3)A 7→ (−z1, z2 + τ/2,−z3 + (1 + τ)/2)A.Moreover, each case really occurs.Proof. By Theorem 3.5.2, X is of the form A/G with G isomorphic to either C2 × C2 orD8. Let C3/Λ be a realization of A as a complex torus. In the case G ∼= C2 × C2, we mayassume that G is generated bya : (z1, z2, z3)A 7→ (z1 + u1,−z2,−z3)A,b : (z1, z2, z3)A 7→ (−z1, z2 + u2,−z3 + u3)A,after changing the origin of A if necessary. Hence Λ is stable under the following actions:a0 : (z1, z2, z3) 7→ (z1,−z2,−z3),b0 : (z1, z2, z3) 7→ (−z1, z2,−z3).From this, we see that there exist lattices Λi ⊂ C for i = 1, 2, 3 such that2Λ ⊂ Λ1 × Λ2 × Λ3 ⊂ Λ.Let e1, e2, e3 be the standard basis of C3. SetΛ′ := Λ′1 × Λ′2 × Λ′3 ⊂ Λ, Λ′i := {z ∈ C∣∣ zei ∈ Λ}.Then Λ/Λ′ is a 2-elementary group, that is, Λ/Λ′ ∼= (Z/2Z)⊕n for some n. Since a2 =b2 = (ab)2 = idA and the G-action is free, we have ui 6∈ Λ′i but 2ui ∈ Λ′i. Let v =(v1, v2, v3) ∈ Λ. Suppose v1 ≡ u1 mod Λ′1, then (z1, v2/2, v3/2) ∈ Aa. Hence we concludethat v1 6≡ u1 mod Λ′1. Similarly, vi 6≡ ui mod Λ′i for i = 2, 3. Therefore, we may assumethat there exist τi ∈ H for i = 1, 2, 3 such that1. Λ′i = Z⊕ Zτi,2. ui ≡ τi/2 mod Λ′i,3. vi ≡ 0 or 1/2 mod Λ′i for all v ∈ Λ,after changing each coordinate zi if necessary. Now that we can check the assertion of thetheorem in this case by a direct computation. In particular, T = Λ/Λ′ coincides with onein the table up to permutation of the coordinates.513.5. Calabi–Yau Threefolds of Type ASimilarly, in the case G ∼= D8, we may assume that G is generated bya : (z1, z2, z3)A 7→ (z1 + u1,−z3, z2)A,b : (z1, z2, z3)A 7→ (−z1, z2 + u2,−z3 + u3)A.We use the same notation as above. It follows that Λ′2 = Λ′3, 4u1 ∈ Λ′1, 2u1 6∈ Λ′1,2ui ∈ Λ′i, ui 6∈ Λ′i for i = 2, 3. We haveab : (z1, z2, z3)A 7→ (−z1 + u1, z3 − u3, z2 + u2)A(ab)2 : (z1, z2, z3)A 7→ (z1, z2 + u2 − u3, z3 + u2 − u3)A.By (ab)2 = 1 and Aab = ∅, it follows that (0, u2− u3, u2− u3) ∈ Λ and u2− u3 6∈ Λ′2. Sincethe action of SL(2,Z) on the set of level 2 structures on an elliptic curve is transitive, wemay assume that τ2 = τ3 =: τ , u2 = τ2/2, u3 = (1 + τ3)/2. By a similar argument to thecase G ∼= C2 × C2, we can check that vi ≡ 0 or 1/2 mod Λ′i for any v = (v1, v2, v3) ∈ Λ. Inparticular, we have (0, 1/2, 1/2) ∈ Λ. Since T = T4 implies that (1/2, 0, 0) ∈ Λ′1, which is acontradiction, it follows that T is either T2 or T3. Moreover, we can check that the actionof G has no fixed point for T = T2, T3.Remark 3.5.4. The above four cases for G ∼= C2 × C2 have previously been studied byDonagi and Wendland [DW].As was mentioned earlier, in contrast to Calabi–Yau threefolds of type K, Calabi–Yau threefolds of type A are not classified by the Galois groups of the minimal splittingcoverings. They are classified by the minimal totally splitting coverings, where abelianthreefolds A which cover X split into the product of three elliptic curves.Together with Theorem 3.2.1, Proposition 3.5.3 finally completes the full classificationof Calabi–Yau threefolds with infinite fundamental group:Theorem 3.5.5. There exist precisely fourteen Calabi–Yau threefolds with infinite funda-mental group, up to deformation equivalence. To be more precise, six of them are of typeA and eight of them are of type K.52Chapter 4Calabi–Yau Threefolds of Type K:Mirror Symmetry4.1 IntroductionThis chapter studies mirror symmetry of Calabi–Yau threefolds of type K. A Calabi–Yauthreefold X is called of type K if admits an e´tale Galois covering by the product of a K3surface and an elliptic curve. Among many candidates of such coverings, we can always takea unique smallest one, up to isomorphism as a covering, and we call it the minimal splittingcovering of X. In the previous chapter, the full classification of Calabi–Yau threefolds oftype K is given.Theorem 4.1.1 (Theorem 3.2.1). There exist exactly eight Calabi–Yau threefolds of typeK, up to deformation equivalence. The equivalence class is uniquely determined by theGalois group G of the minimal splitting covering. Moreover, the Galois group is isomorphicto one of the following combinations of cyclic and dihedral groupsC2, C2 × C2, C2 × C2 × C2, D6, D8, D10, D12, or C2 ×D8.An explicit presentation for the deformation classes of the eight Calabi–Yau threefoldsof type K is also given. Although Calabi–Yau threefolds of type K are special and rareamong all Calabi–Yau threefolds, their explicit nature makes them an exceptionally goodlaboratory for general theories and conjectures. Indeed, the simplest example, known as theEnriques Calabi–Yau threefold (or the FHSV-model in physics [FHSV]), has been the one ofthe most accessiblecompact Calabi–Yau threefolds both in string theory and mathematics[FHSV, Asp, Mar, KM1, MP, PP]. The objective of this chapter is to investigate Calabi–Yau threefolds of type K with a view toward mirror symmetry. In mathematics, there arevarious formulations of mirror symmetry and each one is important in its own way. In thischapter, we are particularly interested in the A- and B-Yukawa couplings and Strominger–Yau–Zaslow conjecture. The main results in this chapter are the following.Theorem 4.1.2 (Theorem 4.3.4). Let X be a Calabi–Yau threefold of type K. The asymp-totic behavior of the A-Yukawa coupling Y XA around the large volume limit coincides withthe asymptotic behavior of the B-Yukawa coupling Y XB around a large complex structurelimit. The identification respects the rational structure of the trilinear forms.534.2. Calabi–Yau Threefolds of Type KTheorem 4.1.3 (Propositions 4.4.6 & 4.4.7). Let X be a Calabi–Yau threefold of typeK. There exists a special Lagrangian T 3-fibration piX : X → B such that each (possiblysingular) fiber has Euler number 0. The base space B is topologically identified as eitherthe 3-sphere S3 or an S1-bundle over RP2.In a certain sense, Calabi–Yau threefolds of type K are close cousins of Borcea–Voisinthreefolds [Bor, Voi], and the above results are, indeed, parallel to what is known forBorcea–Voisin threefolds. Mirror symmetry of Borcea–Voisin threefolds has different flavourthan that for complete intersection Calabi–Yau threefolds in toric varieties and homoge-neous spaces, and hence it has been a very important source of examples beyond theBatyrev–Borisov toric mirror symmetry. We hope that Calabi–Yau threefolds of type Kprovide more examples of interesting mirror symmetry.This chapter is organized as follows: Section 2 starts with a brief review of the clas-sification of Calabi–Yau threefolds of type K obtained in the previous chapter. Section3 is devoted to the study mirror symmetry of Calabi–Yau threefolds of type K. We willprobe the asymptotic behavior of A- and B-Yukawa couplings and confirm that they co-incide. A similar study was previously done by Voisin for Borcea–Voisin threefolds [Voi].The B-model theory will be complemented by an explicit computation of the B-Yukawacouplings via the Picard–Fuchs equation. Section 4 discusses the Strominger–Yau–Zaslowconjecture [SYZ], which in particular claims the existence of special Lagrangian fibrationsof Calabi–Yau threefolds. It will be shown that each Calabi–Yau threefold of type K admitsa natural special Lagrangian T 3-fibration. Our results can be seen as a generalization ofGross and Wilson’s work on Borcea–Voisin threefolds [GW]. We include a brief review ofBorcea–Voisin threefolds [Bor, Voi] in Appendix.4.2 Calabi–Yau Threefolds of Type KWe begin with a review of the construction of Calabi–Yau threefolds of type K obtained inthe previous chapter. Let us first consider the simplest case, Enriques Calabi–Yau threefold(Example 3.2.5). According to the classical theory of Enriques surfaces, any generic K3surface with an Enriques involution is realized as a Horikawa model (Proposition 1.1.16).The Horikawa model thus gives rise to the Enriques Calabi–Yau threefold. In order toobtain more general Calabi–Yau threefolds of type K, we will consider special classes ofHorikawa model as follows. Let ρ1, ρ2 : G→ PSL(2,C) be 2-dimensional complex projectiverepresentations ofG := HoC2, which we do not specify at this point. Let λ be the generatorof the second factor C2. We then get a G-action ρ1 × ρ2 on P1 × P1. Suppose that thereexists a G-invariant smooth curve B of bidegree (4, 4). We then obtain a Horikawa K3surface S as the double covering pi : S → P1 × P1 branching along B and the G-action onP1 × P1 lifts to S as a symplectic G-action (Definition 3.2.6). We further assume that the544.2. Calabi–Yau Threefolds of Type Kcurve B does not pass through any of fixed points of g ∈ G, g 6= 1. With the same notationas in Proposition 1.1.15, it can be checked that the symplectic G-action and the coveringtransformation θ commute.S/〈θ〉id// S/〈θ◦λ〉Gρ1×ρ2!!∃symplectic++P1 × P1 TBy twisting λ by θ, we obtain a new G-action on S, i.e.Aut(S) ⊃ G× 〈θ〉 ⊃ H o 〈θ ◦ λ〉 ∼= GThe new G-action on S turns out to be a Calabi–Yau action. In the previous chapter, it isshown that any generic K3 surface equipped with a Calabi–Yau action is (not necessarilyuniquely) realized in this way. To put it another way, there exist 2-dimensional complexprojective representations ρ1, ρ2 : G → PSL(2,C) which satisfy all the assumptions men-tioned above. In this chapter, we do not need explicit presentations of Calabi–Yau actions,so we will not delve into the details of the construction. We close this section by providingtwo concrete examples.Example 4.2.1. Suppose thatG ∼= D12 := 〈a, b|a6 = b2 = baba = 1〉.For i = 1, 2, we define ρi : D12 → PSL(2,C) bya 7→[ζi12 00 ζ12−i12], b 7→[0 11 0],where ζk denotes a primitive k-th root of unity. A basis of D12-invariant polynomials ofbidegree (4, 4) are given byx4z4 + y4w4, x4zw3 + y4z3w, x2y2z2w2.A generic linear combination of these cuts out a desired smooth curve of bidegree (4, 4).Example 4.2.2. Suppose thatG ∼= D8 × C2 = 〈a, b, c|a4 = b2 = baba = 1, ac = ca, bc = cb〉.For i = 1, 2, we define ρi : C8 × C2 → PSL(2,C) bya 7→[ζ8 00 ζ78], b 7→[0 11 0], c 7→[√−1i−1 00√−11−i].A basis of D8 × C2-invariant polynomials of bidegree (4, 4) are given byx4z4 + y4w4, x4w4 + y4z4, x2y2z2w2.A generic linear combination of these cuts out a desired smooth curve of bidegree (4, 4).554.2. Calabi–Yau Threefolds of Type K4.2.1 Some Topological ComputationThroughout this section, X denotes a Calabi–Yau threefold of type K and pi : S ×E → Xits minimal splitting covering with Galois group G.Definition 4.2.3. The Brauer group of a smooth projective variety Y isBr(Y ) := Tor(H2(Y,O×Y )).By the exact sequenceH2(Y,OY )→ H2(Y,O×Y )→ H3(Y,Z)→ H3(Y,OY )and the universal coefficient theorem, for a Calabi–Yau threefold Y , we haveBr(Y ) ∼= Tor(H3(Y,Z)) ∼= Tor(H2(Y,Z)) ∼= Tor(H4(Y,Z)).Therefore the Brauer group of a Calabi–Yau threefold is topological, in contrast to that ofa K3 surface, which is analytic.Proposition 4.2.4. The Brauer group Br(X) is non-trivial if |H| is odd.Proof. The minimal resolution ˜S/G of the quotient surface S/G is an Enriques surface.Moreover, a standard Mayer–Vietoris sequence argument shows thatH2(˜S/G,Z)/(⊕ji=1ZEi) ∼= H2(S/G,Z),where E1, . . . , Ej are the exceptional divisors of the minimal resolution. Therefore weconclude that Tor(H2(S/G,Z)) contains an element of order 2. The natural projectionpi : X → S/G admits an |H|-section s : S/G→ X. This shows the existence of an elementof order 2 in H2(X,Z) when |H| is odd as we have pi∗ ◦ s∗ = |H| id.Definition 4.2.5. Let M be a free abelian group of finite rank and µ : M⊗3 → Z asymmetric trilinear form on M over Z. Let L be a lattice, which consists of a free abeliangroup L of finite rank and a symmetric bilinear form 〈∗, ∗∗〉L : L × L → Z. We say thatthe trilinear form µ is of type L if the following hold:1. There is a decomposition M ∼= L⊕N as an abelian group such that N ∼= Z.2. For α, β ∈ L and n ∈ N ∼= Z we have µ(α, β, n) = n〈α, β〉L and the remaining valuesare obtained by extending the form symmetrically and linearly and by setting othernon-trivial values to be 0.A scalar extension of the notion is also defined in a similar manner.564.2. Calabi–Yau Threefolds of Type KG C2 D8 D10 D12 C2 ×D8L U ⊕ E8 I1,2 U U U or I1,1Table 4.1: Trilinear intersection formsProposition 4.2.6. The trilinear intersection form µX on H2(X,Z) mod torsion definedby the cup product is of type L for some lattice L. Moreover, when G is isomorphic toC2, D8, D10, D12 or C2 ×D8, the lattice L is given by Table (4.1).Proof. We will only prove the first assertion as we will not need the second in the restof this chapter6. In this proof, we always work with cohomology groups modulo torsion.Assume first that H 6∼= C2 × C2, C2 × C4. Let 0E denote the origin of E and pick a pointptS ∈ SH . Defineα := pi∗([S × 0E ]PD) ∈ H2(X,Z), β := pi∗([ptS × E]PD) ∈ H4(X,Z),where we denote by pi∗ the transfer map and by [C]PD the Poincare´ dual of a cycle C. Wealso setA := [S × 0E ] ∈ H4(S × E,Z), B := [ptS × E] ∈ H2(S × E,Z),The degree of the map pi∗ is computed aspi : A 1:1−→ A/H 2:1−→ A/G, pi : B |H|:1−→ B/H 1:1−→ B/G.It thus follows that 12α ∈ H2(X,Z), 1|H|β ∈ H4(X,Z) and 12α ∪1|H|β = 1. We thenconclude that the trilinear form µX is of type L with N := 1|H|βZ and some lattice L.When H ∼= C2 × C2 (resp. C2 × C4), we have SH = ∅. In this case, we take ptS ∈ SC2(resp. SC4) and a similar argument works. We leave the details to the reader.Proposition 4.2.7. The second Chern class c2(X) is given by c2(X) = 24|G|β, where β isdefined in the proof above.Proof. Let βS be a generator of H4(S,Z) with∫S βS = 1. By the universal property ofcharacteristic classes, we havepi∗c2(X) = c2(S × E) = 24p∗1βS ,where p1 : S × E → S is the first projection. Applying the transfer map pi∗, we obtain|G|c2(X) = 24pi∗p∗1βS = 24β.6 For the second assertion, we need the Poincare´ duality and a classification of indefinite unimodularlattices. Recall that for n,m ≥ 1, there exists a unique odd unimodular lattice In,m := 〈1〉⊕n ⊕ 〈−1〉⊕m ofsignature (n,m), up to isomorphism.574.2. Calabi–Yau Threefolds of Type KH H2(S,Z)HC2 U⊕3 ⊕ E8(−2)C2 × C2 U ⊕ U(2)⊕2 ⊕D4(−2)C2 × C2 × C2 U(2)⊕3 ⊕ 〈−4〉⊕2C3 U ⊕ U(3)⊕2 ⊕A2(−1)⊕2C4 U ⊕ U(4)⊕2 ⊕ 〈2〉⊕2D6 U(3)⊕2 ⊕A2(2)⊕A2(−1)⊕2D8 U ⊕ 〈4〉⊕2 ⊕ 〈−4〉⊕3C5, D10 U ⊕ U(5)2C6, D12 U ⊕ U(6)2C2 × C8, C2 ×D8 U(2)⊕ 〈4〉⊕2 ⊕ 〈−4〉⊕2Table 4.2: Symplectic invariant lattices4.2.2 Lattices H2(S,Z)G and H2(S,Z)HC2In this section, we will determine the lattices MG := H2(S,Z)G and NG := H2(S,Z)HC2 .They will play an essential role in the study of mirror symmetry in the following sections.In the end of this section, as a simple application of our study, we will determine the Picardnumber of Calabi–Yau threefolds of type K, which was previously computed by Oguiso andSakurai via the Lefschetz fixed point formula [OS].Let S be a K3 surface and H a finite group. In [Has] it is shown that the action on theK3 lattice H2(S,Z) induced by a symplectic action of H on S depends only on H up toisomorphism, except for the five groups: Q8, T24, S5, L2(7) and A6.Theorem 4.2.8 (Hashimoto [Has]). Let H be a group listed in the table below. Supposethat H acts on a K3 surface S symplectically. The isomorphism class of the invariantlattice H2(S,Z)H is given by Table (4.2):The main proposition in the present section is the following classification of the latticesNG (we also list MG for future reference).Proposition 4.2.9. Let S be a K3 surface equipped with a Calabi–Yau G-action. For anysemi-direct decomposition G = H o C2, where H := Ker(G → GL(H2,0(S))), the latticesMG and NG are given by Table (4.3).Proof. It suffices to determineNG because the classification ofMG was completed in Propo-sition 3.3.4. Let Λ := H2(S,Z).584.2. Calabi–Yau Threefolds of Type KG MG NGC2 U(2)⊕ E8(−2) U ⊕ U(2)⊕ E8(−2)C2 × C2 U(2)⊕D4(−2) U(2)⊕2 ⊕D4(−2)C2 × C2 × C2 U(2)⊕ 〈−4〉⊕2 U(2)⊕2 ⊕ 〈−4〉⊕2D6 U(2)⊕A2(−2) U(3)⊕ U(6)⊕A2(−2)D8 U(2)⊕ 〈−4〉 U(4)⊕2 ⊕ 〈−4〉D10 U(2) U(5)⊕ U(10)D12 U(2) U(6)⊕2C2 ×D8 U(2) U(4)⊕ 〈4〉 ⊕ 〈−4〉Table 4.3: Lattices H2(S,Z)G and H2(S,Z)HC2Case H ∼= C1. The lattice NG = U ⊕ U(2) ⊕ E8(−1) is well-known in the theory ofEnriques surfaces.Case H ∼= C2. By the result of Ito and Ohashi [IO, Theorem 1.1], we see that NG ∼=U(2)⊕2 ⊕D4(−2).Case H ∼= C2 × C2. Since S is realized as a Horikawa model, we may assume, aftersuitable twists by the covering transformation, that C2 × C2 × C2 acts on the K3 surfaceS symplectically. By Theorem 4.2.8, we see that H2(S,Z)C⊕32 ∼= U(2)⊕3 ⊕ 〈−4〉⊕2. Thenwe conclude thatNG = (U(2))⊥H2(S,Z)C⊕32∼= U(2)⊕2 ⊕ 〈−4〉⊕2because the covering transformation acts trivially on the polarization U(2) and as negationon its orthogonal complement.Case H ∼= C3. We know that, by Theorem 4.2.8,MG ∼= U(2)⊕A2(−2) ⊂ H2(S,Z)C3 ∼= U ⊕ U(3)⊕2 ⊕A2(−1)⊕2.By localizing at p = 2, 3, we see that the discriminant group A(NG) is given by A(NG) ∼=Z⊕42 ⊕ Z⊕33 . Proposition 1.1.3 then implies that the genus gNG is a one-point set and wesee that U(3)⊕ U(6)⊕A2(−2).Case H ∼= C4. By the same reason as Case H ∼= C2 × C2, we see that D8 acts on theK3 surface S symplectically. We refer this action as D′8-action. Then we can prove thatthe lattice N⊥G is index 2 extension ofMG ⊕H2(S,Z)D′8∼= U(2)⊕ U ⊕ 〈4〉⊕2 ⊕ 〈−4〉⊕3.594.2. Calabi–Yau Threefolds of Type Kby the lift of two divisors defined by {z = 1}, where z is the coordinate of P1 in Proposition3.2.11. This determines the discriminant form of NG. We apply Proposition 1.1.3 andconclude that the genus gNG is a one-point set. The representative of gNG is given in theabove table.Case H ∼= C5. An almost identical argument to Case H ∼= C3 applies to this case.Case H ∼= C6. An almost identical argument to Case H ∼= C4 applies to this case.Case H ∼= C2 × C4. An almost identical argument to Case H ∼= C4 applies to this case.We note that〈4〉⊕2 ⊕ 〈−4〉⊕2 ∼= U(4)⊕ 〈4〉 ⊕ 〈−4〉.We refer the reader to Appendix for the orthogonal complements of MG and NG inH2(S,Z).An effective lattice theoretic criterion that provides a necessary and sufficient conditionfor the existence of a Calabi–Yau G-action on a K3 surface S is given by a lattice N⊥G -polarization. The dimension of the moduli space of K3 surfaces equipped with a Calabi–YauG-action is equal to rankNG−2, which is the dimension of the period domain defined by thetranscendental lattice T (S) = NG. Let us make a simple remark on how this is compatiblewith the Horikawa model realization. We observe that rankNG−2 coincides with the naivecount of dimension dimH0(OZ(4, 4))G − 1 of deformation of the H-equivalent Horikawamodel, except for the cases G ∼= C2 or C2 × C2. We refer the reader to Proposition 3.2.11for an explicit generators of the invariant sections H0(OZ(4, 4))G. For example, in the caseG ∼= C2, we must take into account the automorphism group of such a K3 surface S:dimH0(OZ(4, 4))G − 2− 1 = 13− 2− 1 = 10,where 2 is the dimension of the automorphism group of S commuting with the Enriquesinvolution. For G ∼= C2 × C2, such an automorphism group of S is 1-dimensional. Adding1-dimensional contribution coming from the deformation of an elliptic curve, we obtain theformulah2,1(X) = rankNG − 1,for general G, and = 11, 7 for G ∼= C2, C2 × C2 respectively. In their paper [OS], Oguisoand Sakurai computed the Picard number ρ(X), which is equal to h2,1(X) for Calabi–Yauthreefolds of type K, via the Lefschetz fixed point formula. The above observation providesus with an alternative proof of their result.604.3. Yukawa Couplings4.3 Yukawa CouplingsIn this section, we begin our discussion on mirror symmetry. Calabi–Yau threefolds of typeK are, by construction, topological self-mirror threefold, i.e.χ(X) = 2(h1,1(X)− h2,1(X)) = 0.However, mirror symmetry should involve more than the mere exchange of Hodge num-bers. We will see that the mirror symmetry of Calabi–Yau threefolds of type K bears aresemblance to mirror symmetry of Borcea–Voisin threefolds (see Appendix B). The latterrelies on the strange duality of certain involutions of K3 surfaces discovered by Nikulin[Nik3]. Our study, on the other hand, will employ the classification of lattices MG and NG(Proposition 4.2.9).As a warm-up, let us first consider the case G ∼= C2. A generic K3 surface S with anEnriques involution is a self-mirror K3 surface in the sense of Dolgachev [Dol], i.e.U ⊕NS(S) ∼= T (S) (4.3.1)where NS(S) and T (S) are given by MG and NG respectively in this case. Thus thecorresponding Enriques Calabi–Yau threefold (see Example 3.2.5) is an example of a self-mirror threefold. For G 6∼= C2, the corresponding K3 surface is never a self-mirror K3surface because T (S) = NG is of rank smaller than 12. Nevertheless, as we will see below,the corresponding Calabi–Yau threefold X = (E × S)/G of type K turns out to be a self-mirror threefold. In general MG and NG do not manifest symmetry over integers Z, butthe dualityU ⊕MG ∼= NG. (4.3.2)holds over rational numbers Q (or some extension of Z). It is worth noting that MG is notequal to NS(S) but to NS(S)G, while T (S) = NG.Remark 4.3.1. We may think of the duality (4.3.2) as an H-equivariant version of theduality (4.3.1) because we are considering sublattices of H2(S,Z)H : MG = (H2(S,Z)H)ι,NG = (H2(S,Z)H)ι.4.3.1 Moduli Spaces and Mirror MapsThroughout this section, X is a Calabi–Yau threefold of type K and S × E → X is itsminimal splitting covering with Galois group G. We will analyze the moduli space of Xequipped with a complexified Ka¨hler class. Leaving some discrete identifications, such amoduli space locally splits into the product of two deformation spaces: the complex modulispace and the complexified Ka¨hler moduli space.614.3. Yukawa CouplingsLet us begin with the moduli space of marked K3 surfaces S with a Calabi–Yau G-action. The period domain of such K3 surfaces is given byDGS := {CωS ∈ P(NG ⊗ C) | 〈ωS , ωS〉 = 0, 〈ωS , ωS〉 > 0}.The extended G-invariant complexified Ka¨hler moduli space is defined by the tube domainKGS := {BS + iκS ∈MG ⊗ C | 〈κS , κS〉 > 0}.The class BS is what physicists call the B-field, which is usually defined modulo H2(S,Z).This is where the Brauer group comes into the play (see Section 4.3.2), but we for a momentregard KGS as a covering of the G-invariant complexified Ka¨hler moduli space KGS /H2(S,Z),analogous to DGS being a covering of the complex moduli space. We will construct a mirrormap at the level of theses coverings.Proposition 4.3.2. There exists a holomorphic isomorphism qGS : KGS → DGS .Proof. By Proposition 4.2.9 there always exists an isomorphism NG⊗Q ∼= U⊕(MG⊗Q) asquadratic spaces over rational numbers Q. The holomorphic isomorphism qGS : KGS → DGSis then given by the mappingBS + iκS 7→ C(e−12〈BS + iκS , BS + iκS〉f +BS + iκS),where e, f denotes the standard basis of U . This is known as the tube domain realization[Dol, Voi].Let us next recall the mirror symmetry of elliptic curves. Consider an elliptic curveE = C/(Z ⊕ ZτE) with a Calabi–Yau G-action, by which we mean a G-action on Edescribed in Proposition 3.2.3. We can continuously deform the G-action as we vary themoduli parameter τE . The period domain of such elliptic curves is thus given by the upperhalf-planeDE := {τE ∈ C | Im(τE) > 0}.The extended complexified Ka¨hler moduli space of E is identified, through integration overE, with the setKE,C := {BE + iκE ∈ H1,1(E) ∼= C | κE > 0}.There is an isomorphism qE : KE,C → DE given by qE(BE + iκE) := BE + iκE . Weassociate to a pair (E1, BE1 + iκE1) the mirror pair (E2, τE1) with τE2 := BE1 + iκE1 .Given a point (BS + iκS , BE + iκE) on the extended complexified Ka¨hler moduli spaceKX,C := KGS ×KE,C of X, the point(qGS (BS + iκS), qE(BE + iκE)) ∈ DX := DGS ×DE624.3. Yukawa Couplingsdetermines a pair (S∨, E∨) of a K3 surface and an elliptic curve, and hence another Calabi–Yau threefold X∨ := (S∨ ×E∨)/G of type K. In the same manner, a point (ωS , τE) ∈ DXdetermines an extended complexified Ka¨hler structure ((qGS )−1(ωS), q−1E (τE)) ∈ KX,C on thepair (S∨×E∨), which subsequently determines an extended complexified Ka¨hler structureon X∨. We therefore obtain an involution q = (qGS , qE) of DX × KX,C. We identify thisinvolution with the self-mirror map of the family of Calabi–Yau threefolds X of type K.4.3.2 Brauer Group and A-Model Moduli SpaceFor a Calabi–Yau threefold X of type K with odd |H|, we know that the Brauer groupis non-trivial (Proposition 4.2.4). In this section, we explain the role of the Brauer groupin mirror symmetry, following the exposition of Aspinwall and Morrison [AM]7. In the A-model topological string theory, the correlation functions depend on non-trivial instantons,namely holomorphic maps σ : Σg → X from the world-sheet Σg, i.e. genus g curve, to thetarget threefold X. Furthermore, the action is required to depend linearly on the homologyclass of the image of the map:µ ∈ Hom(H2(X,Z),C×) ∼= H2(X,C×).By the exact sequence0→ H2(X,Z)/Tor(H2(X,Z))→ H2(X,C)→ H2(X,C×)→ Tor(H3(X,Z))→ 0we see thatH2(X,C×) ∼=(H2(X,C)/H2(X,Z))⊕ Tor(H3(X,Z)),provided that the sequence splits. We now assume that the B-field represents a class inthe quotient H2(X,R)/H2(X,Z). Then we may think of the complexified Ka¨hler modulispace KX,C lying in the first summand. In this situation, the isomorphism above showsthat we need to incorporate in the A-model moduli space the contribution coming fromTor(H3(X,Z)) ∼= Br(X), when the Brauer group is non-trivial. State it another way, adiscrete parameter α ∈ Br(X) is required in order to determine the A-model correlationfunctions.We may consider the shape of the moduli space of A-model topological field theory asfollows. Suppose that the complexified Ka¨hler class B + iκ ∈ KX,C/H2(X,Z) approachesinfinity, which we call the large volume limit (LVL) of KX,C/H2(X,Z), in the sense that∫β κ → ∞ for all non-zero effective curves class β ∈ H2(X,Z). At the LVL, therefore,the choice of α does not matter. Thus different components of the moduli space, eachparametrized by B + iκ ∈ KX,C/H2(X,Z) but having a different values of α, are joined atthe LVL (Figuer 4.1). For the rest of this chapter, we will ignore the issue of the Brauer7 It is, however, safe to say that there is no consensus as to how the Brauer group comes into the play.Look at [Gro2, Section 3] for another aspect of torsions in cohomology groups.634.3. Yukawa CouplingsLVLα2α3 α4 α1Figure 4.1: A-model moduli space for Br(X) ∼= Z4group without harm, by assuming that such an α is chosen.4.3.3 A- and B-Yukawa CouplingsWe will introduce the A-Yukawa coupling for a Calabi–Yau threefold X with a complexifiedKa¨hler class B + iκ ∈ KX,C/H2(X,Z). The A-Yukawa couplingY XA : H1,1(X)⊗3 → C[[q]]is the symmetric trilinear form given byY XA (H1,H2,H3) = H1 ∪H2 ∪H3 +∑β 6=0nβe2pii∫β(B+iκ)1− e2pii∫β(B+iκ)3∏i=1∫βHi,where the sum is over effective homology classes β ∈ H2(X,Z) mod torsion and C[[q]]is the Novikov ring associated to H2(X,Z). nβ is naively the number of rational curveson X in the homology class β. A mathematical definition of nβ is carried out via theGromov–Witten invariants of X, about which we shall skip the details in this chapter. Atthe LVL, where∫β κ→∞ for all non-zero effective curves class β ∈ H2(X,Z), we say thatthe A-Yukawa coupling YA asymptotically converges to the cup product, which we studiedin Theorem 4.2.6, as all the quantum correction terms vanish.We will next define the B-Yukawa coupling of X. Recall that we have the followingisomorphismsH2,1(X) ∼= H1(X,Ω2X) ∼= H1(X,TX).644.3. Yukawa CouplingsThe differential of the period map on the Kuranishi space for X can be identified with theinterior productH1(X,TX) −→ Hom(Hk,l(X),Hk−1,l+1(X))for k, l ∈ N. By iterating the maps above, we obtain a symmetric trilinear form, which wecall the B-Yukawa coupling of X,Y XB : H1(X,TX)⊗3 −→ Hom(H3,0(X),H0,3(X)) ∼= C,where the last identification depends on a trivialization H3,0(X) ∼= C. Let X be a Calabi–Yau threefold of type K with the minimal splitting covering S ×E → X and Galois groupG. Choose holomorphic volume forms dzE and ωS on E and S respectively. Using theisomorphismϕ : TS → ΩS , v 7→ ιvωS = ωS(v, ∗),we identify the following cohomology groupsH1,1(S)HC2 ⊗H1,0(E) ∼= H1(S, TS), η ∧ θ ⊗ dzE ↔ ϕ−1(η)⊗ θH2,0(S)HC2 ⊗H0,1(E) ∼= H1(E, TE). ωS ⊗ dzE ↔∂∂zE⊗ dzE ,where η (θ) is the (anti-)holomorphic part of η ∧ θ ∈ H1,1(S)HC2 .Lemma 4.3.3. Assume that the trivialization of H3,0(X) ∼= H2,0(S) ⊗ H1,0(E) is givenby a nowhere-vanishing global section ωS ⊗ dzE. Then we haveY XB (ωS ⊗ dzE , η1 ∧ θ1 ⊗ dzE , η2 ∧ θ2 ⊗ dzE) = 〈η1 ∧ θ1, η2 ∧ θ2〉H1,1(S)HC2,where 〈∗, ∗∗〉H1,1(S)HC2denotes the cup product restricted to H1,1(S)HC2. The whole B-Yukawacoupling Y XB is obtained by extending the above form trilinearly and symmetrically and bysetting other non-trivial couplings to be 0.Proof. Using the above identification of cohomology groups, we may show the claim by astraightforward computation.Therefore the B-Yukawa coupling is of type 〈∗, ∗∗〉H1,1(S)HC2after C-extension. On theother hand, by Proposition 4.2.9, there always exists a decompositionNG = U(k)⊕ U(k)⊥NG , U(k)⊥NG ⊗Q ∼= MG ⊗Q.for some k ∈ N. Moreover, U(k)⊥ ∼= MG holds for G ∼= C2, C2×C2 or C2×C2×C2. Sincethe complex moduli space is an arithmetic quotient of a bounded symmetric domain, itcan be compactified to be a projective variety via the Bailey–Borel compactfication [BB].654.3. Yukawa CouplingsThe primitive isotropic sublattices in the transcendental lattice T (S) = NG correspondto the cusps of the Baily–Borel compactification [Sca]. Therefore, once we choose thestandard basis e, f of the hyperbolic lattice U(k), we may think that the isotropic vector ecorresponds to a large complex structure limit (LCSL) in the context of mirror symmetry.We will not delve more into the details of LCSL, but suggest the exposition [CK2] to thereader.Let us now investigate the behavior of the B-Yukawa coupling when the moduli pointof our K3 surface S approaches to the cusp, to put it another way, when the period pointCωS approaches to the line Ce. The Hodge (1, 1)-part of the quadratic space H2(S,C)HC2is then identified with the quotient quadratic space:H1,1(S)HC2 ∼= (ωS)⊥H2(S,C)HC2/CωS .Hence, at the limit as the period ωS approaches to e, the quadratic spaceH1,1(S)HC2 becomesthe C-extension of e⊥/Qe ∼= MG ⊗ Q. We thereby conclude that the quadratic factor ofthe B-Yukawa coupling Y XB asymptotically converges to MG ⊗ C near the LCSL.4.3.4 Asymptotic Behavior of Yukawa CouplingsGiven a pair (X ,Y ) of mirror families of Calabi–Yau threefolds. One feature of mirrorsymmetry is the identification of the Yukawa couplingsY XA (H1,H2,H3) = Y YB (θ1, θ2, θ3)after suitable transformation, so-called the mirror map, of local moduli parameters Hiaround the LVL of the complexified Ka¨hler moduli space KX,C/H2(X,Z) of X and θiaround a LCSL of the complex moduli space MY of Y . Here X is a general member ofX near the LVL and Y is a general member of Y near the LCSL. This highly non-trivialconjecture has been confirmed for a large class of Calabi–Yau threefolds, e.g. completeintersection in toric varieties [CK2].Our claim is that Calabi–Yau threefolds of type K are self-mirror threefolds in thesense that the 2 Yukawa couplings Y XA and Y XB have similar asymptotic behaviors; as thecomplexified Ka¨hler moduli B+iκ ∈ KX,C/H2(X,Z) approaches to the LVL, the A-Yukawacoupling Y XA becomes the classical cup product on H1,1(X), which is the C-extension ofthe trilinear form on H2(X,Z) given in Theorem 4.2.6. As the period ωS of the K3 surfaceS approaches to the cusp of the period domain DGS , the B-Yuakawa coupling Y XB becomesa trilinear form, which has a linear factor and whose residual quadratic form is the C-extension of e⊥/Qe ∼= MG ⊗ Q. Summarizing the above argument, we finally obtain thefollowing theorem.664.3. Yukawa CouplingsTheorem 4.3.4. Let X be a Calabi–Yau threefold of type K. The asymptotic behavior ofthe A-Yukawa coupling Y XA around the LVL coincides with the asymptotic behavior of theB-Yukawa coupling Y XB around a LCSL. The identification respects the rational structureof the trilinear forms.ForG ∼= C2 the identification respects the integral structure. However, sinceH2(X,Z)/Toris in general contained in H2(S ×E,Z) of finite index via pi∗, we cannot really expect theabove theorem to hold over Z, but need some extensions. See also [Voi] for the case ofBorcea–Voisin threefolds.4.3.5 B-Model ComputationIn this section, we will compute the B-models of Calabi–Yau threefolds of type K with 3- or4-dimensional moduli space. Let X be a Calabi–Yau threefold of type K and pi : S×E → Xthe minimal splitting covering with Galois group G. For G ∼= D12, with the same notationas in Example 4.2.1, the period integral of S is given byΦ0 :=∫γRes (xdy − ydx) ∧ (zdw − wdz)√ax2y2z2w2 + b(x4z4 + y4w4) + c(x4zw3 + y4z3w),where [a : b : c] ∈ P2 is the moduli parameter and we choose some γ ∈ H2(S,Z). On theopen set {a 6= 0} ∼= C2 with z1 := (b/2a)2 and z2 := (c/2a)2, the period integral readsΦ0(z1, z2) = 1 + 12(z1 + z2) + 420(z21 + 4z1z2 + z22) + 18480(z1 + z2)(z21 + 8z1z2 + z22) + . . .for an indivisible integral cycle γ invariant under monodromy about (z1, z2) = (0, 0). Thena numerical calculation shows that the Picard–Fuchs system is generated byDi := Θ2i − 4zi(4Θ1 + 4Θ2 + 3)(4Θ1 + 4Θ2 + 1)for i = 1, 2, where Θi := zi ∂∂zi is the Euler differential. We observe that the point (z1, z2) =(0, 0) is a LCSL of this family. The linear-logarithmic solutions areΦ1(z1, z2) := Φ0 log(z1) + 40z1 + 64z2 + 1556z21 + 7904z1z2 + 2816z22 + . . . ,Φ2(z1, z2) := Φ0 log(z2) + 64z1 + 40z2 + 2816z21 + 7904z1z2 + 1556z22 + . . . ,Then the mirror map around the LCSL is given byq1 = exp(Φ1(z1, z2)/Φ0(z1, z2))= z1 + 8z1(5z1 + 8z2) + 4z1(469z21 + 2304z1z2 + 1024z22) + . . . ,q2 = exp(Φ2(z1, z2)/Φ0(z1, z2))= z2 + 8z2(8z1 + 5z2) + 4(1024z21 + 2304z1z2 + 469z22)z2 + . . . ,674.3. Yukawa Couplingswhere qi = e2piiti for i = 1, 2 and (t1, t2) is the complexified Ka¨hler parameter of a mirror K3surface. However, in the present case, we expect (t1, t2) to be the G-invariant complexifiedKa¨hler parameter of S itself so that it descends to the complexified Ka¨hler parameter ofX. The inverse mirror maps read (see for example [KM1])z1(q1, q2) =ϑ82(q1)64(ϑ43(q1) + ϑ44(q1))2(1− ϑ82(q2)(ϑ43(q2) + ϑ44(q2))2),z2(q1, q2) =ϑ82(q2)64(ϑ43(q2) + ϑ44(q2))2(1− ϑ82(q1)(ϑ43(q1) + ϑ44(q1))2),where ϑ2(q), ϑ3(q) and ϑ4(q) are the elliptic theta functions:ϑ2(q) : =∑n∈Zq12 (n+12 )2 = 2q1/8∞∏n=1(1− qn)(1 + qn)2,ϑ3(q) : =∑n∈Zqn22 =∞∏n=1(1− qn)(1 + qn−12 )2ϑ4(q) : =∑n∈Z(−1)nqn22 =∞∏n=1(1− qn)(1− qn−12 )2.The period Φ0 also reads, with respect to the mirror coordinates,Φ0(q1, q2) =12√(ϑ43(q1) + ϑ44(q1))(ϑ43(q2) + ϑ44(q2)).We denote by ∇zi the Gauss–Manin connection with respect to the local moduli parameterzi and defineCzi,...,zj :=∫SωS ∧ (∇zi · · ·∇zjωS).Using the Griffith transversality relationsCzi = 0, 3∂z1Cz1,z1 = 2Cz1,z1,z1 ∂z2Cz1,z1 + 2∂z1Cz1,z2 = 2Cz1,z1,z2 ,we determine the B-Yukawa couplings, up to multiplication by a constant, as follows:Czi,zi(z1, z2) =125zi(1− 27(z1 + z2 + 26z1z2) + 212(z21 + z22)),Cz1,z2(z1, z2) =1− 64(z1 + z2)212z1z2(1− 27(z1 + z2 + 26z1z2) + 212(z21 + z22)).Via the mirror map, we determine the A-Yukawa couplingsKti,tj (q1, q2) =1Φ0(z1, z2)2∑k,lCzk,zl(z1, z2)∂zk∂ti∂zl∂tj.684.3. Yukawa CouplingsThere is no quantum correction for a K3 surface and we see that Kt1,t1 = Kt2,t2 = 0 andKt1,t2 = 1. This recovers the fact H2(S,Z)D12 ∼= U(2), up to multiplication by a constant.An almost identical argument works for G ∼= D10 and C2×D8, and we see that the B-modelcomputation is compatible with the arguments in the previous sections.We now turn to the elliptic curve part and consider the following family of ellipticcurves with level 6 structurex61 + x32 + x23 + z−1/63 x1x2x3 = 0in the weighted projective space P2(1, 2, 3). The Picard–Fuchs operator is given byD3 := Θ23 − 12z3(6Θ3 + 5)(6Θ3 + 1)and the mirror map is given byq3 = z3 + 312z23 + 107604z33 + 39073568z43 + 14645965026z53 + 5609733423408z63 + . . .As in the K3 surface case, the fundamental period can be written in terms of the Eisensteinseries of weight 4Φell0 (q3) = E1/44 (q3).Therefore, we observe that the period integral and the mirror maps of the threefold X areall written in terms of the modular forms for G ∼= D12.Proposition 4.3.5. The Picard–Fuchs differential system for X is generated by the oper-ators Di for i = 1, 2, 3.Proof. By the Griffith transversality, we see that D1 and D2 generate the system for S.Indeed, the 3rd covariant derivatives of ωs are obtained by the covariant derivatives ofD1 and D2. In the same manner, the D3 generate the system for E. Let us now turn toX = (S ×E)/G with a holomorphic 3-form ωS ⊗ dz. Again, by the Griffith transversality,we see thatC〈Ω,∇ziΩ,∇zi∇zjΩ, 〉1≤i,j≤3 = H3,0(X)⊕H2,1(X)⊕H1,2(X).This shows the existence of 3 linear relations, i.e. 3 generators of degree 2. Moreover, wehaveC〈Ω,∇ziΩ,∇zi∇zjΩ,∇zi∇zj∇zkΩ〉1≤i,j,k≤3 = H3(X,C)This shows the existence of 9 more linear relations. However, the observation for S and Ein the beginning confirms that they are all derived from the covariant derivatives of the 3generators of degree 2. This proves the assertion.694.4. Special Lagrangian FibrationsLet us next consider the case G ∼= D8. With a similar notation, the period integral isgiven byΦ0(z1, z2, z3) = 1 + 12(z1 + z2 + z3) + 420(z21 + z22 + z23 + 4z1z2 + 4z2z3 + 4z1z3) + . . .and the Picard–Fuchs system is generated byDa =(a1 − 64a1z1 + 4(3a5 − 16a1 − 16a2)z2 − 12a4z3)Θ21+ (a2 − 12a5z1 − 64a2z2 + 4(3a6 − 16a2 − 16a3)z3)Θ22+ (a3 + 4(3a4 − 16a1 − 16a3)z1 − 12a6z2 − 64a3z3)Θ23− 128(a1z1 + a2z2 + a1z3)(Θ1Θ2 +Θ2Θ3 +Θ3Θ1)− 64(a1z1 + a2z2 + a3z3)(Θ1 +Θ2 +Θ3)− 12(a1z1 + a2z2 + a3z3)for a = [a1, . . . , a6] ∈ C6. In the same manner as above, we determine the A-Yukawacouplings, up to multiplication by a constant,Kt1,t1 Kt1,t2 Kt1,t3Kt2,t1 Kt2,t2 Kt2,t3Kt3,t1 Kt3,t2 Kt3,t3 =0 1 11 0 11 1 0∼= U ⊕ 〈−2〉,which is compatible with H2(S,Z)D8 ∼= U(2)⊕ 〈−4〉.4.4 Special Lagrangian FibrationsLet X be a Calabi–Yau manifold of dimension n with a Ricci-flat Ka¨hler metric g. Let κbe the Ka¨hler form of g. There exists a nowhere-vanishing holomorphic n-form ΩX on Xand we normalize ΩX by setting(−1)n(n−1)2 ( i2)nΩX ∧ ΩX =κnn!,which uniquely determines ΩX up to a phase eiθ.Definition 4.4.1. A submanifold L ⊂ X of real dimension n is called special Lagrangianifκ|L = 0, <(ΩX)|L = eiθLvolLfor some constant θL ∈ R. Here volL denotes the Riemannian volume form on L inducedby g. We say that a surjective map pi : X → B is a special Lagrangian Tn-fibration ifgeneric fibers of pi are special Lagrangian n-tori.704.4. Special Lagrangian FibrationsIn the paper [SYZ], Strominger, Yau and Zaslow proposed that mirror symmetry shouldbe what physicists call T-duality. In this so-called SYZ description, a Calabi–Yau manifoldX of dimension n admits a special Lagrangian Tn-fibration pi : X → B near a LCSL. Amirror Calabi–Yau manifold Y is then obtained by fiberwise dualization of pi, modifiedby suitable instanton corrections. The importance of this conjecture lies in the fact thatit could lead to an intrinsic characterization of the mirror Calabi–Yau manifold Y ; theoriginal Calabi–Yau manifold X may be seen as the (compactified) moduli space of thesespecial Lagrangian fibers Tn ⊂ Y with flat U(1)-connections.Example 4.4.2. Let E := C/(Z+Zτ) be an elliptic curve with τ ∈ H. Fix a holomorphic1-form dz := dx+ idy and a Ka¨hler form κ := dx ∧ dy. A special Lagrangian submanifoldL ⊂ X is a real curve such that κ|L = 0 and dy|L = eiθLvolL for some θL ∈ R. The firstcondition is vacuous as L is of real dimension 1, and the second condition implies that Lmust be a line. For example, the mappiE : E = C/(Z+ Zτ)→ S1 = R/=(τ), z 7→ =(z)is a special Lagrangian smooth T 1-fibration. Note that the Ka¨hler form κ is invariant undertranslations and negation −1E.Given a Calabi–Yau manifold, finding a special Lagrangian fibration is an importantand currently unsolved problem in high dimensions, whereas there are some examplesunder relaxed conditions (see for example [Gro]). Among others, the best result in 3-dimensions may be Gross and Wilson’s work on Borcea–Voisin threefolds [GW]. Recallthat Borcea–Voisin threefolds are obtained from a K3 surface S with anti-symplectic (non-Enriques) involution ι and an elliptic curve E by resolving singularities of the quotient(S × E)/〈(ι,−1E)〉. See Appendix or the original papers [Bor, Voi] for more details. Wewould like to point out that there are two drawbacks in Borcea–Voisin threefolds case[GW]:1. One needs to allow a degenerate Ricci-flat metric.2. One needs to work on a slice of complex moduli space, where the threefold is realizedas a blow-up of an orbifold Calabi–Yau threefold (S × E)/〈(ι,−1E)〉.In this section we will construct special Lagrangian fibrations of Calabi–Yau threefolds oftype K with smooth Ricci-flat metric. We remark that such a fibration for the EnriquesCalabi–Yau threefold was essentially constructed in [GW]. This section is clearly influencedby Gross and Wilson’s work, and most of our arguments below are slight modification oftheirs. We state it here and do not repeat it each time in the sequel.714.4. Special Lagrangian FibrationsComplex Structure Holomorphic 2-form Ka¨hler formI ωI := <ωI + i=ωI κIJ ωJ := κI + i<ωI κJ := =ωIK ωK := =ω + iκI κK := <ωITable 4.4: Complex structure, holomorphic 2-form and Ka¨hler form4.4.1 K3 Surfaces as HyperKa¨hler ManifoldsLet S be a K3 surface with a Ricci-flat Ka¨hler metric g. Then the holonomy group with re-spect to the metric g is SU(2) ∼= Sp(1) and the parallel transport defines complex structuresI, J,K satisfying the quaternion relationsI2 = J2 = K2 = IJK = −1such thatS2 = {aI + bJ + cK ∈ End(TS) | a2 + b2 + c2 = 1}is the possible complex structures for which g is a Ka¨hler metric. The period of S in thecomplex structure I is defined by the normalized holomorphic 2-formωI(∗, ∗∗) := g(J∗, ∗∗) + ig(K∗, ∗∗),and the compatible Ka¨hler form on S is given byκI(∗, ∗∗) := g(I∗, ∗∗).See Table (4.4). We denote, for instance, by SK a K3 surface S equipped with the complexstructure K.Proposition 4.4.3 (Harvey–Lawson [HL]). A real smooth surface L ⊂ SI is a specialLagrangian submanifold if and only if L ⊂ SK is a complex submanifold.This proposition reduces the study of special Lagrangian T 2-fibrations on SI to that ofelliptic fibrations on SK .4.4.2 Calabi–Yau Threefolds of Type KI this section we will construct special Lagnrangian fibrations of Calabi–Yau threefolds oftype K. Our argument will again rely on the lattice structure of MG and NG classified inProposition Special Lagrangian FibrationsLet X be a Calabi–Yau threefold of type K and S × E → X its minimal splittingcovering with Galois group G. We may assume that S is equipped with a G-invariantKa¨hler class κ, which uniquely determines a G-invariant Ricci-flat metric on S [Yau]. Wemay also assume that κ is generic in the sense that κ⊥ ∩ H2(S,Z)G = 0. The productRicci-flat metric on S × E is G-invariant and hence descends to a Ricci-flat metric on thequotient Calabi–Yau threefold X. In what follows, when we speak about a metric on thequotient X, it is understood that the metric is the one obtained in this way.Proposition 4.4.4. There exists a period ωI ∈ DGS of S such that its hyperKa¨hler rotationSK admits an elliptic fibration piS : SK → P1 with a multiple section. In other words, SIadmits a special Lagrangian T 2-fibration piS : SI → P1 with a multiple special Lagrangiansection.Proof. By Proposition 4.2.9, the transcendental lattice NG = H2(S,Z)HC2 always containsU(k) for some k ∈ N. We denote the standard basis of U(k) by e, f . We choose a periodωI ∈ DGS with =ωI ∈ U(k)⊥NG ⊗ R. It then follows that e, f are algebraic with respect thecomplex structure K, i.e. 〈e, ωK〉 = 〈f, ωK〉 = 0. We may assume that e is the class of anelliptic fibration piS : SK → P1, after a sequence of reflection by (−2)-curves in NS(SK).In this situation, the class f yields a k-section of the elliptic fibration.It is possible to prove the existence of special Lagrangian fibration for a generic choiceof complex structure of S with a numerical multiple section (see [GW, Proposition 1.3] forthe argument). We henceforth denote simply by S the K3 surface SI with the complexstructure I obtained above.Proposition 4.4.5. The H-action is holomorphic and that of ι is anti-holomorphic on SK ,where K is the complex structure obtained in Proposition 4.4.4. Moreover the G-action onS descends to the base P1 in such way that the above fibration piS : S → P1 is G-equivariant.Proof. Recall that our Ka¨hler form κ is G-invariant. The metric g is still Ricci-flat after ahyperKa¨hler rotation and hence the G-action is isometry with respect to the metric g onSK . The holomorphic 2-form ωK is harmonic and so is g∗ωK for any g ∈ G. For any h ∈ H,we hence have h∗ωK = ωK as a 2-form. It then follows that the H-action is holomorphicon SK . In the same manner, we can show that ι∗ωK = −ωK as a 2-form and thus ι isan anti-holomorphic. Since H leaves the fiber and multiple section class invariant and ιsimply changes their sign, the action of G on SK descends to the base P1 in such a waythat the fibration pi : SK → P1 is G-equivariant.Let piS : S → P1 ∼= S2 be the special Lagrangian fibration derived above. Combiningthe maps piS and piE , we obtain a special Lagrangian T 3-fibration piS×E : S×E → S2×S1with respect to the Ricci-flat metric on S ×E. By Proposition 4.4.5, piS×E induces a mappiX : X = (S × E)/G→ B = (S2 × S1)/G.734.4. Special Lagrangian FibrationsProposition 4.4.6. The map piX : X → B is a special Lagrangian T 3-fibration such thateach (possibly singular) fiber has Euler number zero.Proof. Let p : S2 × S1 → B be the quotient map. For any generic b ∈ B in the sense thatp−1(b) 6⊂ DpiS×E , the fiber pi−1S×E(p−1(b)) consists of |p−1(b)| copies of T 3. Since G acts onpi−1S×E(p−1(b)) freely, the fiberpi−1X (b) = pi−1S×E(p−1(b))/Gis again T 3. This shows that piX : X → B is a special Lagrangian T 3-fibration becausethe special Lagrangian condition is preserved by the free G-action. Moreover, for anyc = (c1, c2) ∈ S2 × S1, we have the Euler numberχ(pi−1S×E(c)) = χ(pi−1S (c1)× S1) = 0.Therefore, for any b ∈ B, we then haveχ(pi−1X (b)) =χ(pi−1S×E(p−1(b)))|G|= 0.Proposition 4.4.7. The base space B is topologically identified as either the 3-sphere S3or an S1-bundle over RP2.Proof. In the same manner as above, it is easy to see that the map(S × E)/H → (S2 × S1)/His a special Lagrangian T 3-fibration. Since the H-actions on S2 and S1 are rotations, thebase is (S2 × S1)/H ∼= S2 × S1. The induced involution 〈ι〉 ∼= G/H on the first factorS2 ∼= P1 is anti-holomorphic and hence may be identified with eitherz 7→ z with (S2)ι = R ∪∞, or z 7→ −1zwith (S2)ι = ∅.The involution 〈ι〉 ∼= G/H on the second factor S1 ⊂ C is the reflection along the real axiswith (S1)ι = S0. The case where (S2)ι = R ∪∞ was previously studied in [GW, Section3] and the base B = (S2 × S1)/〈ι〉 is topologically identified with S3. On the other hand,when (S2)ι = ∅, the base B = (S2×S1)/〈ι〉 is endowed with a S1-bundle structure via thefirst projectionB = (S2 × S1)/〈ι〉 → S2/〈ι〉 ∼= RP2.744.5. Appendix (Borcea–Voisin Threefolds)It is worth noting that we do not expect that B need be a 3-sphere S3 because Calabi–Yau threefold X of type K is not simply-connected. It would be interesting to study thesingular fibers of the special Lagrangian fibration piX . The classification may be possiblebased on a close study of singular fibers of piS . The various lattices in Proposition 4.2.9will be useful in such an investigation.Another interesting problem is to pursue mirror symmetry in the SYZ description.Due to the lack of a genuine section of piX , it is not clear at this point whether or notwe can obtain a mirror threefold, which are conjecturally original manifold, by dualizingthe special Lagrangian T 3-fibration we constructed. Aspects of this assertion have beenconfirmed for Borcea–Voisin threefolds by Gross and Wilson [GW]. Their method can beapplied to the Enriques Calabi–Yau threefold. They also recovered the mirror maps fromthe Leray spectral sequence associated to special Lagrangian T 3-fibration. For Borcea–Voisin threefolds, this is the first available description of the mirror map. Their argumentwith a mild modification should work for Calabi–Yau threefolds of type K.4.5 Appendix (Borcea–Voisin Threefolds)We briefly review mirror symmetry of Borcea–Voisin threefolds developed in [Bor, Voi]. LetS be a K3 surface with an anti-symplectic involution ι. The fixed locus of ι is a disjointunion of smooth curves C1, . . . , CN . There are 3 cases:1. If H2(S,Z)ι ∼= U(2)⊕E8(−2), then ι is an Enriques involution.2. If H2(S,Z)ι ∼= U ⊕E8(−2), then Sι = C1 t C2, where Ci is an elliptic curve.3. If H2(S,Z)ι is none of above, we may take Ci ∼= P1 for i ≥ 2 andg(C1) = 11−12(rankH2(S,Z)ι + rankF2(H2(S,Z)ι)∨/H2(S,Z)ι)N = 12(rankH2(S,Z)ι − rankF2(H2(S,Z)ι)∨/H2(S,Z)ι) + 1Set N ′ :=∑Ni=1 g(Ci). Let E be an elliptic curve and −1E its negation involution. Weconsider the diagonal involution ι′ := (ι,−1E) on the product threefold S×E. The quotientthreefold (S × E)/〈ι′〉 has A1-singularities along 4N curves, but there exists an explicitcrepant resolution, which yields a smooth Calabi–Yau threefoldX := ˜(S × E)/〈ι′〉754.5. Appendix (Borcea–Voisin Threefolds)Such a Calabi–Yau threefold is called a Borcea–Voisin threefold [Bor, Voi]. Moreover, theHodge numbers are given byh1,1(X) = 11 + 5N −N ′, h2,1(X) = 11 + 5N ′ −N.Assume (N,N ′) 6= (5, 1). According to Nikulin’s classification of involutions of K3 surfaces[Nik3], H2(S,Z)ι contains a hyperbolic plane U and further more there exists another K3surface S∨ with an anti-symplectic involution ι∨ such thatH2(S,Z)ι ∼= U ⊕H2(S∨,Z)ι∨ , (NS , N ′S) = (N ′S∨ , NS∨).Since NS(S) = H2(S,Z)ι and T (S) = H2(S,Z)ι (assuming that S is generic with suchan involution ι), S and S∨ are mirror symmetric in the sense of Dolgachev [Dol]. There-fore, mirror symmetry of Borcea–Voisin threefolds are essentially equivalent to that of K3surfaces. More precisely, Borcea–Voisin threefolds X and X∨, obtained from S and S∨respectively, have mirrored Hodge numbers. The first two cases of the above classificationof anti-symplectic involution yield self-mirror Borcea–Voisin threefolds.76Chapter 5Non-Commutative ProjectiveCalabi–Yau Schemes5.1 IntroductionThe present chapter is concerned with some non-commutative Calabi–Yau projective schemes.The goal of the chapter is twofold: to construct the first examples of non-commutativeprojective Calabi–Yau schemes, in the sense of Artin and Zhang [AZ1], and to introduce avirtual counting theory of stable modules on them.Non-commutative Calabi–Yau algebras have attracted much attention in both math-ematics and physics [BS, Gin, Sze] for the last decades. This is largely because non-commutative geometry plays a fundamental role in quantum geometry, which is not cap-tured solely by the classical commutative geometry. However, almost all known non-commutative Calabi–Yau algebras are quiver algebras, and thus are a non-commutativeanalogue of local Calabi–Yau manifolds. The first objective of this chapter is to constructthe first examples of (non-trivial) non-commutative projective Calabi–Yau schemes.Theorem 5.1.1 (Theorem 5.2.2 & Corollary 5.3.8 ). Let k be an algebraically closed fieldof characteristic zero. We consider the following graded k-algebraAn := k〈x1, . . . , xn〉/(n∑k=1xnk , xixj = qijxjxi)i,j .where qij ∈ k× satisfies qii = qnij = qijqji = 1. Assume further that there exists c ∈ k×such that∏ni=1 qij = c for any 1 ≤ j ≤ n. Then the quotient category Coh(An) :=Gr(An)/Tor(An) is a Calabi–Yau (n− 2) category. Moreover, there exist quantum param-eters qi,j such that the graded k-algebra An is not realized as a twisted coordinate ring ofany commutative Calabi–Yau (n− 2)-fold.We observe that the Calabi–Yau condition imposes strong constrains on the quantumparameters qi,j . A key consequence is that Coh(An) is not obtained as a deformation ofthe homogeneous coordinate ring of a Calabi–Yau manifold. Moreover, Theorem 6.1.5 saysthat there exists a Calabi–Yau (n − 2) category Coh(An) which does not come from the775.1. Introductionconventional commutative geometry [Zha].One motivation to study non-commutative projective Calabi–Yau schemes comes fromDonaldson–Thomas theory [Tho]. Donaldson–Thomas theory is a virtual counting theoryof stable sheaves on a polarized Calabi–Yau threefold (X,L). It can be reformulated purelyin algebra; counting stable modules over the homogeneous coordinate ring⊕∞i=1H0(X,Li)of the threefold X. In [Sze], Szendroi introduced a non-commutative version of the theory,which virtually counts stable modules over a non-commutative resolution of the conifold,or more generally quiver Calabi–Yau 3 algebras. His theory relies on the existence of theglobal Chern–Simon functional on the moduli space of stables modules, and cannot bereadily generalized to the projective case.Next, the moduli scheme Mα of stable modules in Coh(An) with a fixed dimensionvector α will be constructed by the differential graded Lie algebra technique developed byBehrend and his collaborators [BCHR]. The goal of the second part of the present chapteris to show that the moduli scheme Mα for n = 5 comes equipped with a natural symmetricobstruction theory.Theorem 5.1.2 (Theorem 5.5.3). There exists a Calabi–Yau 3 tangent complex on themoduli scheme Mα. Moreover, Mα admits a natural symmetric obstruction theory.The theorem essentially follows from the paper [BCHR] but we present a detailed proofalong with necessary background materials in Section 5.4. When α comes from a numericalpolynomial H(t), that is, αt = H(t) for t ∈ N, the moduli scheme Mα conjecturally givesthe moduli scheme of sheaves of Hilbert polynomial H(t) provided the length of α is takensufficiently large. In this case, the theorem above would represent a non-commutativeversion of the work [Tho] of Thomas and a projective version of the work [Sze] of Szendro¨i.The layout of this chapter is the following. Section 5.2 is devoted to the construction ofprojective Calabi–Yau schemes. We will recall some basics on non-commutative projectiveschemes in the sense of Artin and Zhang [AZ1]. An example is constructed as a hypersurfacein a non-commutative projective space8. Section 5.3 computes some example of Hilbertscheme of points on An. In Section 5.4, we lay the necessary background for the theorydeveloped in [BCHR]. The main idea is to build a finite dimensional gauge theory associatedto a certain differential graded Lie algebra. Section 5.5.2 is the second main part of thischapter. It begin with a brief review of symmetric obstruction theory, and then we showthat the moduli scheme M of stable modules constructed via the finite dimensional gaugetheory naturally comes equipped with a natural symmetric obstruction theory.8 I am very grateful to Michel Van den Bergh for useful correspondence on non-commutative Calabi–Yauprojective schemes.785.2. Non-Commutative Calabi–Yau Projective Schemes5.2 Non-Commutative Calabi–Yau Projective SchemesIn this section, we review the notion of non-commutative projective geometry introducedby Artin and Zhang [AZ1]. The basic idea is that a non-commutative graded noetherian k-algebra A is regarded as a homogeneous coordinate ring of a non-commutative projectivevariety X over k. In this situation, the category of finitely generated graded right A-modules modulo torsion is identified with the category of coherent sheaves on X. Then weintroduce non-commutative projective Calabi–Yau schemes. It will be shown that Calabi–Yau condition imposes quite strong constraints on quantum parameters. Throughout thisarticle, non-commutative means not necessarily commutative.5.2.1 Non-commutative Projective SchemesLet k be a field and A =⊕∞i=0Ai be a connected noetherian graded k-algebra with unit.We assume that each graded piece is finite dimensional and A0 ∼= k. Our main example ofinterest is the homogeneous coordinate ring A of a projective variety over C. We denote byGr(A) the category of graded right A-modules with morphisms the A-module homomor-phisms of degree zero and by gr(A) the subcategory consisting of finitely generated rightA-modules. The augmentation ideal of A is defined by m :=⊕∞i=1Ai.Let M =⊕∞i=1Mi be an graded right A-module. An element x ∈ M is called torsionif xmn = 0 for some n ∈ N. All torsion elements form a submodule of M and we donateit by tor(M). Let Tor(A) denote the subcategory of Gr(A) of torsion modules and tor(A)denote the intersection of Tor(A) and gr(A). For any integer n ∈ Z and graded A-moduleW we define M(n) as the graded A-module that is equal to M as A-module, but withgrading M(n)i := Mn+i. We refer the functor s : Gr(A)→ Gr(A), M 7→M(1) as the shiftfunctor. sn is called the n-th shift functorIn [AZ1], Artin and Zhang introduced the notion of a non-commutative projectivescheme as follows. We define Tails(A) to be the quotient abelian categoryTails(A) := Gr(A)/Tor(A).The canonical exact functor from Gr(A) to Tails(A) is denoted by pi. tails(A) is alsodefined in the same manner. If M ∈ Gr(A), we use the corresponding script letter Mfor pi(M). For example A := pi(AA) where AA is A viewed as a right A-module. Thenon-commutative scheme of a graded right noetherian k-algebra A is defined as the tripleProj(A) := (Tails(A),A , s)In this paper we work on noetherian objects and the tripleproj(A) := (tails(A),A , s),795.2. Non-Commutative Calabi–Yau Projective Schemeswhich we call a projective scheme of A. When X = proj(A), we often denote A by OX ,whether or not A is commutative.Since Tails(A) is an abelian category with enough injectives, we may define the functorsExtiTails(A)(M , ∗) as the i-th right derived functor of HomTails(A)(M , ∗). In particular theglobal section functorH0(X, ∗) := HomTails(A)(OX , ∗) : Tails(A) −→ Vectkinduces the higher cohomologiesH i(X,M ) := ExtiTails(A)(OX ,M ).The bifunctor Extitails(A)(∗, ∗∗) is defined as restriction of ExtiTails(A)(∗, ∗∗) on tails(A).Definition 5.2.1. We say that a noetherian graded k-algebra A satisfies condition χ ifdimk ExtiTails(A)(k,M) <∞ for all i ≥ Calabi–Yau ConditionLet k be an algebraic closed field of characteristic zero. We denote by An the non-commutative graded k-algebraAn := k〈x1, . . . , xn〉/(n∑k=1xnk , xixj = qijxjxi)i,j ,where qij are n-th roots of unity with qii = qijqji = 1. The graded k-algebra An is of theform An = Bn/(fn) whereBn := k〈x1, . . . , xn〉/(xixj = qijxjxi)i,j , fn :=n∑k=1xnk .Bn is a Koszul Artin–Shelter (AS) regular algebra. We observe that fn :=∑ni=1 xni is anormalizing element of degree equal to the global dimension of the k-algebra Bn. Thusinformally tails(An) is a non-commutative Fermat hypersurface in quantum Pn−1. Thisexample was previously studied in physics [BS] without much mathematical justification.Theorem 5.2.2. Let An be the k-algebra defined above. Assume further that there exists aconstant c ∈ k× such that∏ni=1 qij = c for any 1 ≤ j ≤ n. Then tails(An) is a Calabi–Yau(n− 2) projective scheme in the sense thatgl.dim(tails(An)) = n− 2805.2. Non-Commutative Calabi–Yau Projective Schemesand tails(An) has a functorial perfect paringExti(M ,N )⊗k Extn−2−i(N ,M ) −→ kfor all M ,N ∈ tails(An). In particular, the derived category Db(tails(An)) is Calabi–Yau(n− 2) category9.Example 5.2.3. Let X = Proj(C) ⊂ P4 be the commutative Fermat quintic threefold givenbyC := k[x1, x2, x3, x4, x5]/(5∑i=1x5i ).Let qi be a 5-th root of unity for i = 1, . . . , 5. Then the map[x1 : x2 : x3 : x4 : x5] 7→ [q1x1 : q2x2 : q3x3 : q4x4 : q5x5]induces a projective automorphism σ of X. The twisted homogeneous coordinate ring Cσis then given byCσ := k〈x1, x2, x3, x4, x5〉/(5∑i=1x5i , xixj = qijxjxi)i,j ,where qij = qiq−1j . A result of Zhang [Zha] implies an equivalence of categoriesTails(C) ∼= Tails(Cσ).In particular Tails(Cσ) is a Calabi–Yau 3 category. Note that for any 1 ≤ j ≤ 5 we have5∏i=1qij = q1q2q3q4q5q−5j = q1q2q3q4q5,which is compatible with Theorem 5.2.2.If the graded k-algebra An is realized as a twisted coordinate ring of a commutativeprojective scheme X, then tails(An) ∼= Coh(X) as above and thus it is not really inter-esting. In Section 5.3, by computing Hilb1(tails(An)), we will prove that there exists anon-commutative Fermat Calabi–Yau (n− 2)-fold that is not realized as a twisted coordi-nate ring of a Calabi–Yau (n− 2)-fold.9 A Calabi–Yau n category C is a k-linear triangulated category, where k is a field, with a functorialperfect paringHom(E ,F )×Hom(F , E [n]) −→ kfor objects E ,F ∈ C.815.2. Non-Commutative Calabi–Yau Projective SchemesIn the rest of this section, we prove Theorem 5.2.2, assuming that gl.dim(tails(An)) =n−2, the proof of which will be given in the next section. In the following, we write A = Anand B = Bn for notational convenience. Before starting the proof, we need to know aboutthe balanced dualizing complex RA of A. A balanced dualizing complex plays a role ofgood dualizing sheaf in non-commutative graded algebra [Yek]. It behaves better than adualizing complex and corresponds, in the commutative case, to the local duality. SinceA has finite global dimension and is finite over its center Z(A), A satisfies the conditionχ. Then there is a formula [Yek, VdB] for the balanced dualizing complex RA of A as agraded ring10;RA = RΓm(A)′ ∈ Db(Tails(A))where Γm denotes local cohomology of A with respect to the augmentation ideal m. Localcohomology does not depend on the ring with respect to which it is taken so we maycompute it using a B-bimodule resolution of A0 −→ B(−n) ×f−→ B −→ A −→ 0.Here we used the fact that f ∈ Z(B). The exact sequence induces the following trianglein Db(tails(A)).RΓm(B(−n))×f// RΓm(B)yyrrrrrrrrrrRΓm(A)[1]ggOOOOOOOOOOOThis triangle relates RA with RB.We start computing the balanced dualizing complex RB. Let C be a two-sided noethe-rian Koszul AS regular algebra of global dimension n. By a result of Smith [Smi], itsKoszul dual C ! is a Frobenius algebra i.e. (C !)∗ ∼= Cφ! for some automorphism φ! of C !.By functionality, φ! is obtained by dualizing an automorphism φ of C.Theorem 5.2.4 (Van den Bergh [VdB, Theorem 9.2]). Let C be as above and let  theautomorphism of C which is multiplication by (−1)m on the graded piece Cm. Then thebalanced dualizing complex of C is given by Cφn+1 [n](−n).Proposition 5.2.5. Let B as above. The balanced dualizing complex RΓm(B)′ is Bφ[n](−n)as a graded B-bimodule, where φ is the automorphism of B which maps xj 7→∏ni=1 q−1ij xjfor 1 ≤ j ≤ n.10The exponent M′stands for the Matlis dual of a graded ring M .825.2. Non-Commutative Calabi–Yau Projective SchemesProof. First, B is a Koszul AS regular algebra of global dimension n. The Koszul dual B!of B is given by the twisted exterior algebraB! = 〈y1, . . . , yn〉/(qijyiyj + yjyi, y2k)i,j,k,where y1, . . . , yn is the dual basis of x1, . . . , xn. B! is a Frobenius algebra and (B!)∗ ∼= Bφ! ,where φ! is uniquely determined by the property of Frobenius pairing (a, b) = (φ(b), a) forany a, b ∈ B!. We hence obtain ab = φ!(b)a for any a ∈ B!i and b ∈ B!n−i. It then followsimmediately thatφ!(yj) =n∏i=1(−qji)yj .By dualizing φ!, we obtain the desired map φ. This completes the proof.Let C be a graded k-algebra and Cψ be a graded twisted k-algebra of C, where ψis the automorphism of C which acts by multiplication of cm on the graded piece Cmfor some c ∈ k. Such a special automorphism is invisible when passing to the quotientcategory Tails(C). In other words tensoring with such a bimodule is the identity functoron Tails(C). We are now ready to prove Theorem 5.2.2.Proof of Theorem 5.2.2. By Proposition 5.2.5 we obtain the following triangle in the de-rived category Db(Tails(A))RA // Bφ[n](−n)×fyyrrrrrrrrrrBφ[n],[1]bbFFFFFFFFFwhere the automorphism φ of B is given in Proposition 5.2.5. Then it immediately followsthatRA = Aφ′ [n− 1],where φ′ is the automorphism of A induced by φ. Remember that we have one extracondition; there exists a fixed constant c ∈ k× such that∏ni=1 qij = c for all 1 ≤ j ≤ n.Since Tails(A) has finite global dimension, the Serre functor of Tails(A) is induced by thedualizing complex RA of A. Although the functorF (∗) = ∗ ⊗Aφ′ [n− 1]is not (n − 1)-th shift functor in the category Gr(A) (unless c = 1), the Serre functorinduced by RA is the (n− 2)-th shift functor [n− 2] on the quotient category Tails(A).835.2. Non-Commutative Calabi–Yau Projective Schemes5.2.3 Proof of gl.dim(tails(An)) = n− 2We shall prove that tails(An) has global dimension n−2. As before k is an algebraic closedfield of characteristic zero. We begin with some lemmas. Let R is a finitely generatedcommutative ring and C an R-algebra which is finitely generated as an R-module. Assumefurther that R ⊂ Z(C).Lemma 5.2.6. The ring C has finite global dimension if the projective dimension of everysimple module is bounded by some fixed number m. The minimum such m is the globaldimension of C.Proof. This is clear by considering the long exact sequence induced from a short exactsequence.Lemma 5.2.7. Assume that C is a PI ring11. If S is a simple C-module, then its annihi-lator Ann(S) is some maximal ideal m of R. We then havepdimC(S) = pdimCm(Sm).Proof. Since C is a PI affine k-algebra, every simple C-module is finite dimensional [MR,Theorem 13.10.3]. We now have a map f : R → EndC(S) and EndC(S) is both a skewfield (by Schur’s lemma) and finite dimensional. Thus we conclude that EndC(S) = k andthe map f is surjective. Therefore, the kernel of the map f , which is the annihilator of S,is a maximal ideal in R. This proves the first half of the Lemma.Since the localization functor is exact, we havepdimC(S) ≥ pdimCm(Sm).If M and N are finitely generated C-modules, then ExtiC(M,N) is a finitely generatedR-module. Furthermore if m is a maximal ideal in R, thenExtiC(M,N)m = ExtiCm(Mm, Nm).Assume that ExtiCm(Sm, Nm) is zero for all N . Since Sn = 0 for any n maximal ideal inR which is not the annihilator of S, we also have ExtiCn(Sn, Nn) = 0 for such n and anyC-module N . This means that ExtiC(S,N) = 0 and hence pdimC(S) ≤ pdimCm(Sm). Thisproves the second half of the Lemma.Lemma 5.2.8 ([MR, Theorem 7.3.7]). Let S be a right Noetherian ring and f a regularnormal element belonging to the Jacobson radical J(S) of S. If rgld(S/(f)) <∞ thenrgld(S) = rgld(S/(f)) + 1.11The ring C above is a PI ring as it is finite over R ⊂ Z(C)845.2. Non-Commutative Calabi–Yau Projective SchemesLemma 5.2.9 ([MR, Theorem 7.3.5]). Let S be a ring and M an S-module. Take anormalizing non-zero divisor f ∈ Ann(M) and assume that pdimS/(f)(M) is finite. Wethen havepdimS/(f)(M) + 1 = pdimS(M).Let us begin with the proof of gl.dim(tails(An)) = n − 2. Recall that our non-commutative ring An is of the formAn := k〈x1, . . . , xn〉/(n∑k=1xnk , xixj = qijxjxi ∀i, j).We write ti = xni andD := k〈t1, . . . , tn〉/(n∑k=1tk).Then proj(An) may be seen as the category of modules over the sheaf of algebras B asso-ciated to An on the commutative scheme proj(D). The sheaf B is obtained by gluing fiveaffine patches given by inverting new variables t1, . . . , tn respectively.Let us invert for instance tn. Put Ti = ti/tn and Xi = xi/xn (right denominators). Theaffine patch under consideration is given byC := k〈X1, . . . , Xn−1〉/(n∑k=1Xnk + 1, XiXj = QijXjXi ∀i, j),where Qij := qij/(qniqnj). We then must show that gl.dim(C) = n − 2. Note that C is afree R-module withR := k[T1, . . . , Tn−1]/(n∑k=1Tk + 1),which is isomorphic to a polynomial ring in three variables.Let m = (T1 − a1, . . . , Tn−1 − an−1) with∑n−1i=1 ai +1 = 0 be a maximal ideal of R. ByLemmas 5.2.6, it is sufficient to show that gl.dim(Cm) = n− 2.We first consider the case when all ais are different from zero. Then we haveC/m = k〈X1, . . . , Xn−1〉/(XiXj = QijXjXi, Xnk − ak ∀i, j, k).is a twisted group algebra and hence semi-simple. This means that we have gl.dim(C/m) =0.855.3. Hilbert Schemes of PointsThe generators Ti − ai of m in R form a regular sequence in Cm. By the Lemma 5.2.8we conclude that gl.dim(Cm) = n− 2 because Cm/m ∼= C/m has global dimension zero.We may therefore assume that for instance an−1 = 0. Let S be a simple moduleannihilated by m. Since Tn−1 = Xnn−1 and Xn−1 ia a normalizing element, xn−1S is asubmodule of S. We thus conclude that the simple module S is actually annihilated byXn−1. Therefore S may be seen as a C/(Xn−1)-module, whereC/(xn−1) = k〈X1, . . . , Xn−2〉/(XiXj = QijXjXi, Xn1 + · · ·+Xnn−2 + 1, ∀i, j, k).According to Lamma 5.2.9, our problem reduces to showingpdimC/(xn−1)(S) = n− 3.The ring C/(Xn−1) is of the same kind of C and we can repeat the argument above;ultimately it is enough to show that the ringC/(X2, . . . , Xn−1) = k〈X1〉/(Xn1 + 1)has global dimension zero, which is clearly true. From this we also conclude that gl.dim(An) =n− 2. This completes the proof.5.3 Hilbert Schemes of PointsIn this section, we study abstract Hilbert schemes of points on non-commutative projectiveschemes. A way to assign geometric objects to a non-commutative scheme is to considerthe moduli problem. Recall that any variety is a moduli space of itself, i.e. X ∼= Hilb1(X).In the case where X is not a commutative scheme, the abstract Hilbert scheme Hilb1(X)is no longer a trivial object and has been studied well in non-commutative geometry. Herewe think of Hilbn(X) as the moduli space of n-point modules on X:Definition 5.3.1. A graded right A-module M is called an m-point module if1. M is generated in degree 0 with Hilbert series hM (t) = m1−t .2. There exists a surjection A→M of A-modules.The isomorphism classes of m-point module on A is denoted by Hilbm(A).Example 5.3.2. Let Fn := k〈x1, . . . , xn〉 be the free associative algebra in n variables.The abstract Hilbert scheme Hilb1(Fn) is an N-indexed sequences of points in the projectivespace Pn−1. In other words, the point modules are parametrized by the infinite product865.3. Hilbert Schemes of Points∏∞i=0 Pn−1. This can be seen as follows. First fix a graded k-vector space of Hilbert series11−t ,M = ⊕∞i=0kmiwhere mi is a basis of the degree i piece Mi. If M is an A-module, then we have mixj =λi,jmi+1 for some λi,j ∈ k. It is clear that giving M an A-module structure is equivalentto giving a sequence λi,j ∈ k. Since a point module is by definition cyclic, we need λi,j 6= 0for some j for a fixed i. Moreover, two point modules determined by sequences {λi,j} and{λ′i,j} are isomorphic if and only if for each i the vectors (λi,1, . . . , λi,n) and (λ′i,1, . . . , λ′i,n)are scalar multiples. This amounts to considering each vector (λi,1, . . . , λi,n) as a point inthe projective space Pn−1.For a finitely presented graded algebra A = Fn/I, Hilb1(A) corresponds to a subsetZ ⊂∏∞i=0 Pn−1 ∼= Hilb1(Fn) determined by an infinite set of equivalence relations. In factwe can take Zk to be the projection of Z onto the first k copies of Pn−1 and we see thatHilb1(A) = lim←−Zk. If A is a strongly noetherian graded k-algebra, Hilb1(A) is a projectivescheme parametrizing the point modules for the algebra [AZ2].Proposition 5.3.3. For the ambient AS regular algebraBn = 〈x1, . . . , xn〉/(xixj = qijxjxi)i,j ,the abstract Hilbert scheme Hilb1(Bn) is isomorphic to either Pn−1 or the union of somefaces of the fundamental (n − 1)-simplex Pn−1 containing all P1’s making up the 1-faces.The generic case corresponds to the 1-skelton of Pn−1 consisting of all P1’s.Proof. We begin with n = 2 case. LetA = k〈x, y, z〉/(xy − pyx, yz = qzy, zx = rxz)be the quantum P2 with some p, q, r 6= 0 (we use p, q, r to avoid too many indices). By theanalysis above the point modules correspond to sequence of points in P2 such thatλi,1λi+1,2 = pλi,2λi+1,1, λi,2λi+1,3 = qλi,3λi+1,2, λi,3λi+1,1 = rλi,1λi+1,3for all i ≥ 0. Multiplying the RHSs and LHSs above, we getλi,1λi,2λi,3λi+1,1λi+1,2λi+1,3 = pqrλi,1λi,2λi,3λi+1,1λi+1,2λi+1,3.There are two cases, pqr = 1 or pqr 6= 1.875.3. Hilbert Schemes of PointsCase pqr = 1. We easily solve the equation on the first pair of points [λ0,1 : λ0,2 :λ0,3], [λ1,1 : λ1,2 : λ1,3] and obtain a linear automorphism σ of P2 sending[a, b, c] 7→ [a : pb : pqc]such that the set of solutions is the graph of σ: {(p, σ(p))} ⊂ P2 × P2. Since the otherequations are just the index shift of the first set, it follows that the complete set of solutionsto the equations is given by{(p, σ(p), σ2(p), . . . )} ⊂∞∏i=0P2.This shows that the isomorphism classes of point modules are parametrized by P2.Case pqr 6= 1. Consider the equation on the first pair of points [λ0,1 : λ0,2 : λ0,3], [λ1,1 :λ1,2 : λ1,3]. It is easy to see that one of λ0,1, λ0,2, λ0,3 must be zero. LetE = {[λ0,1 : λ0,2 : λ0,3] ∈ P2 | λ0,1λ0,2λ0,3 = 0}.be a union of three projective lines. The solution is again given by{(p, σ(p)) | p ∈ E} ⊂ P2 × P2Observe that the image of σ|E is again E ⊂ P2. The full set of solution is{(p, σ(p), σ2(p), . . . ) | p ∈ E} ⊂∞∏i=0P2.and the isomorphism classes of point modules are parametrized by three lines E ⊂ P2. Forgeneral n ≥ 2, a similar argument works. More precisely, for any choice of three quantumcommutative relations of the form xy = pyx, we can show that the isomorphism classes ofpoint modules of the corresponding quantum P2 are parametrized by three lines E ⊂ P2.This proves the assertion for the general case.It is not difficult to see that Hilb1(Bn) ∼= Pn−1 if and only if Bn is a twisted homogeneouscoordinate ring of Pn−1.Definition 5.3.4. We call the quantum parameters are generic if any choice of relationsxy − pyx, yz = qzy, zx = rxz, the condition pqr 6= 1 holds.Note that this notion depends on the expression of the generators of relations.885.3. Hilbert Schemes of PointsProposition 5.3.5. Let S = proj(A4) be a non-commutative Fermat quartic K3 surface,whereA4 = 〈x1, . . . , x4〉/(4∑k=1x4k, xixj = qijxjxi)i,jfor some qij ∈ C. Then Hilb1(A4) is either a quartic K3 surface or 24 distinct points.In particular, the Euler number of Hilb1(A4) is always 24, independent of the value of thequantum parameters qij.Proof. On case by case basis, it can be checked that Hilb1(B4) is isomorphic to either P3or 1-skelton of P3 under the K3 constraints on qij in Theorem 5.2.2. In the former case,the equation∑4k=1 x4k = 0 cuts out a (not necessarily Fermat) quartic K3 surface in P3. Inthe latter case, the equation∑4k=1 x4k = 0 cuts out four distinct points in each line P1, soHilb1(A4) consists of 6× 4 distinct points.Proposition 5.3.6. There are 24 non-commutative K3 surfaces proj(A4) for genericchoice of quantum parameters 12.Proof. This can be proven by a direct computation as follows. LetB4 := k〈x, y, z, w〉/(xy − pyx, yz − qzy, zx− rxz, yw − swy,wx− txw,wz − uzw)be the quantum P3 with quantum parameters p, q, r, s, t, u ∈ C×. Since they are 4-th rootsof unity are are 46 = 4096 choices and 1312 of them satisfy generic assumption:pqr 6= 1, rtu 6= 1, pst 6= 1, qsu 6= 1.We now would like to impose the Calabi–Yau conditionprt= pqs = qru= tsu= c.for some fixed c ∈ C. The numbers of solution to the equation above depends on the valueof c.1. (c = 1) There are 20 solutions.2. (c = i) (p, q, r, s, t, u) is given by either(i,−1, 1,−1, 1, i), (−1,−1, i, i, 1, 1), or (−i,−1, 1, 1,−1, i).3. (c = −1) (p, q, r, s, t, u) is given by (−i,−1, 1, i, i, 1)12We ignore the permutation abundance because genericness depends of the expression of generators.895.3. Hilbert Schemes of Points4. (c = −i) There is no solution.Let us next take a look at our non-commitative quintic threefold X = proj(A5).Proposition 5.3.7. Let proj(A4) be a non-commutative Calabi–Yau threefold. If the quan-tum parameters qij are generic, then∏ni=1 qij = 1 for any 1 ≤ j ≤ n.Proof. This is proven by the aid of computer. There are 3000 choices of the quantumparameters qij with∏ni=1 qij = 1 for any 1 ≤ j ≤ n.Corollary 5.3.8. For a generic choice of the quantum parameters, proj(A5) admits adeformation in the direction of∏ni=1 xi. More precisely, A5 can take the form ofA5 := C〈x1, . . . , xn〉/(n∑k=1xnk + φn∏l=1xl, xixj = qijxjxi)i,jwith some φ ∈ C.An almost identical argument for the K3 surface case applies to the threefold case.When Hilb1(B5) ∼= P4, Hilb1(A5) is isomorphic to a certain smooth quintic threefold andwe getχ(Hilb1(A5)) = χ(quintic threefold) = −200.On the other hand, in the generic case, Hilb1(B5) consists of 10 lines and the equationf5 =∑5k=1 x5k = 0 cuts out five distinct points in each line P1 to getχ(Hilb1(A5)) = 10× 5 = 50.The argument above easily generalize to an arbitrary dimension.Theorem 5.3.9. There exists a non-commutative Fermat Calabi–Yau n-fold that is notrealized as a twisted coordinate ring of any commutative Calabi–Yau n-fold.Let us next consider Hilbm(Fn) for the free associative algebra in (n + 1) variablesFn := k〈x1, . . . , xn〉. This needs the following infinite quiver Q := {E, V } with E = Nnand V = N;•0xi_*4 •1xi_*4 •2xi_*4 •3xi_*4 . . .Let N be the moduli space of representation Q-modules with constant dimension vector(m)N, then Hilbm(Fn) is given by the open subset of N defined by the condition thatV0 generated the the whole module. In particular when m = 1, we recover the formulaHilb1(Fn) ∼=∏∞i=0 Pn−1.905.4. Moduli Spaces via Differential Graded Lie AlgebraFor a finitely presented graded algebra A = Fn/I, the Hilbert scheme Hilbm(A) is givenas a subset Z ⊂ Hilbm(B) determined by an infinite set of equivalence relations. In thissituation, we have the following general result [AZ2].Theorem 5.3.10 (Artin and Zhang [AZ2]). Let A be a connected graded finitely generatedalgebra. Given a numerical polynomial H : Z→ N, we define a functor T : k-Alg → Set byassigning to k-algebra R the set of homogeneous right ideals I ⊂ A⊗kR/I with (A⊗kR/I)tflat over R of constant rank H(t) for t ≥ 0. If A is strongly noetherian, the functor T isrepresented by a projective scheme X for any Hilbert function H(t).Recall that a k-algebra is called strongly noetherian if A⊗k R is noetherian ring for allcommutative noetherian rings R. A Calabi–Yau graded k-algebra An is strongly noetherianas it is a quotient of a strongly noetherian ring Bn. However, at this point, we do not knowhow to compute this abstract projective scheme Hilbm(An) for m > 1.5.4 Moduli Spaces via Differential Graded Lie AlgebraIn [BCHR], the authors constructed the moduli scheme of stable sheaves on a projectivescheme via a certain differential graded Lie algebra (Hochschild chain complex). Theirmethod can be thought of as a finite dimensional analogue of the gauge theory. An advan-tage of the method is that it is readily generalized to non-commutative projective schemesproj(A) under a mild assumption. Another advantage is that the method is more directand avoids the use of Quot schemes. In this section, a sheaf on X means an element ofCoh(X) ∼= tails(A) and so it makes sense even if A is not commutative. Notation is thesame as the previous section; A is a connected noetherian graded k-algebra with augmen-tation ideal m.5.4.1 Differential Graded Lie AlgebraLet us briefly review some basics of differential graded Lie algebras (DGLAs).Definition 5.4.1. A differential graded Lie algebra (DGLA) consists of the following:1. A Z-graded vector space L =⊕∞i=0 Li. Each Li is usually finite dimensional. For ahomogeneous element x ∈ Li, we define |x| := i.2. A graded alternating linear map [∗, ∗∗] : ∧2L→ L[Li, Lj ] ⊂ Li+j , [x, y] + (−1)|x||y|[y, x] = 0that satisfies the graded Jacobi identity[x, [y, z]] = [[x, y], z] + (−1)xy[y, [x, z]].This means that [x, ∗] is a derivation for [∗, ∗∗].915.4. Moduli Spaces via Differential Graded Lie Algebra3. A derivation d : L→ L[1] such that d2 = 0.By a slight abuse of notation, we often denote DGLA (L, [∗, ∗∗], d) simply by L.Definition 5.4.2. The curvature map C : L1 → L2 is defined by x 7→ dx + 12 [x, x]. ThesetMC(L) := {x ∈ L1 | C(x) = 0}is called the Maurer-Cartan set of a DGLA L.There is an obvious notion of morphisms of DGLAs and they induces morphisms ofcohomology groups and the Maurer–Cartan sets.Proposition 5.4.3. For each x ∈ L1 we define the twisted differential bydx := d+ [x, ∗] : L→ L.If x ∈MC(L), then (dx)2 = 0 and moreover (L, [∗, ∗∗], dx) is a DGLA.Definition 5.4.4. A gauge group for a DGLA L is an algebraic group G such that1. The Lie algebra of G is L0, i.e. TeG = L0.2. There is a linear action G × Li → Li of G on each Li, denoted (g, x) = gx, whosederivative is the Lie algebra action of L0 on Li[∗, ∗∗] : TeG× Li = L0 × Li → Li.3. The G-action is compatible with the bracket [∗, ∗∗] in the sense that g[x, y] = [gx, gy].4. There is a group cocycle γ : G→ L1 such that the derivative of γ is dd = dγ|e : TeG = L0 → L1.5. −γ(g) is a Maurer-Cartan element for all g ∈ G and gd = d−γ(g), i.e.g(d(g−1x)) = dx− [γ(g), x], ∀x ∈ LpGiven a gauge group G for L, the gauge action is the non-linear action of G on L1defined by g · x = gx− γ(g). This action preserves MC(L).If γ = 0, then the gauge action is linear and we haveg[x, y] = [gx, gy], g(dx) = d(gx).925.4. Moduli Spaces via Differential Graded Lie AlgebraThis shows that G acts on L as an automorphism of a DGLA.Note that we can recover a DGLA L =⊕∞i=0 Li from the gauge group G and theaugmentation ideal m :=⊕∞i=1 Li. On the other hand, we can construct a gauge group G(not uniquely) by the integration:L0 d //expL1G.γ>>||||||||Definition 5.4.5. Given a DGLA L with a gauge group G, then the quotient stack [MC(L)/G]is called the moduli stack of (L,G).We shall next explain the construction of a DGLA from a graded vector space, following[BCHR]. Let H(t) ∈ Q[t] be a numerical polynomial and V be a graded vector spaceV =⊕qi=p Vi with dimension vector α = (H(i))qi=p. By End(V ) we denote the algebra ofk-linear endomorphisms of V . It inherits a grading from V . Only Endi(V ) in the rangei ∈ [p− q, q− p] are non-zero. The Hochschild cochain complex associated to a graded ringA and a graded vector space V is L :=⊕∞n=0 Ln, whereLn := Homgr(m⊗n,End(V )).Note that every Ln is finite dimensional and that Ln = 0 unless n ∈ [0, q− p] because m ispositively graded. Each Ln is graded positively:Ln =⊕p≤j≤i≤qi−j≤nLni,j ,whereLni,j := Hom((m⊗n)i−j ,Hom(Vi, Vj)).We let G =∏qi=pGL(Vi) be the group of degree preserving linear automorphisms of V ,which we call the gauge group. It acts from the left on L via conjugation. The action ofG on Ln preserves the double grading; if g = (gp, . . . , gq) and µ ∈ Ln then(g · µ)i,j = giµi,jg−1j .Define the differential d : Ln → Ln+1 bydµ(a1, . . . , an+1) :=n∑i=1(−1)n−iµ(a1, . . . , aiai+1, . . . , an).935.4. Moduli Spaces via Differential Graded Lie AlgebraFor example d : L0 → L1 is equal to zero and d : L1 → L2 is given by dµ(a, b) = µ(ab). It iseasy to check that d2 = 0 and the gauge action preserves the differential. The differentialpreserves the projective double grading. For µ ∈ Lm and ν ∈ Ln define an associativemultiplication µ ◦ ν ∈ Lm+n byµ ◦ ν(a1, . . . , am+n) = (−1)m+nµ(a1, . . . , am) ◦ ν(am+1, . . . , am+n).The bracket [∗, ∗∗] on L is given by the graded commutator with respect to this associativeproduct ◦. For example if µ, ν ∈ L1 we have[µ, ν](a, b) = µ(a) ◦ ν(b)− ν(a) ◦ µ(b).The differential d acts as a derivation with respect to the bracket [∗, ∗∗] and hence thetriple (L, [∗, ∗∗], d) is a DGLA.The scheme theoretic zero locus of the curvature map C : L1 → L2, µ 7→ dµ+ 12 [µ, µ],MC(L) :={µ ∈ L1 | dµ+ 12[µ, µ] = 0}is the Maurer–Cartan set of L.Proposition 5.4.6. There exists an equivalence of groupoids[MC(L)/G] ∼= (m-modules)α,where the latter is the groupoids consisting of graded m-modules whose underlying vectorspace has dimension vector α.Proof. Since we have 12 [µ, µ] = µ ◦µ for µ ∈ L1, we see that µ is a Maurer–Cartan elementif and only ifµi,j(ab) = µj,k(a) ◦ µk,j(b)holds for all i > k > j and a ∈ mi−k, b ∈ mk−j . Therefore µ ∈ MC(L) if and only ifit defines a left action of the graded vector space m on V . Dividing by the gauge actionremoves the choice of basis in V , we obtain the desired equivalence of groupoids.5.4.2 Deformation TheoryIn this section, we shall show that the cohomology groups of the DGLA (L, [∗, ∗∗], dx)govern the deformation theory of the moduli [MC(L)/G].Lemma 5.4.7. For x ∈ L1, the orbit map G → L1 by g 7→ g · x derives to the twisteddifferential −dx : L0 → L1.945.4. Moduli Spaces via Differential Graded Lie AlgebraProof. By definition, g 7→ gx− γ(g) derives to a 7→ [a, x]− da = −dx(a).Corollary 5.4.8. We have Lie(StabG(x)) ∼= H0(L, dx). That is, the cohomology groupH0(L, dx) represents the infinitesimal stabilizers of G at x.Proof. The Cartesian diagramStabG(x)// Spec(C)0G // L1.induces another Cartesian diagram on tangent spaces at the origin e ∈ G.Lie(StabG(x))// 00L0 −dx// L1.Lemma 5.4.9. There is a natural isomorphism TxMC(L) ∼= Z1(L, dx) for x ∈MC(L).Proof. The Cartesian diagramMC(L)// Spec(C)0L1 F // L2indices an exact sequence 0 → Tx(MC(L)) → L1 = TxL1 → L2 = TF (x)L2. Moreover, adirect computation shows that F (x+ z) = F (x) + dx(z), i.e. dF |x = dx.Corollary 5.4.10. Suppose that pi : L1 → L1/G is smooth at x ∈MC(L), then H1(L, dx) ∼=Tx(MC(L)/G).Proof. By the smoothness assumption, we have an exact sequence0→ TxGx→ TxL1 → Tpi(x)(L1/G)→ 0.The orbit map G→ L1 by g 7→ g · x induces the surjection TeG→ Tx(Gx), and so that wehave the exact sequence TeG→ TxL1 → Tpi(x)(L1/G)→ 0. Then we can conclude thatTeG→ Tx(MC(L))→ Tpi(x)(MC(L)/G)→ 0955.4. Moduli Spaces via Differential Graded Lie Algebrais also exact, assuming thatMC(L)pi// L1piMC(L)/G // L1/Gis commutative. Therefore we obtainTeG=// Tx(MC(L))∼=// Tx(MC(L)/G)∼=// 0L0 −dx// Z1(L, dx) // H1(L, dx) // 0We now turn to obstruction theory of DGLAs. Let A′ → A be a square zero extensionof differential graded algebras, i.e.0→ I → A′ → A→ 0is exact and I2 = 0. Then I is a differential graded A-module. Then we get an inducedshort exact sequence of DGLAs0→ L⊗ I → L⊗A′ → L⊗A→ 0and L⊗ I is a DGLA ideal in L⊗A′ such that the bracket vanishes [L⊗ I, L⊗ I] = 0. Inother words, L⊗ I is an abelian Lie algebra. Note that dx = d+ [x, ∗] defines a differentialon L⊗ I for x ∈MC(L).Let y be an arbitrary lift of x ∈ MC(L) to (L ⊗ A′)1, then the general lift of x to(L⊗A′)1 is given by y − z for z ∈ (L⊗ I)1.Lemma 5.4.11. We have y − z ∈MC(L⊗A′) if and only if dx(z) = dy + 12 [y, y].Proof. In order to obtain a Maurer–Cartan element y− z ∈MC(L⊗A′), we need to solvethe equationd(y − z) + 12[y − z, y − z] = dy + 12[y, y]− dx(z) = 0.This clearly shows the assertion.Lemma 5.4.12. The element dy + 12 [y, y] defines an element of H2(L ⊗ I, dx) and is inindependent of the choice of the lift y.965.4. Moduli Spaces via Differential Graded Lie AlgebraProof. The first assertion follows from the assumption that x ∈ MC(L). The secondassertion follows from a direct computation.Definition 5.4.13. The element h := dy+ 12 [y, y] ∈ H2(L⊗ I, dx) is called the obstructionto lifting x.In summary, an element x ∈MC(L⊗A) can be lifted to an element of MC(L⊗A′) ifand only if the obstruction h vanishes.5.4.3 Geometric Invariant TheoryWe would rather want moduli schemes than moduli stacks [MC(L)/G)] ⊂ [L1/G]. A keyobservation is that L1 is a simple affine space; it is a representation of a quiver whosevertices p, . . . , q and there are dimAi−j arrows from j to i. This leads to the standardmoduli problem of representation of the quiver L1 with dimensional vector α. The gaugegroup G however contains the diagonal scalars, which act trivially on L1, so we needconsider the reduced gauge group ˜G = G/C×. This amounts to killing the automorphismgroups of m-modules the moduli scheme represents.As a stability θ we consider the extremal character defined byθ := (−H(q), 0, . . . , 0,H(p)),so the corresponding character isχ : ˜G→ C×, g 7→ det(gq)H(p)det(gp)H(q).We denote by Ls and Lss the open subsets of L1 of stable and semi-stable points re-spectively. Similarly we denote by MC(L)s and MC(L)ss the open subsets of stable andsemi-stable points inside MC(L). The G.I.T. quotient of L1 by ˜G is the projective schemegiven byL1// ˜G := Proj∞⊕n=0[L1]χn ,where [L1]χn is the space of χn-twisted invariants of G in [L1]. Since the curvature map Cis G-invariant, it descends to the quotient L1// ˜G and the moduli schemeMC(L)// ˜G ⊂ L// ˜Gis the scheme theoretic zero locus of C on L// ˜G. Wa say that a [p, q]-graded A-module Mis called (semi-)stable if the corresponding point µ in L1 is (semi-)stable with respect to975.4. Moduli Spaces via Differential Graded Lie Algebrathe extreme character. Equivalence classes of stable families of graded A-modules are thengiven by[MC(L)s/ ˜G] ∼= MC(L)s// ˜Gbecause the stabilizer of ˜G is trivial on the stable locus.In [BCHR], it is shown that the geometric invariant theory associated to the Hochschildcochain complex os equivalent to the Gieseker stability of coherent sheaves. More precisely,the following two theorems hold when A is the homogeneous coordinate ring of a smoothprojective scheme.Theorem 5.4.14 (Behrend–Ciocan-Fontanine–Hwang–Rose [BCHR]). Let U be a finitetype open substack of the algebraic stack of coherent sheaves with a Hilbert polynomialH(t). Take sufficiently large q > p > 0, then there exists an open immersion of algebraicstacksΓ[p,q] : U −→ [MC(L)s/ ˜G]Moreover, if E is a U-sheaf, then E is (semi-)stable sheaf if and only if its image Γ[p,q]E is(semi-)stable graded A-module.This morphism is given as follows. Let p ∈ N be large enough so that every sheaf inU is Castelnuovo–Mumford p-regular, then for any sheaf in U, say E on pi : X × T → T ,all the higher direct images Ripi∗E vanish and hence pi∗E gives a locally free sheaf on T ofrank α.Theorem 5.4.15 (Behrend–Ciocan-Fontanine–Hwang–Rose [BCHR]). Let M sα ⊂ M ssα bethe moduli schemes of stable and semi-stable sheaves on X with respect to H(t) in thesense of Simpson and Maruyama. Take sufficiently large q > p > 0, then there exists acommutative diagram of open immersions of schemes.M ssαΓ[p,q]// MC(L)// ˜GM sαOOΓ[p,q]// MC(L)s// ˜GOOThe two scheme in the top row are projective and hence M ssα is a union of connectedcomponents of Xss// ˜G. Theorems 5.4.14 and 5.4.15 heavily rely on the theory of moduliof sheaves of a smooth projective variety. At this point, we do not known whether or notwe can build a parallel theory for the non-commutative Calabi–Yau algebra An. In thefollowing, we content to work with the moduli scheme of graded A-modules with a fixeddimension vector α.985.5. Symmetric Obstruction Theory5.5 Symmetric Obstruction TheoryIn the first section we briefly review the technique of symmetric obstruction theory [Beh?]. Just like the smooth moduli schemes of stable sheaves on a K3 surface are holomorphicsymplectic, symmetric obstruction theory may be thought of an odd symplectic structureon the moduli scheme of stable sheaves on a Calabi–Yau threefold. We show that ourmoduli scheme comes with differential graded structure and is naturally endowed with asymmetric obstruction theory. Throughout this section we work over complex numbersk = C.5.5.1 Odd Symplectic StructureLet M be a Deligne–Mumford stack. A local embedding of M is a diagramUfg// NMwhere f : U → M is etale local chart of M and g : U → N is a closed embedding into asmooth scheme N . The intrinsic normal cone CM of M is a cone stack of pure dimension 0characterized as follows; for any local embedding of M as above we have an isomorphismCM |U ∼= [CU/N/g∗TN ],where CU/N is the normal cone of the embedding g : U → N . The intrinsic normal sheafNM is the minimal abelian cone stack containing CU/N and is described in the same manner(replacing CU/N with NU/N ).Definition 5.5.1 (Behrend–Fantechi [BF]). Let F = [F−1 → F 0] be an object of D[−1,0]perf (M)and LM be the contangent complex of M . A homomorphism φ : F → τ≥−1LM is called aperfect obstruction theory for M if1. h0(φ) is an isomorphism.2. h−1(φ) is an epimorphism.The perfect obstruction theory is called symmetric if we are given an isomorphism θ : F→F∨[1] such that θ∨[1] = θ.We denote the sheaf h1(F∨) by ob and call it the obstruction sheaf of the perfectobstruction theory. It contains the obstructions to the smoothness of M . Note that ob is995.5. Symmetric Obstruction Theoryby no means an intrinsic invariant of M .A perfect obstruction theory φ : F → τ≥−1LM induces a closed immersion of conestack13CM ↪→ NM = h1/h0(τ≥−1LM ) ↪→ F = h1/h0(F),where F is a vector bundle stack. The virtual fundamental class for the perfect obstructiontheory φ : F→ τ≥−1LX is then given by[M ]vir := 0!F[CM ] ∈ Arank(F)(M).If F has a global resolution, i.e. quasi-isomorphic to some [G−1 → G 0], where G−1,G 0 arelocally free sheaves, since CM is a closed substack of F, it induces a closed subcone C of avector bundleC(G−1) := SpecOM (SymG−1)and we may also define[M ]vir = 0!C(G−1)[C] ∈ Arank(F)(M).Assume that φ : F → τ≥−1LX is symmetric, then we have rank(F) = 0 and it defines[M ]vir ∈ A0(M). We also have a good description of the obstruction sheafob = h1(F∨) = h0(F∨[1]) = h0(F) = ΩM .Let M → N be a closed embedding into a smooth Deligne–Mumford stack. We get acanonical epimorphism ΩN |M → ΩM = ob, which pulls back the cone cv ↪→ ob consists ofcurvilinear obstructions to get a cone C ⊂ ΩN |M ⊂ ΩN , i.e. the Cartesin diagramC//cvΩN |M // ob.The key observation made in [Beh] is that C is a conic Lagrangian subvariety in thesymplectic manifold ΩN . Combining this fact with microlocal technique developed byKashiwara and MacPherson, we finally obtain the following useful result.13 Recall that there exists an equivalence of categoriesh : D[−1,0]coh (M)opp ∼= H0(Abelian Cone Stack/M)given byF = [F−1 → F 0] 7→ h1/h0(F) = [SpecOM (SymF−1)/SpecOM (SymF0)].1005.5. Symmetric Obstruction TheoryTheorem 5.5.2 (Behrend [Beh]). Let M be a Deligne–Mumford stack with a symmetricperfect obstruction theory. Then there exists a natural constructible function νM : M → Zdepending only on M , and the following hold:∫[M ]vir1 =∑n∈Znχ(ν−1M (n)).5.5.2 Calabi–Yau 3 Higher Tangent ComplexThe goal of this section is to prove Theorem 5.1.2. We continue our discussion fromSection 5.4.3 about the moduli scheme of A-modules with a dimensions vector α. A basicexample the reader should keep in mind is An on Theorem 5.2.2. In the case where A is thehomogeneous coordinate ring of a smooth projective variety, we see that the moduli schemeof A-modules is really the moduli of sheaves assuming that sufficiently large q > p > 0 inTheorems 5.4.14 and 5.4.15 are chosen. We first have L1 = Spec(Sym(L1)∨). We considerthe trivial vector bundlesL i := Li × L1on L1 for i ≥ 2. The curvature map C gives rise to a section C : L1 → L 2. The twisteddifferential δ : L i → L i+1 is defined by the formulaδ(x) := dµ(x) = dx+ [µ, x]in the fiber over µ ∈ L1. Since δ2 = 0 over the Maurer-Cartan locus MC(L) = Z(C), thetangent complexΘ := [L 0 δ−→ L 1 δ−→ L 2 δ−→ . . . ]is a complex of locally free sheaves over MC(L) in the interval [0, q− p]. By construction,the gauge group G-action on L lifts to an action of the tangent complex Θ and thus itdescends14 to the quotient substackM = [MC(L)/ ˜G] ⊂ N = [L1/ ˜G].Remember that the quotient of the stable sublocusM = [MC(L)s/ ˜G] ⊂ N = [(L1)s/ ˜G]is the moduli scheme of our interest. The complex Θ is called the tangent complex of M(or M).As we saw in Section 5.4.2, the deformation theory of the Hochschild cochain complex Lis governed by its cohomology groups. In this situation, we can see the deformation theory14By an abuse of notation we use the same Θ for its descendant.1015.5. Symmetric Obstruction Theoryvery concretely as follows. Let a C-valued point p : SpecC→M be represented by a gradedA-module E = (V, µ). The complex p∗Θ is the graded normalized Hochschild complexHC∗(A,Endµ(V )), where End(V ) is endowed with the structure of an A-bimodules viaµ ∈ L1. The higher tangent spaces of M at p are defined as the cohomologies of thiscomplex15:T iM (p) := H i+1(p∗Θ) ∼= ExtiA(E,E)The existence of the higher tangent space is a glimpse of a differential graded structureon our moduli scheme M , but we will not discuss this subject in this thesis. We refer thereader to [BCHR] for more details.Let us now assume that Coh(X) = tails(A) is Calabi–Yau 3 category so that we havea functorial perfect pairingExtiA(E,E)⊗ Ext3−iA (E,E) −→ C.The stability implies that E is a Schur object;Ext3A(E,E) ∼= Ext0A(E,E) = C,but we already killed Ext0A(E,E) by reducing the gauge group (i.e. G˜ = G/C×). To killthe higher obstruction Ext3A(E,E) we truncate the complex Θ to getF∨ = (τ[1,2]Θ)[1] = [F 0 −→ F 1].Theorem 5.5.3. There exists a symmetric perfect obstruct theory θ : F→ τ≥−1LM on themoduli scheme M .Proof. We first note that the complex F∨ is explicitly given by[L 1/δL 0 −→ Ker(L 2 → L 3)][1].Since the derivative of the gauge action on L1 is given byL0 → gl(L1) : x 7→ [x, ∗],it follows immediately thatF 0 = L 1/δL 0 ∼= TN |M .The derivative of the curvature section C : L1 → L 2 gives rise to a mapdC = δ|NM/N : NM/N → L215ExtiA(∗, ∗∗) denotes graded Ext groups in the abelian category gr(A).1025.5. Symmetric Obstruction Theoryand hence descends to dC : NM/N → F 1 over M . We therefore get the following commu-tative diagramTN |M //∼=NM/NF 0 δ // F 1.Moreover, since Ker(δ : L 0 → L 1) has a constant rank, F 0 is locally free. Likewise, sincethe third tangent space T 3M is a locally free and L i = 0 for i > q − p, we conclude thatF 1 is also locally free. By dualizing the diagram above, we obtain a perfect obstructiontheory F → τ≥−1LX . It is symmetric because of the Calabi–Yau perfect paring on ourCoh(X).By virtue of of being the Maurer–Cartan locus, up to gauge equivalence, the modulispace is naturally equipped with a derived structure.103Chapter 6Conclusion6.1 Summary of ResultsChapter 2 is concerned with the interplay of the symmetric trilinear intersection form µ onH2(X,Z) and the Chern classes of a Calabi–Yau threefold X. This research is motivatedby the essential role of the K3 lattice in the study of K3 surfaces. We demonstrate somenumerical relations between Chern classes and µ, by applying the Fulton–Lazarsfeld theoryfor nef vector bundles [DPS]. Together with Wall’s classification theorem for real sixfolds[Wal], our results impose constraints on the underlying differentiable structures of Calabi–Yau threefolds16. The following simple formula gives a range of the Euler numbers ofCalabi–Yau threfolds.Theorem 6.1.1 (Theorem 2.3.7). Let (X,H) be a very amply polarized Calabi–Yau three-fold, i.e. x = H is a very ample divisor on X. Then the following inequality holds:−36µ(x, x, x)− 80 ≤ c3(X)2= h1,1(X)− h2,1(X) ≤ 6µ(x, x, x) + 40.Moreover, the above inequality can be sharpened by replacing the left hand side by −80,−180 and right hand side by 28, 54 when µ(x, x, x) = 1, 3 respectively.The formula bounds the Euler number by the intersection number of ample divisors17.It provides a new insight to the conjecture that the Euler numbers of Calabi–Yau threefoldsare bounded.Chapters 3 provides the full classification of Calabi–Yau threefolds with infinite fun-damental group. Let X be a Calabi–Yau threefold with infinite fundamental group. TheBogomolov decomposition theorem states that X admits an e´tale Galois covering either byan abelian threefold (type A) or by the product of a K3 surface and an elliptic curve (typeK). Calabi–Yau threefolds with infinite fundamental group were first studied by Oguisoand Sakurai [OS]. In chapters 3, we complement their work and also develop the basictheory thereof.16 For Wall’s theorem, they must be simply-connected and the cohomology groups must be torsion free.17 Recall that [OP] shows that we can always pass from an ample divisor H on a Calabi–Yau threefoldto a very ample one 10H.1046.1. Summary of ResultsTheorem 6.1.2 ( Theorem 3.2.1, Proposition 3.5.3, Theorem 3.5.5). 1. There exist ex-actly 14 Calabi–Yau 3-folds with infinite fundamental group, up to deformation equiv-alence. More precisely, 6 of them are of type A and 8 of them are of type K.2. For type K, the equivalence classes are uniquely determined by the Galois group G ofthe minimal splitting covering18. Moreover, the Galois group G is isomorphic to oneof the following combinations of cyclic and dihedral groupsC2, C2 × C2, C2 × C2 × C2, D6, D8, D10, D12, or C2 ×D8.3. For type A, the equivalence classes are classified by the full splitting covering of theformE1 × E2 × E3 −→ A −→ X,where the right map is the minimal splitting covering and the left map is the minimalfinite covering of A from the product of three elliptic curves19.Chapters 4, we investigate Calabi–Yau threefolds of type K from the viewpoint ofmirror symmetry. We obtain several results parallel to what is known for Borcea–Voisinthreefolds: Voisin’s work on Yukawa couplings [Voi], and Gross and Wilson’s work onspecial Lagrangian fibrations [GW].Theorem 6.1.3 (Theorem 4.3.4). Let X be a Calabi–Yau threefold of type K. The asymp-totic behavior of the A-Yukawa coupling Y XA around a large volume limit coincides with theasymptotic behavior of the B-Yukawa coupling Y XB around a complex structure limit.Theorem 6.1.4 (Propositions 4.4.6 & 4.4.7). Any Calabi–Yau threefold X of type K admitsa special Lagrangian T 3-fibration pi : X → B, where the base space B is topologically eitherS3 or an S1-bundle over RP2.According to the Strominger–Yau–Zaslow proposal, every Calabi–Yau threefold X ad-mits a special Lagrangian T 3-fibration f : X → B, and mirror threefold Y is then obtainedas the dual fibration f∨ : Y → B. It is however an extremely difficult problem to finda special Lagrangian T 3-fibration for a given Calabi–Yau threefold. The theorem aboveprovides new examples of Calabi–Yau threefolds (with smooth metric) that admit a specialLagrangian fibration20.18 By assumption, X admits an e´tale Galois covering by the product of a K3 surface and an elliptic curve.Among many candidates of such coverings, there always exists a unique smallest one and we call it theminimal splitting covering of X.19 We refer the reader to Proposition 3.5.3 for the precise assertion.20 Before our result, there was only one example, namely the Enriques Calabi–Yau threefold [GW].1056.2. Future Research DirectionChapter 5 constructs the first examples of non-commutative projective Calabi–Yauschemes, in the sense of Artin and Zhang [AZ1], and introduces a virtual counting theoryof stable modules on them.Theorem 6.1.5 (Theorem 5.2.2 & Corollary 5.3.8 ). Let k be an algebraically closed fieldof characteristic zero. We consider the following graded k-algebraAn := k〈x1, . . . , xn〉/(n∑k=1xnk , xixj = qijxjxi)i,j .where qij ∈ k× satisfies qii = qnij = qijqji = 1. Assume further that there exists c ∈ k×such that∏ni=1 qij = c for any 1 ≤ j ≤ n. Then the quotient category Coh(An) :=Gr(An)/Tor(An) is a Calabi–Yau (n−2) category. Moreover, there exists quantum param-eters qi,j such that the graded k-algebra An is not realized as a twisted coordinate ring ofany commutative Calabi–Yau (n− 2)-fold.One motivation to study non-commutative projective Calabi–Yau schemes comes from avirtual counting theory of stable sheaves on a polarized Calabi–Yau threefold (X,L). It canbe reformulated purely in algebra; counting stable modules over the homogeneous coordi-nate ring⊕∞i=1H0(X,Li) of X. The moduli schemeMα of stable modules in Coh(An) witha fixed dimension vector α is constructed by the differential graded Lie algebra techniquedeveloped in [BCHR]. The fundamental theorem we proved is the following:Theorem 6.1.6 (Theorem 5.5.3). There exists a Calabi–Yau 3 tangent complex on themoduli scheme Mα. Moreover, Mα admits a natural symmetric obstruct theory.6.2 Future Research DirectionIn the future research in the theory of trilinear forms, I would like to develop a furtheranalogy with the theory of K3 surfaces, where the Ka¨hler cone and (symplectic) auto-morphism groups are completely understood in terms of the K3 lattice. A project I amworking on is the classification of automorphism groups of Calabi–Yau threefolds in goodsituations, for instance, when µ has a linear factor. The shape and rational points of thecubic hypersurface {µ(x, x, x) = 0} ⊂ H2(X,R) also have important information about theunderlying Calabi–Yau threefold [Wil1], and much progress is expected via, for example,the theory of Diophantine equations. Describing the Ka¨hler cone with the help of µ will becrucial in all what I mentioned above. I believe that the trilinear form µ and the Ka¨hlercone are the key to the study of Calabi–Yau threefolds. Despite the fundamental natureof the trilinear form µ, many basic questions are still unanswered.1066.2. Future Research DirectionThere are a number of natural questions that arise from our study of Calabi–Yau three-folds of type K. One question would be about Kontsevich’s homological mirror symmetryconjecture [Kon2]. It claims that, for a mirror pair of Calabi–Yau threefolds X and Y , thebounded derived category Db(Coh(X)) of the coherent sheaves on X is equivalent to thederived Fukaya category DFuk(Y ) of Lagrangian manifolds with unitary local system, andvice versa. In this homological viewpoint, any Calabi–Yau threefold X of type K shouldyield a derived equivalenceDb(Coh(X)) ∼= DFuk(X).As X has a relatively simple description, we expect that the categories Db(Coh(X)) andDFuk(Y ) are computable after some work.It would also be interesting to see how the modern enumerative geometry (GW/DT/PTtheory) can be applied to Calabi–Yau threefolds of type K. For the Enriques Calabi–Yauthreefold, the Gromov–Witten invariants of low genera have been computed both in thecontext of topological string theory and of algebraic geometry [KM1, MP]. Other Calabi–Yau threefolds of type K should provide more examples of tractable compact Calabi–Yauthreefolds with non-trivial enumerative geometry. The degeneration technique employedin [MP] may be generalized to other cases. We also remark that one of particularly niceproperties of Calabi–Yau threefolds of type K is their fibration structure; they admit a K3fibration, an abelian surface fibration, and an elliptic fibration. We believe that, thanksto this rich structure, they play an important role in dualities among various superstringtheories. We hope to come back to these points in a future work.Another interesting research subject is the BCOV holomorphic anomaly equations ofCalabi–Yau threefolds of type K (see [BCOV1, BCOV2]). The BCOV theory presents abeautiful generalization of the classical mirror symmetry and is capable of computing thehigher genus Gromov–Witten invariants of Calabi–Yau threefolds, up to some holomorphicambiguities. The theory relies on the special Ka¨hler geometry of the complex modulispace of Calabi–Yau threefolds. The rich fibration structures of Calabi–Yau threefolds oftype K make the study of the special Ka¨hler geometry very promising. Lattice polarizedK3 surfaces, elliptic curves with level structure and modular forms naturally come intoplay. Since Calabi–Yau threefolds of type K are self-mirror symmetric, we expect that thecomplex moduli space has multiple LCSLs when a Calabi–Yau threefold has non-trivialBrauer group. Although multiple mirror symmetry is still far from understood, the BOCVtheory would provide a new insight to the field.In chapter 5, we constructed the moduli scheme Mα of stable modules in Coh(An) witha fixed dimension vector α. When α comes from a numerical polynomial H(t), (that is,αt = H(t) for t ∈ N), the moduli scheme Mα conjecturally gives the moduli scheme of1076.2. Future Research Directionstable graded modules on An with Hilbert polynomial H(t) (provided the length of α istaken sufficiently large). 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