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Donaldson-Thomas theory of quantum Fermat quintic threefolds Liu, Yu-Hsiang 2020

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DONALDSON–THOMAS THEORYOF QUANTUM FERMAT QUINTICTHREEFOLDSbyYU-HSIANG LIUA thesis submitted in partial fulfillmentof the requirements for the degree ofDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Mathematics)The University of British Columbia(Vancouver)July 2020© Yu-Hsiang Liu, 2020The following individuals certify that they have read, and recommend to theFaculty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled:Donaldson–Thomas theory of quantum Fermat quintic threefoldssubmitted by Yu-Hsiang Liu in partial fulfillment of the requirements for thedegree of Doctor of Philosophy in Mathematics.Examining Committee:Kai Behrend, MathematicsSupervisorJim Bryan, MathematicsSupervisory Committee MemberKalle Karu, MathematicsSupervisory Committee MemberBen Williams, MathematicsUniversity ExaminerJoanna Karczmarek, PhysicsUniversity ExamineriiAbstractIn this thesis, we study non-commutative projective schemes whose associatedgraded algebras are finite over their centers. We construct symmetric obstructiontheories for their moduli spaces of stable sheaves in the Calabi–Yau-3 case. Thisallows us to define Donaldson–Thomas (DT) type deformation invariants.As an application, we study the quantum Fermat quintic threefold which isthe quintic threefold in a quantum projective space. We give an explicit descrip-tion of its local models in terms of quivers with potential. We then give a fullcomputation of its degree zero DT invariants.iiiLay SummaryEnumerative geometry is a study of counting numbers of geometric objects.One of the important subjects is Donaldson–Thomas theory, which deals withcounting curves on certain (Calabi–Yau) three-dimensional spaces. The mostwell-known example is that there are precisely 2875 lines on a general quinticthreefold.Our goal is to develop and study Donaldson–Thomas theory for “non-commutative” spaces. Classically, algebraic geometry study zeros of multivariatepolynomials, and spaces are considered to be “commutative” since variables ofa polynomial are commutative (i.e., xy = yx). What we call non-commutativespaces are zeros of “quantized” polynomials, where variables only commute upto a scalar (i.e., xy = qyx for some q ∈ C∗).In this thesis, we focus on the simplest and only known example of non-commutative Calabi–Yau space, called the quantum Fermat quintic threefold.We define Donaldson–Thomas theory on this space and give some explicit com-putation.ivPrefaceThis thesis is original, unpublished, independent work of the author Yu-HsiangLiu, with the guidance of the author’s advisor Prof. Kai Behrend.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Symmetric obstruction theory and the Behrend function . . . . . 72.2 Quivers with potential . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Plane partitions and Hilbert schemes of points . . . . . . . . . . . 123 Quantum Fermat quintic threefolds . . . . . . . . . . . . . . . . . . 143.1 Non-commutative projective schemes . . . . . . . . . . . . . . . . 143.2 Sheaves of non-commutative algebras . . . . . . . . . . . . . . . . 183.3 Simpson moduli spaces . . . . . . . . . . . . . . . . . . . . . . . . 213.4 Quantum Fermat quintic threefolds . . . . . . . . . . . . . . . . . 224 Donaldson–Thomas invariants of (X,A) . . . . . . . . . . . . . . . 274.1 Obstruction theories for A-modules . . . . . . . . . . . . . . . . . 274.2 Proof of Theorem 4.1.2 . . . . . . . . . . . . . . . . . . . . . . . . 32vi4.3 Donaldson–Thomas invariants for Coh(A) . . . . . . . . . . . . . 395 Local models of (X,A) . . . . . . . . . . . . . . . . . . . . . . . . . . 445.1 Finite-dimensional A-modules . . . . . . . . . . . . . . . . . . . . 445.2 Simple A-modules and the Ext-quiver . . . . . . . . . . . . . . . . 465.3 Local models of (X,A) . . . . . . . . . . . . . . . . . . . . . . . . 486 DT invariants of the quantum Fermat quintic threefold . . . . . . 556.1 Analytification and weighted Euler characteristics . . . . . . . . . 556.2 Stratifications of Hilbert schemes . . . . . . . . . . . . . . . . . . 596.3 Calculation of DT invariants . . . . . . . . . . . . . . . . . . . . . 617 The quiver Q and colored plane partitions . . . . . . . . . . . . . . 667.1 Colored plane partitions . . . . . . . . . . . . . . . . . . . . . . . 667.2 Multi-colored plane partitions . . . . . . . . . . . . . . . . . . . . 687.3 Proof of Theorem 7.2.2 . . . . . . . . . . . . . . . . . . . . . . . . 728 Related problems and future directions . . . . . . . . . . . . . . . . 758.1 Topological invariants . . . . . . . . . . . . . . . . . . . . . . . . . 758.2 Cohomology theories of (X,A) . . . . . . . . . . . . . . . . . . . 768.3 Motivic DT invariants . . . . . . . . . . . . . . . . . . . . . . . . . 788.4 Joyce–Song’s generalized DT invariants . . . . . . . . . . . . . . . 808.5 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84viiAcknowledgmentsFirst I would like to express my deepest gratitude to my advisor Kai Behrendfor his guidance and support. Without his profound mathematical insight, thisthesis would not have been possible.I would like to thank Jim Bryan for his generosity with his invaluable ideas.I would also like to thank my examiners for their time and feedback.My thanks also go to my fellow graduate students for many helpful conver-sations and advice. I especially want to thank all my friends for making my timein Vancouver much more enjoyable.Lastly, I would like to thank my parents for their endless support throughoutmy life.viiiChapter 1IntroductionDonaldson–Thomas (DT) invariants, first introduced by Thomas [33], are integer-valued deformation invariants on a compact Calabi–Yau 3-fold. They provide avirtual count of curves embedded into a Calabi–Yau 3-fold, and are conjecturallyequivalent to other enumerative invariants, such as Gromov–Witten invariants[25], Pandharipande–Thomas invariants [29], and Gopakumar–Vafa invariants[24]. All of which are closely related to BPS invariants, which are motivated bycounting BPS states in the string theory.DT invariants were originally defined by constructing virtual fundamentalclasses on moduli spaces of stable sheaves over a 3-fold. One obtains a de-formation invariant by integrating the virtual fundamental class whenever themoduli space is proper. Later, Behrend [5] showed that these DT invariants canbe expressed as the Euler characteristics of the moduli spaces weighted by theBehrend constructible function. This discovery not only gives a definition of DTinvariants for non-compact Calabi–Yau 3-fold, it also provides a motivic pointof view of DT invariants, which has been further developed by many authors.In [19], Joyce and Song used Hall algebra techniques to define generalizedDT invariants in the presence of strictly semistable objects. Around the sametime, Kontsevich and Soibelman [21] proposed a refinement of DT invariants,called motivic DT invariants, which are conjecturally defined for any Calabi–Yau-3 (CY3) category. One main class of examples is given by quivers withpotential, which in some sense are local models for any CY3 category. More1importantly, the technical difficulties in defining motivic DT invariants disappearin this situation. For these reasons, quivers with potential have been a centralobject in the study of non-commutative DT theory. Motivic DT invariants ofvarious examples have been computed ([27], [13], [11], ...). They also have beenstudied further within other contexts (such as [28] and [12]).The motivation of this thesis begins with a special example of CY3 cate-gory called the quantum Fermat quintic threefold introduced in [20]. Roughlyspeaking, the quantum Fermat quintic threefold is the Fermat quintic hypersur-face in the quantum projective 4-space, using the language of non-commutativeprojective schemes developed by Artin and Zhang [2]. The quantum Fermatquintic threefold has two crucial features. First, it is truly non-commutative, asit is not equivalent to any Calabi–Yau 3-fold. Second, it is projective, contraryto quivers with potential which are non-commutative analogues of local (affine)Calabi–Yau 3-folds. Here by “projective”, we mean its moduli spaces are ex-pected to projective. This allows us to define DT invariants via integrating thevirtual fundamental class, which is essential for the deformation invariance ofDT invariants.To this end, the quantum Fermat quintic threefold and, more generally, non-commutative Calabi–Yau projective schemes should provide an interesting classof examples in non-commutative DT theory. However, there are some difficul-ties. The existence of moduli spaces of semistable sheaves on non-commutativeprojective schemes is unknown. Even in the case where the moduli spaces havebeen constructed (for example, Artin and Zhang proved the representability ofHilbert schemes [3]), the projectivity remains open in general. At the same time,the quantum Fermat quintic threefold is the only known example of (non-trivial)non-commutative Calabi–Yau projective schemes. Therefore we will restrict our-selves to this particular example.In this thesis, we will first define DT invariants of the quantum Fermat quinticthreefold. Then we will give an explicit description of its local models, which isgiven by certain quivers with potential. Finally, we will use these local models tocompute of degree zero DT invariants on the quantum Fermat quintic threefold.2Overview of the thesisIn chapter 2, we go over various background material and set up some notationthat will be used throughout the whole thesis.In chapter 3, we start with a brief introduction to non-commutative pro-jective schemes. The key observation is that if a graded algebra A is finiteover its center, then A is naturally associated to a smooth projective varietyX with a coherent sheaf A of non-commutative OX-algebras on X , and thenon-commutative projective scheme qgr(A) defined by A is equivalent to thecategory Coh(A) of coherent A-modules on X . Moduli spaces of A-moduleson X have been studied by Simpson [31] in the context of Higgs bundles. Wealso study the general properties of the category Coh(A). In particular, we givean alternative proof that the quantum Fermat quintic threefold is Calabi–Yau.In chapter 4, we consider a general pair (X,A) as above and the Simpsonmoduli space M of stable A-modules on X . We first follow the method in [17]to construct an obstruction theoryE :=(RpiM∗RHomAM (F ,F))∨[−1]→ LMfor the moduli space M . Then we show that if the category Coh(A) is CY3, thetruncated complex τ [−1,0]E is a symmetric obstruction theory for M . Further-more, under suitable conditions, the Hilbert scheme Hilbh(A) can be embeddedinto M as an open subscheme. This allows us to define DT invariants via inte-grating the virtual fundamental class on Hilbh(A), which equals to the weightedEuler characteristic:DTh(A) =∫[Hilbh(A)]vir1 = χvir(Hilbh(A)).We will focus on Hilbert schemes of points, and denote byZA(t) =∞∑n=0χvir(Hilbn(A)) tnthe generating function of degree zero DT invariants on the quantum Fermat3quintic threefold.In chapter 5, we give an explicit description of local models of the quantumFermat quintic threefold (X,A).Theorem A. There exist a stratification X = X(0)∐. . .∐X(3) of X andcoherent sheaves J(i) of non-commutative algebras on C3 such that for anypoint p ∈ X(i), there is an analytic local chart U → C3 of p with a (non-unique)isomorphismA|U ∼= J(i)|Uof sheaves of non-commutative algebras. These sheaves J(i)’s of algebras are (upto Morita equivalence) Jacobi algebras of quivers with potential.More specifically, J(i)’s are (again, up to Morita equivalence) just copies ofC[x, y, z] for i 6= 0, whose DT invariants are well-studied. The Jacobi algebraJ(0) is defined by the quiver Q•• •••   ooiiiiTTNNNN ;;4444##with certain potential W . Its DT invariants will be discussed later.In chapter 6, we use the local models J(i)’s to compute the generating func-tion ZA(t). The idea is that Hilbert schemes of points can be stratified withrespect to the stratification of X .Theorem B. We haveZA(t) =3∏i=0( ∞∑n=0χvir(Hilbn(J(i)),Hilbn(J(i))0)tn),where Hilbn(J(i))0 can be regarded as an analogue of punctual Hilbert schemeof points.4Combing with the known result on DT invariants of C3, it eventually leadstoZA(t) = ZQ,W (t)10 ·M(−t5)−50,where ZQ,W (t) is the generating function of DT invariants of the quiver (Q,W )with potential, and M(t) is the MacMahon function.In chapter 7, we give a computation of the DT invariants ZQ,W (t). Onemight notice that our quiver Q is the McKay quiver of the µ5-action on C3with weight (1, 1, 3), which is associated to an orbifold [C3/µ5]. A computationof DT invariants on an orbifold [C3/µ5] was given in [9] using colored planepartitions [35]. However, our DT invariants ZQ,W use a different framing vector(stability condition) than the orbifold. We introduce the notion of Q-multi-coloredplane partitions associated to a quiver Q. Each Q-multi-colored plane partitionhas an associated dimension vector d, and we denote by nd(Q) the number ofQ-multi-colored plane partitions.Theorem C. We haveZQ,W (t) =∞∑n=1∑|d|=n(−1)|d|+〈d,d〉Qnd(Q) tn,and the numbers nd(Q) can be computed from µ5(1, 1, 3)-colored plane parti-tions.Unfortunately, we do not have a closed formula for the generating functionZQ,W , which is probably expected since there is still no closed formula for theDT invariants on the orbifold [C3/µ5].In chapter 8, we will discuss several subjects related to our work, and proposefuture research directions.NotationsIn this thesis, we work over the field C of complex numbers. All schemes oralgebras are separated and noetherian over C. All (sheaves of) algebras areassociative and unital.5By “non-commutative”, we mean not necessarily commutative, and we as-sume that non-commutative rings are both left and right noetherian. For a(sheaf of) non-commutative ring A, an A-module is always a left A-module.We will use refer right A-modules to Aop-modules. All rings without specifiednon-commutative are commutative.We are particularly interested in a special class of non-commutative rings,quantum polynomial rings. They are polynomial rings with variables only com-mute up to a non-zero scalar. We will use the notationC〈x1, . . . , xn〉(qij) := C〈x1, . . . , xn〉/(xixj − qijxjxi) ,where qij ∈ C∗ and qii = qijqji = 1 for all i, j.For any scheme X of finite type (over C), we write χ(X) for the topologicalEuler characteristic of X with analytic topology. The topological Euler charac-teristic has the excision property: let Z ⊂ X be an algebraic closed subschemeand U its open complement. Then χ(X) = χ(Z) + χ(U).We will study various of generating functions. We will use the notationZ(t1, . . . , tn), which is an element in the formal power series Z[[t1, . . . , tn]]. Theconstant term Z(0, . . . , 0) will always be 1, hence divisions between generatingfunctions make sense.6Chapter 2PreliminariesIn this chapter, we go over various background material and set up some nota-tion that will be used throughout the whole thesis. All results in this chapter aretaken from the literature.2.1 Symmetric obstruction theory and the BehrendfunctionLet X be a scheme of finite type. We consider D(X) = D(OXét) the derivedcategory of coherent sheaves on the small étale site of X . Let LX ∈ D(X) bethe cotangent complex of X . We will only consider complexes E concentratedin negative degrees, that is, hi(E) = 0 for all i > 0.Definition 2.1.1 ([6]). An obstruction theory for X is a morphism φ : E→ LX inD(X) such that h0(φ) is an isomorphism and h−1(φ) is surjective.We will often omit the morphism φ and simply call E an obstruction theoryfor X . The following is the main criterion and one might take it as a moreintuitive definition of an obstruction theory.Theorem 2.1.2 ([6, Theorem 9.7]). Let φ : E→ LX be a morphism. The followingstatements are equivalent:(a) φ : E→ LX is an obstruction theory.7(b) For any scheme S, a square-zero extension S of S, a morphism f : S → X ,there is an extension f : S → X of f if and only if the class(f∗E f∗φ−−→ f∗LX → LS κ(S)−−−→ I[1])∈ Ext1S(f∗E, I)vanishes, in which case the extensions form a torsor over Ext0(f∗E, I).Here κ(S) is the Kodaira–Spencer class of the square-zero extension S. Onemay see [18] for more about this class (and contangent complexes).Obstruction theories play a crucial role in enumerative geometry as they areused to construct virtual fundamental classes. This is the main result of [6] and wewon’t discuss further. What we are interested in is a special type of obstructiontheories.Definition 2.1.3 ([5]). A symmetric obstruction theory for X is an obstructiontheory φ : E→ LX with a morphism θ : E→ E∨[1] such that(i) the obstruction theory φ : E → LX is perfect, i.e., hi(E) = 0 excepti = 0, 1.(ii) θ is a non-degenerate symmetric bilinear form. To be more precise, thebilinear form induced by θEL⊗ E→ OX [1]is non-degenerate and symmetric.Roughly speaking, an obstruction theory is symmetric when there are(canonical) isomorphisms between “deformation spaces” (h0(E)) and “obstruc-tion spaces” (h−1(E)). Any scheme X carrying a symmetric obstruction theoryadmits a virtual fundamental class [X]vir ∈ A0(X) of virtual dimension 0.Theorem 2.1.4 ([5]). If X carries a symmetric obstruction theory and is proper, then∫[X]vir1 = χ(X, νX) :=∑c∈Zc · χ(ν−1X (c)),8where νX : X → C is the Behrend constructible function.The construction function νX is defined for any scheme of finite type (overC). One important consequence is that the number χ(X, νX) is intrinsic to thescheme X and does not depend on the choice of symmetric obstruction theory.We list few properties of the Behrend function νX .(1) If X is smooth, then νX = (−1)dimX is a constant function.(2) Let X,Y be schemes of finite type. Then νX×Y (x, y) = νX(x)νY (y).(3) If f : X → Y is étale, then f ◦ νY = νX . In particular, νX(x) onlydepends on the singularity at x ∈ X .(4) If X ⊂ M is a critical locus of a smooth function f on a smooth schemeM , thenνX(x) = (−1)dimM(1−MF(x)),where MF(x) is the Milnor fiber of f at x.(5) If X carries a symmetric obstruction theory, then νX only depends on theanalytic topology of X ([5, Proposition 4.22]).For any scheme X of finite type, we will writeχvir(X) = χ(X, νX).More generally, if Z ⊂ X is a locally closed subscheme, thenχvir(X,Z) = χ(Z, νX).2.2 Quivers with potentialA quiver Q consists of a pair of finite sets Q0 and Q1 with a pair of mapss, t : Q1 → Q0. We call Q0 the set of vertices, Q1 the set of arrows, and s and ttaking an arrow to its source and target respectively.A path in Q is a sequence of arrows an · · · a1 such that s(ai+1) = t(ai) forall i, and n is called the length of this path. We also allow the path of length 09at each vertex i ∈ Q0, which we denote by ei. The path algebra CQ is the freealgebra generated by all paths in Q, where the product of two paths p and q isdefined to be pq if s(p) = t(q), and 0 otherwise.A representation V of Q is given by a set of finite-dimensional vector spaces{Vi}i∈Q0 and linear maps {Ta : Vs(a) → Vt(a)}a∈Q1 . Alternatively, a represen-tation of Q is a finite-dimensional left CQ-module V (take Vi = eiV ). We writedim(V ) = (dimVi)i∈Q0 , called the dimension vector of V .We will denote NQ := Z⊕Q0 the free abelian group of dimension vectors,and N+Q := Z⊕Q0≥0 . There is a bilinear form on N defined by〈d,d′〉Q =∑i∈Q0did′i −∑a∈Q1ds(a)d′t(a).The Euler pairing is given byχQ(d,d′) = 〈d,d′〉Q − 〈d′,d〉Q.For a dimension vector d = (di)i ∈ N+Q , we consider the affine spaceAd(Q) =∏a∈Q1HomC(Cds(a) ,Cdt(a))and the gauge groupGLd =∏i∈Q0GLdi(C),which acts on Ad(Q) by conjugation. The quotient stackMd(Q) = [Ad(Q)/GLd]is the moduli stack of representations of Q with dimension vector d.More generally, let f ∈ N+Q be a framing vector, f 6= 0. A f -framedrepresentation of Q is a representation V = (Vi, Ta)i∈Q0,a∈Q1 of Q with vectorsv1i , . . . , vfii in Vi for each i which generates V (as a left CQ-module). We considerAf ,d(Q) = Ad(Q)×∏i∈Q0(Cdi)fi10with a natural GLd-action.Theorem 2.2.1 ([32]). There is a linearization of GLd on Af ,d(Q) such thatsemistable points are precisely f -framed representations of Q. In particular, the GITquotientMf ,d(Q) := Af ,d(Q) //GLdis a quasi-projective smooth variety and is a fine moduli space of f -framed represen-tations of Q.Let W be a potential of Q, i.e., a linear combination of cyclic paths. Wedefine the Jacobi algebraJac(Q,W ) = CQ/(∂aW )a∈Q1 ,where ∂a is the non-commutative derivative defined by∂a(a′) =1, a = a′,0, a 6= a′,and∂a(a1 · · · an) =n∑i=1∂a(ai)ai+1 · · · ana1 · · · ai−1.We define a representation of (Q,W ) to be a finite-dimensional left Jac(Q,W )-module, and a framed representation of (Q,W ) similarly.The potential W associates a smooth functionTr(W ) : Mf ,d(Q)→ C.The critical locusMf ,d(Q,W ) :=(dTr(W ) = 0) ⊂Mf ,d(Q)is the fine moduli space of f -framed representations of (Q,W ). Therefore it11makes sense to define DT invariants via weighted Euler characteristics. LetZQ,W,f (t) =∑d∈N+Qχvir(Mf ,d(Q,W ))tdbe the generating function of DT invariants of (Q,W ) with framing vector f ,where t = (ti)i∈Q0 and td =∏i∈Q0 tdii . The framing vector f should beinterpreted as a choice of stability condition.If the framing vector f = (1, . . . , 1), we will simply write ZQ,Wfor ZQ,W,(1,...,1). In this case, (1, . . . , 1)-framed representations are finite-dimensional cyclic Jac(Q,W )-modules. We writeHilbd(Q,W ) := M (1,...,1),d(Q,W )which can be viewed as Hilbert schemes of points on the non-commutative affinespace Jac(Q,W ). For integer n, letHilbn(Q,W ) =∏|d|=nHilbd(Q,W )We will abuse the notation and writeZQ,W (t) := ZQ,W (t, . . . , t) =∞∑n=0χvir(Hilbn(Q,W ))tn.2.3 Plane partitions and Hilbert schemes of pointsA plane partition is a finite subset pi of Z⊕3≥0 such that if any of (i + 1, j, k),(i, j + 1, k), (i, j, k + 1) are in pi, then so is (i, j, k). We will refer points in aplane partition as “boxes”, and a plane partition can be viewed as a pile of boxesstacked in the positive octant. The size |pi| is the number of boxes.We denote by P the set of plane partitions. LetZPL(t) =∑pi∈Pt|pi|12be the generating function of plane partitions. It can be expressed asZPL(t) = M(t) :=∞∏n=11(1− tn)n ,which is now called the MacMahon function.Plane partitions are linked with topological Euler characteristic of Hilbertschemes of points on C3 in the following way.We consider the standard torus T = (C∗)3 action on C3. It induces a naturalT -action on the Hilbert scheme Hilbn(C3). Then T -fixed points of Hilbn(C3)are corresponding to C3-invariant closed subschemes, which are given by T -invariant ideals of C[x, y, z]. It is well-known that T -invariant ideals are gener-ated by monomials.Lemma 2.3.1. There is a one-to-one correspondence between monomial ideals I ⊂C[x, y, z] of index n and plane partitions of size n.Proof. The correspondence is given by sending a monomial I to the plane par-titionpi ={(i, j, k) : xiyjzk /∈ I}.Thus Hilbn(C3) has finitely many T -fixed points andχ(Hilbn(C3))= #(T -fixed points) = #(plane partitions of size n).The generating functionZC3top(t) =∞∑n=0χ(Hilbn(C3))tn = M(t)is given by the MacMahon function.13Chapter 3Quantum Fermat quinticthreefolds3.1 Non-commutative projective schemesWe first review the notion of non-commutative projective schemes defined byArtin and Zhang ([2]).Let A be a locally finite Z≥0-graded C-algebra, by which we mean A =⊕i≥0Ai and each Ai is finite-dimensional. We defineqgr(A) = gr(A)/tor(A)to be the quotient abelian category, where gr(A) is the category of finitely-generated graded A-modules, and tor(A) is its Serre subcategory consisting oftorsion modules, here a graded A-module M is said to be torsion if each elementm ∈M is annihilated by A≥n for some n.Definition 3.1.1 ([2]). The non-commutative projective scheme defined by A is atriple(qgr(A),A, [1]),where A is the object in qgr(A) corresponding to A as an A-module, and [1]is the functor induced by sending any graded module M to M [1] defined by14(M [1])i = Mi+1.More generally, we consider a triple (C,O, s) of an abelian category C, anobject O, and a natural equivalence s : C → C. A morphism (C1,O1, s1) →(C2,O2, s2) between two such triples consists of a functor F : C1 → C2, anisomorphism F (O1) ∼= O2, and a natural isomorphism s2 ◦ F ∼= F ◦ s1. Thismorphism is an isomorphism if F is an equivalence of categories.Definition 3.1.2 ([2]). A non-commutative projective scheme is a triple (C,O, s)which is isomorphic to (qgr(A),A, [1]) for some graded algebra A.For example, if A is commutative and generated by degree 1 elements, thenby Serre’s theorem, (qgr(A),A, [1]) is isomorphic to the triple (X,OX ,− ⊗OX(1)), where X = Proj(A).More generally, let X be a (smooth) projective variety with a polarizationOX(1), and A a coherent sheaf of non-commutative OX-algebras on X . Wemay consider the homogeneous coordinate ring of (X,A)A =∞⊕n=0H0(X,A(n)),which is naturally a graded algebra via the multiplication A⊗A → A.Proposition 3.1.3. The triple (Coh(A),A,− ⊗ OX(1)) is isomorphic to the non-commutative projective scheme defined by the graded algebra A. In particular, thereis an equivalence of (abelian) categories Coh(A) ∼= qgr(A).Proof. This follows directly from [2, Theorem 4.5].If the graded algebra A is given by such (X,A), then the homogeneouscoordinate ring B := ⊕nH0(X,OX(n)) of X is a graded subalgebra of A,which is contained in the center Z(A). Since A is coherent, A(n) is generatedby global sections for sufficiently large n. This implies that A is a finite B-module.15Algebras finite over their centersLet A be a non-commutative graded algebra. For simplicity, we assume bothA and Z(A) are finitely-generated. Suppose there exists a graded subalgebraB ⊂ Z(A) of A such that A is a finite B-module. We consider X = Proj(B)andA = B˜Athe coherent sheaf on X corresponding to the graded B-module BA. Then Ais naturally a sheaf of non-commutative OX-algebras.From now on we assume that A is of finite global dimension. In fact, thisforces A to be an Artin–Schelter regular algebra [2] since A is finite over its cen-ter Z(A). We omit the details and only mention two important consequencesthat A satisfies the technical χ condition in the study of non-commutative pro-jective schemes, and the category qgr(A) has finite cohomological dimension(which equals to the global dimension of A minus 1).Proposition 3.1.4. Suppose A is generated by degree one elements. Then there is anequivalence of (abelian) categories qgr(A) ∼= Coh(A).Proof. If the chosen B is also generated by degree one elements, then we haveHomqgr(A)(AA,M) = Homqgr(B)(BB,BM) = H0(X, B˜M)for any graded A-module M . Thus [2], Theorem 4.5(2) states that there is amorphismA→ A′ :=∞⊕n=0Homqgr(A)(AA,AA[n]) =∞⊕n=0H0(X,A(n))of graded algebras which is an isomorphism at sufficiently large degrees. Thisthen implies that qgr(A) ∼= qgr(A′) ∼= Coh(A).In general, we choose k such that the graded algebra B(k) := ⊕iBki isgenerated by degree one elements. Then it is well-known that X = Proj(B) =Proj(B(k)), and BA and B(k)A(k) define the same coherent sheaf on X . In fact,the same result also holds non-commutative projective schemes ([2, Proposition165.10]). Therefore we have equivalences of categoriesqgr(A) ∼= qgr(A(k)) ∼= Coh(A).Remark 3.1.5. In [3], Artin and Zhang define the notion of (flat) familiesof objects in any abelian category. One may check that the equivalenceqgr(A) ∼= Coh(A) in Proposition 3.1.4 induces equivalence between families,which also preserves flatness. This implies that the Hilbert schemes Hilbh(A)constructed in [3] agree with Simpson’s Hilbert schemes Hilbh(X,A) (see 3.3.2).In particular, this proves the projectivity of Hilbh(A) under the assumption ofProposition 3.1.4.Remark 3.1.6. If A is not generated by degree one elements, one may take thestacky ProjX = Proj(B) =[(Spec(B) \ {0})/C∗],where C∗ acts on Spec(B) via the grading. Then X is a Deligne–Mumford stackwith a projective coarse moduli scheme X = Proj(B). The graded algebra Aalso defines a sheaf A of OX-algebras on X. Then there is also an equivalenceof categories Coh(A) ∼= qgr(A) by the same argument.Finally, since Z(A) is a finitely-generated commutative algebra, by Noethernormalization lemma, there exists a regular subalgebra B ⊂ Z(A) such thatZ(A) is finite over B, hence A is finite over B. Thus we can always choose Bso that X = Proj(B) is smooth (in fact, a projective space).Example 3.1.7. Consider quantum projective spaces which are non-commutativeprojective schemes defined by quantum polynomial ringsA = C〈x0, . . . , xn〉(qij).If qij ’s are roots of unity, then A is finite over its center Z(A).173.2 Sheaves of non-commutative algebrasMotivated by the previous section, we consider a smooth variety X and a coher-ent sheaf A of non-commutative OX-algebras. We may view (X,A) as a ringedspace, and pi : (X,A)→ (X,OX) is a morphism of ringed spaces defined by theunit map OX → A. We denote Coh(A) the category of coherent A-modules,that is, coherent sheaves F on X with a left A-actions A⊗F → F .We begin with a few facts about ringed spaces. There are adjoint functorspi∗ a pi∗ a pi!, where pi∗ = A ⊗ − : Coh(X) → Coh(A), pi∗ : Coh(A) →Coh(X) is the forgetful functor, and pi! = HomOX (A,−) : Coh(X)→ Coh(A).More generally, we have natural isomorphismsHomA(F ⊗ G,H) ∼= HomOX (G,HomA(F ,H)),for coherent A-modules F ,H and OX-module G, andHomOX (G ⊗A F ,H) ∼= HomA(F ,HomOX (G,H))for coherent A-module F , Aop-module G, and OX-module H.Global dimensionOne defines the global dimension of A similarly to the case of algebras.Definition 3.2.1. An A-module F is locally projective if for any x ∈ X , thereexists an affine open set U ⊂ X containing x such that F|U is a projectiveA|U -module.For any locally free sheaf P on X , A ⊗ P is naturally a locally projectiveA-module. Thus any coherent A-module F admits a resolution by locally pro-jective A-modules. We define the projective dimension pd(F) of F to be theshortest length of a projective resolution. Then it is a standard fact in homolog-ical algebra to show thatProposition 3.2.2. The following two numbers (possibly∞) are the same:1. the supremum of pd(F) for all coherent A-modules F ;182. the (co)homological dimension of the category Coh(A), that is, the supremumof n ∈ N such that ExtnA(F ,G) 6= 0 for some coherent A-modules F and G.We call this number the global dimension of A, and denote it by dim(A).We say A is smooth if dim(A) < ∞. Note that it does not automaticallyimply dim(A) = dim(X).For simplicity, from now on we will assume the sheaf A is locally free on X .One immediate consequence is that any locally projective (injective) A-moduleis locally free (injective) over OX . In particular, we have dim(A) ≥ dim(X).Using the local-to-global spectral sequence with some basic properties ofnon-commutative rings (see for example, [26, Theorem 4.4]), we see that thedimension of A can be computed locally using the following lemma.Lemma 3.2.3. The dimension ofA is equal to the supremum of the global dimensionsof algebras Ax for all (closed) points x ∈ X .Serre dualitySince X is a smooth variety, the derived category D(Coh(X)) admits a Serrefunctor (−)⊗ ωX [n], where ωX is the dualizing sheaf and n = dim(X).Proposition 3.2.4. If A is smooth, then the derived category D(Coh(A)) admits aSerre functor ωAL⊗A(−)[n], where ωA = pi!ωX = HomOX (A, ωX) is the dualizingA-bimodule.Proof. For any perfect complexes F and G of A-modules, we have natural iso-morphismsHomA(F ,G) = HomOX (OX , RHomA(F ,G))∼= HomOX (RHomA(F ,G), ωX [n])∗∼= HomOX (RHomA(F ,A)⊗A G, ωX [n])∗∼= HomA(G, RHomA(F ,A)∨ ⊗ ωX [n])∗∼= HomA(G,A∨ ⊗A F ⊗ ωX [n])∗= HomA(G, ωA ⊗A F [n])∗.19Definition 3.2.5. We say A is Calabi–Yau of dimension n if the derived categoryD(Coh(A)) is a Calabi–Yau-n category, i.e., the Serre functor is equivalent to(−)[n].In particular if A is Calabi–Yau, then A is smooth and dim(A) = dim(X).Proposition 3.2.6. Suppose A is smooth, then the followings are equivalent:1. A is Calabi–Yau;2. There is an isomorphism A → ωA of A-bimodules;3. There is a non-degenerate bilinear formσ : A⊗A → ωXof OX -modules such that σ is symmetric and the diagramA⊗A⊗Am⊗id //id⊗mA⊗AσA⊗A σ // ωXcommutes. In other words, (A, σ) is a family of symmetric Frobenius algebrasover X .Proof. It is clear that (1) and (2) are equivalent. For (2) and (3), a bilinear formσ : A⊗A → ωX is non-degenerate if and only if it induces an isomorphismA → A∨ ⊗ ωX ∼= HomOX (A, ωX) = ωAof OX-modules. To check it is a morphism of A-bimodules, we may reduce toaffine open sets Spec(R), where R is regular. Then it is straightforward to verifythat two statements are equivalent.Example 3.2.7. Let X be a Calabi–Yau variety. Then any Azumaya algebra Aon X is Calabi–Yau.203.3 Simpson moduli spacesFrom now on we assume that X is projective and fix a polarization OX(1).Stability conditions for coherent A-modules and their moduli spaces have beenstudied by Simpson ([31]) for general sheaf A of non-commutative algebras onX . We recall their definitions and main results.For a coherent A-module, we define its Hilbert polynomial, rank, slope, andsupport to be the same as its underlying coherent sheaf on X (with respect toOX(1)). In particular, we say a coherent A-module is pure (of dimension d) ifits underlying coherent sheaf is so.Definition 3.3.1 ([31]). A coherent A-module F is (semi)stable if it is pure, andfor any non-trivial A-submodule G ⊂ F ,pX(G)(m)r(G) (≤)pX(F)(m)r(F) ,for sufficiently large m, where pX is the Hilbert polynomial, and r is the rank.It was shown in [31] that all standard facts for semistable sheaves (cf. [16]) arealso true for semistable A-modules, such as• Any pure coherent A-module F has a unique filtration, called the Harder–Narasimhan filtration,0 = F0 ⊂ F1 ⊂ . . . ⊂ Fk = Fof coherent A-modules such that the quotients Fi/Fi−1’s are semistableA-modules with strictly decreasing reduced Hilbert polynomials.• Any semistable A-module F has a filtration, called a Jordan–Hölder filtra-tion,0 = F0 ⊂ F1 ⊂ . . . ⊂ Fk = Fof coherent A-modules such that the quotients JHi := Fi/Fi−1’s arestable A-modules with the same reduced Hilbert polynomial (as F ). Fur-thermore, the polystable A-module JH(F) := ⊕iJHi does not depend on21the filtration (up to isomorphic). We say two semistable A-modules F andG are S-equivalent if JH(F) ∼= JH(G) as A-modules.• If A is a stable A-module, then HomA(F ,F) = C.We fix a polynomial h.Proposition 3.3.2 ([31]). The Hilbert scheme Hilbh(X,A) parameterizing quotientsA → F as coherent A-modules with pX(F) = h is representable by a projectivescheme. In fact, it is the closed subscheme of the Quot scheme QuotpX(A) (who pa-rameterizes OX -module quotients of A) given by the locus that the universal quotientis a morphism of A-modules.The moduli spaces of (semi)stable A-modules were constructed in the sameway as the ones for (semi)stable sheaves, which is via GIT quotient on certainHilbert (Quot) schemes. We omit the details and state the main results.Theorem 3.3.3 ([31]). Let M(s)s,h(X,A) be the moduli stack of (semi)stable A-modules with Hilbert polynomial h. Then(a) The moduli stack M(s)s,h(X,A) is an Artin stack of finite type, and admitsa good moduli spaceM (s)s,h(X,A).(b) The coarse moduli scheme M ss,h(X,A) is projective, whose points are in one-to-one correspondence with S-equivalent classes of semistable A-modules.(c) The morphism Ms,h(X,A) → M s,h(X,A) is a C∗-gerbe, and M s,h(X,A)is the open subscheme ofM ss,h(X,A) whose points corresponds to isomorphismclasses of stable A-modules.We will often simply write Hilbh(A) and Mss,h(A) if the base space X isclear.3.4 Quantum Fermat quintic threefoldsIn this section, we will define the main subject of this thesis — quantum Fermatquintic threefolds introduced in [20].22Definition 3.4.1. A quantum Fermat quintic threefold is a non-commutative pro-jective scheme associated to a graded algebraA = C〈t0, . . . , t4〉(qij)/( 4∑k=0t5k),with deg(ti) = 1 for all i and qij ’s satisfying(i) qij is a 5-th root of unity.(ii) qii = qijqij = 1 for all i, j.(iii)∏j qij is independent of i.The condition (iii) is equivalent to the category qgr(A) being CY3 [20, The-orem 2.1].In this thesis, we will add an additional condition:(iv) qijqjk 6= qik for all distinct i, j, k.This is to ensure that the non-commutative projective scheme has the “maximalnon-commutativity”. To see this, we recall Zhang’s twisted graded algebras [36]:Let A = ⊕iAi be a graded algebra. For any automorphism σ : A → A ofgraded algebras, Zhang defines a new multiplication on A byx ∗ y = x · σm(y)for any x ∈ Am and y ∈ An. The graded algebra with the new multiplication∗ is called a twisted graded algebra of A, denote by Aσ . Then it is shown thatnon-commutative projective schemes associated to A and any twisted algebraAσ are isomorphic.Now, if qijqjk = qik for some distinct i, j, k, then we take the automorphismσ : A→ A defined by xj 7→ qijxj and xk 7→ qikxk. The twisted algebra Aσ hasquantum parameters qij = qjk = qik = 1, which means the non-commutativeprojective scheme qgr(A) contains a commutative closed subscheme of positivedimension.23Theorem 3.4.2. Up to a possible change of the primitive root q ∈ µ5, all quan-tum Fermat quintic threefolds (satisfying (i)–(iv)) are isomorphic as non-commutativeprojective schemes.Proof. We fix a primitive 5-th root q of unity. Any quantum Fermat quinticthreefold is determined by a skew-symmetric matrix N = (nij)i,j ∈ M5(Z/5Z)such that (1, 1, 1, 1, 1)ᵀ is an eigenvector of N . We denote by AN the gradedalgebra with quantum parameters qij = qnij .Consider the following actions on the set of above matrices N ’s:(a) Change of the primitive root q: For any element a ∈ (Z/5Z)×, changingq to qa is equivalent to multiple all elements nij in N by a.(b) Change of variables ti’s: For any permutation σ ∈ S5, we consider thechange of variables t˜i = tσ(i). Then the new graded algebra has quantumparameters given by n˜ij = nσ(i),σ(j).(c) Twisted graded algebras: For any (a0, . . . , a4) ∈ (Z/5Z)5, let σ : AN →AN be the automorphism defined by σ(xi) = qaixi. Then the twistedalgebra (AN )σ has quantum parameters given byNσ := (nij + ai − aj)i,j .The proof is done with the aid of computer. There are precisely 3000 choices ofN ’s. All of them are equivalent under the three actions above.For computation purpose, we will fix a particular choice of quantum param-eters(qij)i,j =1 q q−1 q q−1q−1 1 q q−1 qq q−1 1 q q−1q−1 q q−1 1 qq q−1 q q−1 1 (3.1)where q ∈ µ5 is a (fixed) primitive 5-th root of unity, and we will call it thequantum Fermat quintic threefold.24To associate a pair (X,A), we takeB = C[x0, . . . , x4]/( 4∑k=0xk),where xi = t5i for each i, and X = Proj(B) ∼= P3. Since A is a graded-free B-module, the sheaf A of non-commutative OX-algebras induced by A is locallyfree. In fact, A is a graded free B-module, and we can writeA = OX ⊕OX(−1)⊕121 ⊕OX(−2)⊕381 ⊕OX(−3)⊕121 ⊕OX(−4)as a OX-module.It is shown in [20] that the graded algebra A is of finite global dimension,so A also has finite global dimension. Here we give an alternative proof that(X,A) is Calabi–Yau.Lemma 3.4.3. The sheaf A of OX -algebras is Fobenius via(−,−) : A⊗OX A → A→ ωX ,where the first arrow is the multiplication map, and the second arrow is the projectionto the component OX(−4) ∼= ωX . If∏j qij = 1 for all i, then the pairing issymmetric.Proof. We write down the multiplication maps of A explicitly. ConsiderI ={a = (a0, a1, . . . , a4) ∈ {0, 1, . . . , 4}5, a0+a1+. . .+a4 is a multiple of 5},a basis of A(5) over B = B(5). For simplicity, we will write a0 + a1 + . . .+ a4 =5 |a|. Note that I is naturally an abelian group as a subgroup of (Z/5Z)5 (butthe function | − | is not linear). Then as a OX-module, we may writeA =⊕a∈IOX(−|a|).25We denote the multiplication map A⊗A → A on each component byOX(−|a|)⊗OX(−|b|)φa,b−−→ OX(−|a+ b|),whereφa,b = qa,bxc00 xc11 xc22 xc33 xc44is the section in H0(X,OX(|a|+ |b| − |a+ b|))given byqa,b =∏i>jqaibjij , ci =5, ai + bi ≥ 5;0, ai + bi < 5.Write 4 = (4, 4, 4, 4, 4) ∈ I , OX(−|4|) = OX(−4) is the component cor-responding to ωX . Since for each component OX(−|a|), there is a uniquecomponent OX(−|4− a|) such that the multiplication mapOX(−|a|)⊗OX(−|4− a|)→ OX(−|4|) ∼= ωXis an isomorphism, the induced map A → HomOX (A, ωX) is an isomorphismof OX-modules.The pairing (−,−) is symmetric if and only if qa,4−a = q4−a,a for all a ∈ I .That is, ∏i>jqai(4−aj)ij =∏i>jq(4−ai)ajij ⇐⇒∏i>jqai−ajij = 1,which is equivalent to that∏j qij = 1 for all i.26Chapter 4Donaldson–Thomas invariantsof (X,A)The purpose of this chapter is to define DT invariants on a CY3 pair (X,A).We begin with a study of deformation-obstruction theory of A-modules and con-struct an obstruction theory for the moduli space of stable A-modules. Then wewill use it to construct a symmetric obstruction theory and define DT invariantsusing the Hilbert schemes of (X,A).4.1 Obstruction theories for A-modulesLet X be a smooth projective variety and A a locally free sheaf of non-commutative OX-algebras. Let S be a scheme, and F a coherent AS-moduleon X × S, flat over S. Suppose S ⊂ S is a square-zero extension with idealsheaf I .Definition 4.1.1. An A-module extension of F over S is a coherent AS-moduleF on X × S, flat over S, such that F|X×S ∼= F as AS-modules.It is a general fact ([23]) that existence of such extensions must be governedby an obstruction class in Ext2AS (F ,F ⊗ pi∗SI). However, to obtain an obstruc-tion theory on the moduli spaces, it requires a more explicit description of theobstruction class. We generalize the result in [17], showing that the obstructionclass is the product of Atiyah and Kodiara–Spencer classes.27Theorem 4.1.2. There exists a natural classatA(F) ∈ Ext1AS (F ,F ⊗ pi∗SLS),called the Atiyah class, such that for any square-zero extension S ⊂ S with idealsheaf I , an A-module extension of F over S exists if and only if the obstruction classob =(F atA(F)−−−−→ F⊗pi∗SLS [1]idF⊗pi∗Sκ(S/S)[1]−−−−−−−−−−−→ F⊗pi∗SI[2])∈ Ext2AS (F ,F⊗pi∗SI)vanishes, where κ(S/S) ∈ Ext1S(S, I) is the Kodiara–Spencer class for the extensionS ⊂ S. Moreover, if an extension of F over S exists, then all (equivalence classes of )extensions form an affine space over Ext1A(F ,F ⊗ pi∗SI).We follow closely the method in [17]. The key idea is that the obstructionclasses are given universally by a morphism of Fourier–Mukai transforms. Theproof will be given in the next section. We set up some notations for this chapter.For any morphism f : S → T of schemes, we will abuse the notation andwrite f also for the induced morphism idX × f : X × S → X × T . So there arenatural functorsf∗ : Coh(AT )→ Coh(AS), f∗ : Coh(AS)→ Coh(AT ),where the latter one is induced by the natural morphism AT = A  OT →f∗AS = A  f∗OS . We denote by D(b)(AS) the (bounded) derived categoryof Coh(AS). In this section, tensor products ⊗ will always be derived tensorproducts over O unless stated otherwise.We briefly recall the definition of Fourier–Mukai transforms. For any schemesS and T , and a complex P ∈ D(OS×T ) of coherent OS×T -modules, we maydefine the Fourier–Mukai functorΦP : D(AS)pi∗S−→ D(AS×T ) −⊗P−−−→ D(AS×T ) RpiT∗−−−→ D(AT ),where P is called the Fourier–Mukai kernel. Given any morphism φ : P1 → P2in D(OS×T ), it induces a natural transformation ΦP1 → ΦP2 between functors.28For any object F ∈ D(AS), we writeφ(F) : ΦP1(F)→ ΦP2(F)for the induced morphism in D(AT ).Now, we use the Atiyah class to construct an obstruction theory on themoduli space. Let S be any scheme, and F be a coherent AS-module, flat overS. Let α be a class in Ext1AS (F ,F ⊗ pi∗SLS). Then α defines a morphismα : F → F ⊗ pi∗SLS [1] in D(AS).Since F is flat over S, F is perfect. The natural map F → F∨∨ is an isomor-phism in D(AS). Thus there is an isomorphismF ⊗ pi∗SLS ∼= RHomOX (F∨, pi∗SLS) in D(AS)By adjunction, α defines a morphismF∨ ⊗AS F → pi∗SLS [1] in D(OX×S).We then apply Verdier duality, it yields a morphismRpiS∗((F∨ ⊗AS F)⊗ pi∗XωX)[n− 1]→ LS in D(OS).Lemma 4.1.3. For any perfect complexes F and G of coherent AM -modules, there isa canonical isomorphism(RpiM∗RHomAM (F ,G))∨ ∼= RpiM∗ (G∨ ⊗AM F ⊗ pi∗XωX) [n].Proof. Use the same argument as above, with pi∗SLS replaced by pi∗SOS = OX×S .We conclude that any class α ∈ Ext1AS (F ,F ⊗ pi∗SLS) defines a morphism(RpiS∗RHomAS (F ,F))∨[−1]→ LS . (4.1)29It is almost by definition to see that if S = M is a fine moduli space and F isthe universal family of coherent A-modules, then the morphism (4.1) induced bya class α is an obstruction theory for M if and only if α is the Atiyah class.In general, the moduli space M := M s,pX (A) is not a fine moduli spacebecause the C∗-gerbe M → M is not trivial so the universal family on M doesnot descend to M . We use the fact that any C∗-gerbe is étale locally trivial, thatis, there is an étale cover U → X with a sectionMU //==M.We denote by F the pullback of the universal family of stable A-modules toX × U .Consider the natural transformationΦ : HomSch(−, U)→M,where M is the moduli functor for stable A-modules on X , and Φ sends anymorphism f : S → U to the AT -module (idX × f)∗F on X × S.Lemma 4.1.4. The natural transformation Φ satisfies1. Φ(SpecC) is surjective;2. For any f : S → U and any square-zero extension S ⊂ S, the map Φ(S)induces a bijection between subsets{f : S → U : f |S = f}⊂ HomSch(S,U)and {A-module extensions of F over S}⊂M(S).In other words, U has the same deformation-obstruction theory as a fine moduli space.Proof. This is essentially the definition of the morphism U →M being étale.30Theorem 4.1.5. Let U and F be described as above. Then the Atiyah class atA(F)defines an obstruction theoryE :=(RpiU∗RHomAU (F ,F))∨[−1]→ LU (4.2)for U .Proof. Let f : S → U be a morphism, and S ⊂ S be a square-zero extensionwith ideal sheaf I . We consider the classo =(Lf∗E→ Lf∗LU → LS κ(S/S)−−−−→ I[1])∈ Ext1OS (Lf∗E, I).Observe that there are natural isomorphismsHomOS (Lf∗E,LS) ∼= HomOU (E, Rf∗LS)∼= HomOX×U (F∨ ⊗AU F , pi∗URf∗LS [1])∼= HomOX×U (F∨ ⊗AU F , Rg∗pi∗SLS [1])∼= HomOX×S (g∗(F∨ ⊗AU F), pi∗SLS [1])∼= HomOX×S (g∗F∨ ⊗g∗AU g∗F , pi∗SLS)∼= HomOX×S ((g∗F)∨ ⊗AS (g∗F), pi∗SLS)∼= HomAS (g∗F , g∗F ⊗ pi∗SLS)where g = idX × f : X × S → X × U . By the functoriality of Atiyah classes, itsends the composition (Lf∗E → Lf∗LU → LS) to the Atiyah class atA(g∗F)on X ×S. Therefore the class o ∈ Ext1OS (Lf∗E, I) corresponds to the obstruc-tion class ob ∈ Ext2AS (g∗F , g∗F ⊗ pi∗SLS) for the coherent AS-module g∗F onX × S.The rest of the proof follows from Lemma 4.1.4: there exists an extension off : S → U to S if and only if there exists an A-module deformation of g∗Fover S.The coherent AU -module F on X × U is pulled back from the C∗-gerbeM→M , so it can be regarded as a twisted AM -module on X ×M .31Lemma 4.1.6. The complex RHomAU (F ,F) on X × U descends to a complex inDét(OX×M ), which we denote by RHomAM (F ,F).Proof. Note that for any coherent A-module G, HomA(G,G) is the equalizer ofHomOX (G,G) //// HomOX (A⊗ G,G).Then the proof follows from the fact that for any twisted sheaf F ,HomOX (F ,F)is a (untwisted) coherent sheaf (see for instance [10]).By functoriality of the Atiyah class, the obstruction theory E → LU alsodescends to a morphism in Dét(OM )(RpiM∗RHomAM (F ,F))∨[−1]→ LM in Dét(OM ). (4.3)Since being an obstruction theory is an étale local property, we conclude thatCorollary 4.1.7. The morphism (4.3) is an obstruction theory for the moduli spaceM .4.2 Proof of Theorem 4.1.2We fix a scheme S and a square-zero extension i : S ⊂ S with ideal sheaf I . LetF be a coherent AS-module on X × S, flat over S. Suppose F is an A-moduleextension of F over S. The isomorphism Li∗F = i∗F → F induces an exacttriangleRi∗(F ⊗ pi∗SI)→ F → Ri∗F e−→ Ri∗(F ⊗ pi∗SI)[1],which gives a class e ∈ Ext1AS (Ri∗F , Ri∗(F ⊗ pi∗SI)). For any F , we have anexact triangleQF → Li∗Ri∗F → F (4.4)in D(AS) given by adjunction.Lemma 4.2.1. A class e ∈ Ext1AS (Ri∗F , Ri∗(F ⊗ pi∗SI)) is given by an A-moduledeformations of F over S if and only if the compositionΦe : QF → Li∗Ri∗F Li∗e−−−→ Li∗Ri∗(F ⊗ pi∗SI)32is an isomorphism in D(AS).Proof. Observe the diagramQFΦe((Li∗Ri∗(F ⊗ pi∗SI) // Li∗F //r%%Li∗Ri∗F //Li∗Ri∗(F ⊗ pi∗SI)[1]F(4.5)in D(AS). The morphism r : Li∗F → F is an isomorphism if and only if Φe is.Then F is an A-module deformation of F .Now, we apply HomAS (−,F ⊗ pi∗SI) to the exact triangle (4.4), it yieldsExt1AS (Ri∗F , Ri∗(F ⊗ pi∗SI))Ext1AS (F ,F ⊗ pi∗SI) // Ext1AS (Li∗Ri∗F ,F ⊗ pi∗SI) // Ext1AS (QF ,F ⊗ pi∗SI)δ // Ext2AS (F ,F ⊗ pi∗SI)The second arrow sends a class e to the morphismΨe : QFΦe−→ Li∗Ri∗(F ⊗ pi∗SI)[1]→ F ⊗ pi∗SI[1],where the second map is the adjunction map. The proof consists of followingsteps.(a) There exists a class piF ∈ Ext1AS (QF ,F ⊗ pi∗SIS) such that Ψe = piF forany class e given by a deformation.(b) The obstruction class δ(piF ) ∈ Ext2AS (F ,F ⊗ pi∗SI) is the product ofAtiyah class and Kodaira–Spencer class.(c) If the obstruction class δ(piF ) vanishes, then there exists a class e given bya deformation such that Ψe = piF .(d) Suppose e and e′ are two classes such that Ψe = Ψe′ . If a class e is givenby a deformation, then so is e′.33Construction of piFWe first consider the trivial case A = OX , F = OX×S , and the class e given bythe deformation OX×S . Then the morphism Ψe is determined by the square-zero extension S ⊂ S, which we denote bypi : QOX×S → Li∗Ri∗pi∗SI[1]→ pi∗SI[1]. (4.6)Note that it is the pull-back of the morphism QOS → I[1] in D(OS) defined inthe same way.Assume that an A-module deformation F of F over S exists. Then for anycoherent sheaf P on X × S, there are canonical isomorphismsF ⊗ Li∗Ri∗P ∼= Li∗(F ⊗Ri∗P) ∼= Li∗Ri∗(F ⊗ P).This implies that Ψe is equal to piF := F ⊗ pi, up to a canonical isomorphismQF ∼= F ⊗QOX×S .The obstruction class δ(piF)Observe that QF and F ⊗QOX×S may not be isomorphic for general coherentA-module F , so we need an alternative definition of piF . To define piF and studythe obstruction class δ(piF ), we recall several facts proved in [17].(i) There exists an exact triangleQ → H → ∆∗OS δ0−→ Q[1] in D(OS×S),where ∆ : S → S × S is the diagonal map, such that the exact triangle(4.4) is given by Fourier–Mukai transformsΦpi∗S×SQ(F)→ Φpi∗S×SH(F)→ F(pi∗S×Sδ0)(F)−−−−−−−−→ Φpi∗S×SQ(F)[1].In particular, the map δ : Ext1A(QF ,F ⊗ pi∗SI) → Ext2A(F ,F ⊗ pi∗SI) isthe composition with (pi∗S×Sδ0)(F).34(ii) There exists a natural morphismpi0 : Q → ∆∗I[1] in D(OS×S)such that (pi∗S×Spi0)(F) = piF for any F admitting a deformation. There-fore we define piF = (pi∗S×Spi0)(F) for general coherent AS-module F .(iii) The universal obstruction classω0 := pi0[1] ◦ δ0 : ∆∗OS → ∆∗I[2] in D(OS×S)decomposes intoω0 : ∆∗OS αS−−→ ∆∗LS [1] ∆∗κ(S/S)[1]−−−−−−−→ ∆∗I[2],where αS ∈ Ext1S×S(∆∗OS ,∆∗LS) is the (truncated) universal Atiyahclass, which is intrinsic and functorial to S, and κ(S/S) ∈ Ext1S(LS , I) isthe (truncated) Kodaira–Spencer class of the square-zero extension S ⊂ S.(See [17], Definition 2.3 and 2.7 for details.)Definition 4.2.2. We define the Atiyah class for a coherent AS-module F to be(pi∗S×SαS)(F) ∈ Ext1AS (F ,F ⊗ pi∗SLS).The proofs can be found in [17, Section 2.5 and Section 3.1]. We remark thatwhile [17] only consider coherent sheaves, the proofs are all done at the level ofFourier–Mukai kernels. Therefore it also works for any coherent A-modules.Existence of deformationsWe will show that if the obstruction class δ(piF ) = (pi∗S×Sω0)(F) vanishes, thenthere exists an A-module extension of F over S.First we remark that we may assume both X and S are affine as in [17],Section 3.3. Although the existence of deformations is not a local property, forany given class e ∈ Ext1AS (Li∗Ri∗F ,F ⊗pi∗SI), the conditions that Φe being anisomorphism and Ψe = piF can be checked locally. In other words, if δ(piF ) = 035but there is no deformation of F , then there is a class e such that Ψe = piF bute is not given by a deformation. Then we can find an affine open set U such thatΦe|U is not an isomorphism but Ψe|U = piF |U , which is a contraction.Therefore we may assume that X = Spec(B), A = A is an R-algebra,S = Spec(R), and S = Spec(R). Let M be a (left) AR := A⊗C R-module, flatover R. For convenience, a tensor product ⊗ without subscript is over R. Wefirst recall the standard obstruction theory for modules (cf. [22]).Choose a (possibly-infinite) free resolution of M. . .→ A⊕n−3Rd−3−−→ A⊕n−2Rd−2−−→ A⊕n−1R →M → 0.Consider the trivial deformation A⊕n•Rof A⊕n•R . Then we choose an arbitrarylifting d′• : A⊕n•R→ A⊕n•+1Rof d•. Since (d′•+1 ◦d′•)|R = d•+1 ◦d• = 0, the mapd′•+1 ◦ d′• factors intod′•+1 ◦ d′• : A⊕n•R → A⊕n•Rob•−−→ A⊕n•+2R ⊗ I → A⊕n•+2R ,where the first and third arrows are given by the extension 0→ I → R→ R→0. It is well-known that the class{ob•} ∈ HomAR(A⊕n•R , A⊕n•+2R ⊗ I) (4.7)is a 2-cocycle defining a class in Ext2AR(M,M) which is independent of thechoice of the resolution (A⊕n•R , d•) and lifting d′•. We will show that this ob-struction class is the same as δ(piF ).Recall that the universal obstruction class ω0 ∈ Ext2S×S(∆∗OS ,∆∗I) isrepresented by the 2-extension0→ ∆∗I → J |S×S → OS×S → ∆∗OS → 0,where J is the ideal sheaf defining S ⊂ S × S. We omit the details but acrucial consequence is that if M = A⊕nR is free, then the obstruction class(pi∗S×Sω0)(M) ∈ Ext2AR(M,M ⊗ I) is represented by the 2-extension (of AR-36modules)0→ A⊕nR ⊗ I → K⊕n|R → Γ⊕n|R → A⊕nR → 0,where the restriction −|R is the tensor product − ⊗R R, Γ = AR ⊗C R is thefree AR-module, the arrowΓ = AR ⊗C R = A⊗C R⊗C R→ AR = A⊗C Ris induced by the R-linear evaluation map R ⊗C R → R via R → R, andK = ker(Γ → AR) is the kernel. Since R ⊗C R is a free R-module, we maychoose a (non-canonical) splittingR⊗C R ∼= L⊕Rsuch that the evaluation map R⊗C R is given by the short exact sequence0→ L⊕ I → L⊕R→ R→ 0.Then Γ ∼= N⊕AR, K ∼= N⊕(AR⊗I), where N = A⊗CL is a free AR-module.Let A⊕n•R →M be a free resolution. Then it associates a 2-extension0→ A⊕n•R ⊗ I → K⊕n• |R → Γ⊕n• |R → A⊕n•R → 0 (4.8)of AR-modules, and a short exact sequence0→ K⊕n• → Γ⊕n• → A⊕n•R → 0of complexes of AR-modules, where the differentials are arbitrarily chosen liftingof the differentials in (4.8). We write down the differentials explicitly with respectto the splitting Γ ∼= N ⊕AR and K ∼= N ⊕ (AR ⊗ I):0 // N⊕n• ⊕ (AR ⊗ I)⊕n• //( ∗ ∗η• ∗ )N⊕n• ⊕A⊕n•R//( ∗ γ•η• d′•)A⊕n•R //d•00 // N⊕n•+1 ⊕ (AR ⊗ I)⊕n•+1 // N⊕n•+1 ⊕A⊕n•+1R // A⊕n•+1R// 037and0 // (AR ⊗ I)⊕n• //N⊕n• |R ⊕ (AR ⊗ I)⊕n• //( ∗ ∗σ• ∗ )N⊕n• |R ⊕A⊕n•R //( ∗ β•∗ d•)A⊕n•R //d•00 // (AR ⊗ I)⊕n•+1 // N⊕n•+1 |R ⊕ (AR ⊗ I)⊕n•+1 // N⊕n•+1 |R ⊕A⊕n•+1R // A⊕n•+1R // 0Observe that{A⊕n•Rβ•−→ N⊕n•+1 |R σ•+1−−−→ (AR ⊗ I)⊕n•+2}∈ HomAR(A⊕n•R , A⊕n•+2R ⊗ I)defines a 2-cocycle which corresponds to the class of the 2-extension (4.8) inExt2AR(M,M ⊗ I), i.e., the obstruction class δ(piF ).On the other hand, d′• : A⊕n•R→ A⊕n•+1Ris a lifting of d•. The differentialson Γ⊕n• implies that−d′•+1 ◦ d′• = η•+1 ◦ γ• : A⊕n•R → N⊕n•+1 → A⊕n•+2R.Since η•+1 factors through N⊕n•+1 → (AR ⊗ I)⊕n•+2 → A⊕n•+2R , the composi-tion can be decomposed intoη•+1 ◦ γ• : A⊕n•R → A⊕n•Rγ•|R−−−→ N⊕n•+1 |R η•+1|R−−−−→ (AR ⊗ I)⊕n•+2 → A⊕n•+2R .By definition, γ•|R = β• and η•+1|R = σ•+1. This shows that the classicalobstruction class (4.7) defined by the lifting d′• is the class −δ(piF ).Finally, suppose e is the class corresponding to a deformation (A⊕n•R, d′•),and e′ is another class such that Ψe = Ψe′ . Then e − e′ is in the image of a1-cocycle{f•} ∈ HomAR(A⊕n•R , A⊕n•+1R ⊗ I).Then it is a standard fact that (A⊕n•R, d′•+ f˜•) also defines a deformation, wheref˜• : A⊕n•R → A⊕n•Rf•−→ A⊕n•+1R → A⊕n•+1R ,which then corresponds to the class e′.384.3 Donaldson–Thomas invariants for Coh(A)We now restrict our attention to (X,A) being CY3. Let M be a quasi-projectivecoarse moduli scheme of stable A-modules with a universal twisted AM -moduleF on X ×M .First we construct a symmetric bilinear formθ : RpiM∗RHomAM (F ,F)→(RpiM∗RHomAM (F ,F))∨[1].We write F = RHomAM (F ,F). For a given isomorphism ωA → A of A-bimodules, it induces an isomorphismRHomAM (F , ωA ⊗A F)→ F in D(OX×M ).Taking (−)∨[−1] on both sides, it gives an isomorphism(RpiM∗F)∨ [−1]→(RpiM∗RHomAM (F , ωA ⊗A F))∨[−1] in D(OM ),(4.9)and the right hand side is isomorphic toRpiM∗((pi∗XωA ⊗AM F)∨ ⊗AM F ⊗ pi∗XωX)[2].Note that(pi∗XωA ⊗AM F)∨= RHomOX×M (pi∗XωA ⊗AM F ,OX×M )∼= RHomAM (F , RHomOX×M (pi∗XωA,OX×M ))∼= RHomAM (F ,AM ⊗ pi∗Xω∨X) in D(AopM ),where the last isomorphism is given by pi∗XωA ∼= HomOX×M (pi∗XA, pi∗XωX) ∼=A∨M ⊗ pi∗XωX . Therefore(pi∗XωA⊗AMF)∨⊗AM (F⊗pi∗XωX) ∼= RHomAM (F , (F⊗pi∗XωX)⊗pi∗Xω∨X) ∼= F.39Thus (4.9) is an isomorphismθ : (RpiM∗F)∨[−1]→ RpiM∗F [2] =((RpiM∗F)∨[−1])∨[1].In fact, we have θ∨[1] = θ, so θ is symmetric.Theorem 4.3.1. The moduli schemeM carries a symmetric obstruction theory(τ [1,2]RpiM∗RHomAM (F ,F))∨[−1]→ LMwhich in particular is a perfect obstruction theory.Proof. Consider the map σ : OM → RpiM∗F induced by the scalar mapRpi∗MOM = OX×M → F. For any point m ∈ M , this map induces the scalarmap σm : C → HomA(Fm,Fm), which is an isomorphism since Fm is stable.Thus the cone of σ is the truncated complex τ≥1RpiM∗E.On the other hand, we may considerRpiM∗Fθ−1[−2]−−−−−→ (RpiM∗F)∨[−3] σ∨[−3]−−−−→ OM [−3].The induced map Ext3A(Fm,Fm) → C on each point m ∈ M is dualto the scalar map, which is an isomorphism. The cone of the compositionτ≥1RpiM∗F→ OM [−3] is the truncated complex τ [1,2]RpiM∗F.Therefore, τ [1,2]RpiM∗F is perfect of amplitude in degree 1 and 2, and theobstruction theory (RpiM∗F)∨[−1]→ LM induces a morphism(τ [1,2]RpiM∗F)∨[−1]→ LM .Furthermore, by the construction, θ induces a symmetric bilinear formθ : (τ [1,2]RpiM∗F)∨[−1]→((τ [1,2]RpiM∗F)∨[−1])∨[1].If the moduli space M = M s,h(X,A) is projective (for example, when theHilbert polynomial h has coprime coefficients), then we may define the DT in-40variant via integration ∫[M ]vir1,which equals to the Behrend function weighted Euler characteristic χ(M,νM )by [5]. In particular, this invariant depends only on the scheme structure of themoduli space M , which depends only on the abelian category Coh(A) with achosen stability condition.Donaldson–Thomas invariantsRecall that in classical Donaldson–Thomas theory, the Hilbert schemes (with afixed curve class) are isomorphic to the moduli spaces of stable sheaves withfixed determinant. Therefore one may study Donaldson–Thomas invariants onHilbert schemes. We do not have the notion of determinant for A-modules, butwe can still define Donaldson–Thomas type invariants on Hilbert schemes undersome suitable assumptions.Lemma 4.3.2. Let h be a polynomial of degree ≤ 1. Suppose the stalk Aη at genericpoint η ∈ X is a division algebra, then there is a natural morphismHilbh(A)→M s,p−h(A), (4.10)sending any quotient A → F to its kernel, where p is the Hilbert polynomial of A.Proof. This is the analogue of the classical result that all torsion-free rank 1sheaves are stable. Since Aη is a division algebra, any A-submodule of A hasthe same rank as A.Proposition 4.3.3. Under the assumptions in Lemma 4.3.2, if H1(X,A) = 0, thenthe natural morphism (4.10) is an open immersion.Proof. Let 0 → I → A → F → 0 be a quotient in Hilbh(A). Since F hascodimension ≥ 2 support, the map I → A induces an isomorphismI∨∨ → A∨∨ ∼= A41of A-modules. This shows that the quotient A → F is uniquely determined byits kernel I , which means that the morphism (4.10) is injective.Next, we show the morphism is étale. Let S be a scheme and S ⊂ S asquare-zero extension. Fix a flat family of A-module quotient0→ I → AS → F → 0on X × S. Suppose there exists an A-module extension I of I over S, then Iinduces a short exact sequence0→ I → I∨∨ → F → 0of AS-modules. Since I∨∨|S = I∨∨ ∼= AS , I∨∨ is a deformation of AS over S,which by our assumption, it must be AS . Note that taking double dual (−)∨∨in general does not preserve flatness, but it is flat over an open subset. So inour situation, I∨∨ is flat over S, which implies it is flat over S. Thus AS → Fis a deformation (as A modules) of the quotient AS → F . This is easy to seethat the converse is also true: any deformation of the quotient AS → F gives adeformation of I . The proof is completed.Consequently, the Hilbert scheme Hilbh(A) for polynomial h with deg(h) ≤1 carries a symmetric obstruction theory. We define the DT invariantDTh(A) =∫[Hilbh(A)]vir1.which is equal to the weighted Euler characteristic χvir(Hilbh(A).Example 4.3.4. Recall that any Azumaya algebra on a Calabi–Yau variety isagain Calabi–Yau. Via the correspondence between Azumaya algebras andgerbes, this gives a definition of Donaldson–Thomas invariants on gerbesX→ X over Calabi–Yau threefolds, which has been studied by [15].Remark 4.3.5. While we only construct a perfect obstruction theory when A isCalabi–Yau, it is expected that (some truncation of) the obstruction theory (4.2)is perfect for more general (X,A). For example, if A is of global dimension42≤ 2, then Ext3A(F ,F) = 0 for all F , thus the obstruction theory is auto-matically perfect. This gives an analogue of Donaldson invariants for certainnon-commutative surfaces.Another example is if A is of global dimension 3 and H i(A) = 0 for alli > 0, then it is easy to see that Ext3A(I, I) = 0 for any ideal sheaf I inHilbh(A) (see [25, Lemma 2]). This implies that Hilbh(A) carries a perfectobstruction theory via Proposition 4.3.3.Example 4.3.6. We consider the case X = Spec(C) is a point with A = A afinite-dimensional algebra of finite global dimension. Then (stable) coherent A-modules are exactly finite-dimensional (irreducible) representations of A. TheseDT invariants give virtual counts of irreducible representations. Unfortunately,the moduli space M s,n(A) is in general not projective unless n = 1.Our main interest is the pair (X,A) given by the quantum Fermat quinticthreefold described in 3.4. We verify that A satisfies the assumptions in Propo-sition 4.3.3. Since the graded algebra A is a domain, the stalk Aη at the genericpoint η ∈ X is also a domain. Combining with the fact that Aη is a finite-dimensional algebra over the function field k(X), Aη is a division algebra. AlsoX ∼= P3 and A is a sum of line bundles, thus H1(X,A) = 0.We will focus on Hilbert schemes of points on the quantum Fermat quinticthreefold. That is, the Hilbert polynomial h is a constant. We will writeZA(t) =∞∑n=0χvir(Hilbn(A)) tnfor the generating function of degree zero DT invariants of the quantum Fermatquintic threefold.43Chapter 5Local models of (X,A)In this chapter, we study zero-dimensional coherent A-modules, which we willsimply call finite-dimensional A-modules. Let Coh(A)fd be the category offinite-dimensional A-modules.5.1 Finite-dimensional A-modulesWe first note that the category Coh(A)fd can be studied locally. Any finite-dimensional A-module is supported (as a coherent sheaf on X ) at finitely manypoints.Lemma 5.1.1. For any finite-dimensional A-module F ,F =⊕x∈supp(F)Fx.Proof. We have F = ⊕x∈supp(F)Fx as coherent sheaves on X . Since the A-action on F is defined locally, Fx is naturally an A-submodule for each x.Furthermore, it is clear that the projection F → Fx is a morphism of A-modules.We see that F = ⊕x∈supp(F)Fx in Coh(A)fd.If we choose an affine open cover {Ui} of X , then for each Ui, A|Ui is givenby a non-commutative algebra, and Coh(A|Ui) is the category Mod(A|Ui) ofhonest finite-dimensional modules. Then the categories {Coh(A|Ui)}i can beregarded as an affine open cover of Coh(A).44For the quantum Fermat quintic threefold (X,A),X = Proj(C[x0, . . . , x4]/(x0 + . . .+ x4))is a hyperplane in P4. Let {Uij}i 6=j be the affine open cover of X defined byUij = (xixj 6= 0).Lemma 5.1.2. For each i 6= j, there is an isomorphism f : U01 → Uij such thatf∗(A|Uij ) is isomorphic to A, up to a possible change of primitive root q ∈ µ5.Proof. This is proved by an explicit computation. For each i 6= j, let σ ∈ S5 be apermutation mapping {i, j} to {0, 1}. Then σ defines a change of variables andinduces an automorphism f : U01 → Uij . In this case we can compute the non-commutative algebra f∗(A|Uij ) as below. We see that there exists a permutationσ so that f∗(A|Uij ) is equal to A, after a possible change of primitive rootq ∈ µ5.Therefore it is sufficient to study the non-commutative algebra A := A|U01 .Now we write down the non-commutative algebra A explicitly. By definition,A is the degree zero part of the graded algebra(C〈t0, . . . , t4〉/(∑kt5k, titj − qijtjti))[ 1t50,1t51].Then we can writeA = C〈u1,1u1, u2, u3, u4〉/(1 +4∑k=1u5k, uiuj − qijujui), (5.1)where ui = (tit40)/(t50) and(qij)=1 q3 q4 q3q2 1 q4 q4q q 1 q3q2 q q2 1 . (5.2)455.2 Simple A-modules and the Ext-quiverAs seen in [21], semistable objects in a Calabi–Yau-3 category should be locallygiven by representations of quivers with potential, and the quivers are obtainedby the Ext-quivers of stable objects. See also [34] for the case of the category ofcoherent sheaves on a Calabi–Yau threefold.For the category Coh(A)fd, a finite-dimensional A-module is alwayssemistable, and it is stable if and only if it is simple. We consider the natu-ral forgetful mapHilbn(A)→Mn(A) := M ss,n(A).This is the analogue of Hilbert–Chow map. The closed points of the coarsemoduli scheme Mn(A) correspond to polystable A-modules, that is, semisimpleA-modules.As shown in the previous section, we only need to consider finite-dimensionalA-modules.Lemma 5.2.1. All simple A-modules have dimension 1 or a multiple of 5. Further-more, there are exactly 5 one-dimensional A-modules given by(u1, u2, u3, u4) = (ξ, 0, 0, 0),where ξ ∈ C and 1 + ξ5 = 0.Proof. Let V be a d-dimensional simple A-module. We abuse the notation andwrite ui ∈ EndC(V ) for the action of ui ∈ A. The relations uiuj = qijujuiimply that for each i, ker(ui) ⊂ V is an invariant subspace. Thus for each i, uiis either 0 or invertible. Next, taking determinants of the relations yieldsdet(ui) det(uj) = qdij det(uj) det(ui).If d is not a multiple of 5, then qdij 6= 1. Since u0 is invertible, det(ui) = 0 forall i 6= 0 and thus ui = 0. We conclude that d = 1 and u0 acts as a scalar ξ ∈ Cwith 1 + ξ5 = 0.We observe that the 5 one-dimensional simple A-modules are supported atthe point p01 = [1 : −1 : 0 : 0 : 0] ∈ X , and they are all simple A-modules46supported at p01. We denote these 5 simple modules by Ei’s, which correspondsto ξ = −qi, for i ∈ Z/5.Definition 5.2.2. The Ext-quiver Q associated to {Ei}i is the quiver whosevertex set Q0 = {Ei}i and the number of arrows from Ei to Ej is equal to thedimension of Ext1A(Ei, Ej).Proposition 5.2.3. The Ext-quiver Q associated to (E0, E1, . . . , E4) isE0E2 E1E3E4  ooooiiTTTTOO;;;;44## ##(5.3)Proof. The group Ext1A(Ei, Ej) is classified by extensions 0 → Ej → F →Ei → 0. The ui-action on F is of the formu0 =(−qj 00 −qi)and uk =(0 ak0 0)for k > 0.The relations implies that ak(q1kqi − qj) = 0. Thusdim Ext1A(Ei, Ej) = the number of k’s such that q1k = qj−i.We denote the arrows of Q by ai, ci : Ei → Ei+3 and bi : Ei → Ei+4. Fromthe construction of the Ext-quiver Q, we see that the arrows ai’s, bi’s, and ci’scorrespond to the actions of u2, u3, and u4 respectively. Then the q-commutingrelations translate intoaibi+1 − q4bi+4ai+1 = 0,bici+2 − q3ci+1bi+2 = 0,aici+2 − q4ciai+2 = 0.47As expected, these relations can be patched into a potentialW = bac− q4bca,wherea = a0 + a1 + a2 + a3 + a4,b = q4b0 + b1 + qb2 + q2b3 + q3b4,c = c0 + c1 + c2 + c3 + c4.In the next section, we will show that the quiver (Q,W ) with potential in factgives a local model of A near the point p01 ∈ X .5.3 Local models of (X,A)Let (Q,W ) be the quiver with potential from the previous section.Lemma 5.3.1. The Jacobi algebra Jac(Q,W ) is isomorphic toC〈e, u, v, w〉(qij)/(e5 − 1),where (qij) is the quantum parameters (5.2).Proof. Consider the element e = e0 + qe1 + q2e2 + q3e3 + q4e4 ∈ CQ, where eiis the idempotent corresponding to the vertex i, and u = a0 +a1 +a2 +a3 +a4,v = b0+b1+b2+b3+b4, and w = c0+c1+c2+c3+c4. It is clear that e5 = 1 andthe elements e, u, v, w satisfy the appropriated q-commuting relations. ThereforeC〈e, u, v, w〉(qij)/(e5 − 1) is naturally a subalgebra of Jac(Q,W ).To show that they are isomorphic, it is sufficient to show that the element egenerates ei for all i. For each k, we have ek = e0 +qke1 +q2ke2 +q3ke3 +q4ke4.So 1ee2e3e4 =1 1 1 1 11 q q2 q3 q41 q2 q4 q q31 q3 q q4 q21 q4 q3 q2 qe0e1e2e3e4 .48The matrix in the middle is a Vandermonde matrix, which is invertible. Thiscompletes the proof.Remark 5.3.2. One alternative description of the Jacobi algebra Jac(Q,W ) isas follows. Consider the algebraC〈u, v, w〉q := C〈u, v, w〉/ (uv − q4vu, vw − q4wv,wu− q4wu) ,which is the Jacobi algebra of a quantized affine 3-space ([11]). Let G = µ5 withan action on C〈u, v, w〉q defined byq · (u, v, w) = (q3u, q4v, q3w).Then the Jacobi algebra Jac(Q,W ) is isomorphic to the crossed productC〈u, v, w〉qoµ5, note that here the v corresponds to q4b0+b1+qb2+q2b3+q3b4.From this perspective, the Jacobi algebra Jac(Q,W ) is a quantization of the orb-ifold [C3/µ5].The Jacobi algebra Jac(Q,W ) contains a subalgebraC[x, y, z] := C[u5, v5, w5] ⊂ Z(Jac(Q,W ))and is a finite C[x, y, z]-module. Thus Jac(Q,W ) can be regarded as a sheafJ of non-commutative algebras on C3 = SpecC[x, y, z]. There is a canonicalembeddingU01 ↪→ C3,(x2x0,x3x0,x4x0)= (x, y, z) (5.4)which maps the special point p01 to the origin. For any subset U ⊂ U01, we willsimply identify it with its image in C3 without further comment.Theorem 5.3.3. For any point p ∈ U01, there is an analytic open neighborhoodU ⊂ U01 of p such that there is a (non-unique) isomorphismA|U ∼= J |Uof sheaves of non-commutative algebras on U .49Proof. Both sheaves A and J are locally free of rank 625, and we can write themdown explicitly,A|U = OU 〈u1, u2, u3, u4〉(qij)/(1 +4∑k=1u5k, u52 − x, u53 − y, u54 − z)andJ |U = OU 〈e, u, v, w〉(qij)/ (e5 − 1, u5 − x, v5 − y, w5 − z) ,where x, y, z ∈ H0(U,OU ) are (holomorphic) functions corresponding to thecoordinates of U ⊂ C3.Suppose U ⊂ C3 is an analytic open subset such that 5-th roots of theholomorphic function −1 − x − y − z are well-defined, that is, there exists anelementf(x, y, z) ∈ H0(U,OU ) such that f(x, y, z)5 = −1− x− y − z.Then we define a morphism of sheaves of non-commutative algebrasA|U → J |U , (u1, u2, u3, u4) 7→(f(x, y, z)e, u, v, w).This defines an isomorphism since f(x, y, z) is non-vanishing on U and thus isinvertible.Finally, U01 is the open subset of C3 defined by x+ y + z 6= −1. Thereforeit can be covered by analytic open subsets satisfying the property above.Next, we analyze the Jacobi algebra Jac(Q,W ) in more detail.Proposition 5.3.4. Let p = (x0, x1, x2) ∈ C3 with x0 6= 0. Then there is ananalytic open neighborhood U ⊂ C3 of p such thatJ |U ∼= M5(C)⊗C(OU [v, w]/ (v5 − y, w5 − x) ),whereM5(C) is the ring of 5-by-5 matrices. Similar results also hold for points withy0 6= 0 or z0 6= 0.50Proof. Recall thatJ |U = OU 〈e, u, v, w〉(qij)/ (e5 − 1, u5 − x, v5 − y, w5 − z) ,Since x0 6= 0, we can choose an analytic open neighborhood U of p such thatthere exists a holomorphic function f(x, y, z) on U such that f(x, y, z)5 = x.We consider a change of coordinatese1 = e, e2 =uf, v = (e31e22)v, w = (e41e32)w.Then J |U is generated by e1, e2, v, w since e52 = 1. A straightforward computa-tion shows that the elements v, w are, in fact, lying in the center Z(J |U ).Consequently, we can writeJ |U = OU 〈e1, e2〉(q12)[v, w]/ (e51 − 1, e52 − 1, v5 − y, w5 − z)∼=(C〈e1, e2〉(q12)/(e51 − 1, e52 − 1))⊗C(OU [v, w]/(v5 − y, w5 − z)).To see that the finite-dimensional Frobenius algebraC〈e1, e2〉/ (e1e2 − q3e2e1, e51 − 1, e2 − 1)is isomorphic to M5(C), one may verify that the morphism defined bye1 7→1 0 0 0 00 q 0 0 00 0 q2 0 00 0 0 q3 00 0 0 0 q4 , e2 7→0 0 1 0 00 0 0 1 00 0 0 0 11 0 0 0 00 1 0 0 0is an isomorphism.Remark 5.3.5. We may consider the finite covering pi : C3 → C3, pi(x, y, z) =51(x, y5, z5). Then for any U ⊂ C3, we haveM5(C)⊗C(OU [v, w]/(v5 − y, w5 − z))∼=M5(C)⊗Cpi∗Opi−1(U) = pi∗(M5(C)⊗COpi−1(U))=pi∗(M5(OC3)|pi−1(U)).This gives an equivalence between coherent J -modules on U and coherentM5(OC3)-modules on pi−1(U).Now we consider a stratificationC3 = C3(0)∐C3(1)∐C3(2)∐C3(3),where C3(i) consists of points with exactly i coordinates being non-zero. Letp(x0, y0, z0) ∈ C3(2). For simplicity we assume z0 = 0, then Proposition 5.3.4shows that there is an analytic neighborhood U of p such thatJ |U ∼=(M5(C)⊕5)⊗C(OU [w]/ (w5 − z) ).Similarly, for point p ∈ C3(3), there is an analytic neighborhood U of p such thatJ |U ∼=(M5(C)⊕25)⊗COU .Finally, we recall that there is an open cover {Uij}i 6=j such that all sheavesA|Uij of non-commutative algebras are (canonically) isomorphic. There is anatural stratificationX = X3(0)∐X(1)∐X(2)∐X(3), (5.5)where X(i) consists of points with exactly i + 2 coordinates (of P4) being non-zero. It is clear that the canonical isomorphism U01 ∼= Uij and the canonicalembedding (5.4) preserve the strata.Definition 5.3.6. Let p ∈ X . An analytic chart U of p is an analytic open52neighborhood U ⊂ X of p with an embedding U → C3 mapping p to theorigin.Putting all above results together, we obtain the following theorem.Theorem 5.3.7. We define the sheaves of non-commutative algebras on C3J(0) = J ,J(1) = M5(C)⊗C(OC3 [v, w]/ (v5 − y, w5 − z) ),J(2) =(M5(C)⊕5)⊗C(OC3 [w]/ (w5 − z) ),J(3) =(M5(C)⊕25)⊗COC3 .Then for all i and any point p ∈ X(i), there is an analytic chart U → C3 of p suchthatA|U ∼= J(i)|U .Moreover, if p ∈ X(1), then the chart U → C3 maps U ∩X(1) to the locus (y = z =0); and if p ∈ X(2), then the chart U → C3 maps U ∩X(2) to the locus (z = 0).Remark 5.3.8. From the construction of the chart U → C3, we see that ifp ∈ X(1), then the chart maps U ∩ X(1) to the locus (y = z = 0). Similarly,if p ∈ X(2), then the chart maps U ∩X(2) to the locus (z = 0). Besides, sincethe charts are constructed to make 5-th roots of certain holomorphic functionwell-defined, we can choose finitely many charts to cover X(i)’s for each i.In particular, if p ∈ X(i), then the category Coh(A)p of coherent A-modulessupported at p is equivalent to the category of J(i)-modules supported at theorigin.Corollary 5.3.9. Let p ∈ X(i), i 6= 0. There are 5i−1 simple A-modules supportedat p and all of them are of dimension 5. Furthermore, the Ext-quiver associated tothese simple modules is the quiver consisting of 5i−1 vertices, and three loops at eachvertex.Remark 5.3.10. For i 6= 0, the sheaves J(i) of algebras on C3 are given by the53non-commutative C[x, y, z]-algebrasM5(C[x, y, z])⊕5i−1,which are Morita equivalent to the commutative algebras C[x, y, z]⊕5i−1 . More-over, these algebras are Jacobi algebras of a quiver with potential. Therefore,one may interpret Theorem 5.3.7 as an explicit (analytic) local model of (X,A)by quivers with potential.54Chapter 6DT invariants of the quantumFermat quintic threefoldIn this chapter, we aim to give a computation of the generating functionZA(t) =∞∑n=0χvir(Hilbn(A)) tnof degree zero DT invariants of the quantum Fermat quintic threefold.6.1 Analytification and weighted Euler characteristicsFor technical reasons, we consider the category C of analytic schemes carryingan algebraic constructible function. Objects in C are pairs (U, ν) such that U isan analytic open subset of an algebraic scheme X and ν is a function U → Zthat extends to an algebraic constructible function ν : X → Z. Morphisms from(U1, ν1) to (U2, ν2) are analytic morphisms f : U1 → U2 such that ν2 ◦ f = ν1.Also, the product exists in the category C given by(U1, ν1)× (U2, ν2) = (U1 × U2, ν1 × ν2),where (ν1 × ν2)(x1, x2) = ν1(x1)ν2(x2).It is clear that if (U1, ν1) and (U2, ν2) are isomorphic in C, then χ(U1, ν1) =55χ(U2, ν2).Lemma 6.1.1. Let X and Y be schemes. If X and Y are analytic local isomorphic,that is, there exist analytic open covers {Uα}α of X and {Vα}α of Y such that foreach α, there is an analytic isomorphism fα : Uα → Vα. Thenχvir(X) = χvir(Y ).Proof. Since the Behrend function depends only on the analytic topology of ascheme, the analytic isomorphism fα induces an isomorphismfα : (Uα, νUα)→ (Vα, νVα)in C. Then the result follows from the fact that Euler characteristics can becomputed from an open cover.Definition 6.1.2. Let f : X → Y be a morphism of schemes, F a scheme withconstructible functions µ : X → Z, ν : F → Z. We sayf : (X,µ)→ Yis an analytic local fibration with fibre (F, ν) if there is an analytic open cover{Uα}α of Y such that for each α, there is an isomorphism(f−1(Uα), µ) ∼= (Uα, 1)× (F, ν)in C. Note that f : (X,µ)→ (Y, 1) is generally not a morphism in C.Lemma 6.1.3. If f : (X,µ) → Y is an analytic local fibration with fibre (F, ν),thenχ(X,µ) = χ(Y ) · χ(F, ν).Proof. It is easy to see that for any c ∈ Z, f : µ−1(c) → Y is an analytic-localfibration with fibre ν−1(c), thusχ(µ−1(c)) = χ(Y ) · χ(ν−1(c)).56Since our local models of (X,A) are given in analytic topology, it is notobvious that it gives an analytic isomorphism between algebraic moduli spaces.For the rest of the section, let X be a quasi-projective smooth variety and Aa locally free sheaf of non-commutative algebras on X .Definition 6.1.4. The Hilbert–Chow map is the compositionHilbn(A)→Mn(A)→Mn(X) ∼= Symn(X),where the middle morphism sends a finite-dimension A-module to its underlyingcoherent sheaves on X , which has zero-dimensional support and is of length n.For any analytic or algebraic subset S ⊂ X , we define the fiber productHilbn(A)S   //Hilbn(A)Symn(S)  // Symn(X)in the appropriate category. In other words, Hilbn(X,A)S parameterizes A-module quotients supported in S. Clearly if S is analytic open in X , thenHilbn(A) is also analytic open in Hilbn(A). Also if S is a locally closed sub-scheme of X , then Hilbn(A) is also a locally closed subscheme in Hilbn(A)We first prove the equivalence between algebraic and analytic finite-dimensional A-modules.Lemma 6.1.5. The analytification defines an equivalenceCoh(A)fd ∼= Coh(Aan)fdof categories, where Coh(Aan)fd is the category of analytic coherent sheaves on Xwith an Aan-action.Proof. As mentioned before, this statement is Zariski local onX . We may assumeX = Spec(R) is affine, and then A is a non-commutative R-algebra.57First, we may choose a compactification X ⊂ X so we can apply GAGAtheorem [30]. Since analytification preserves supports of coherent sheaves, itinduces an equivalence between Coh(OX)fd and Coh(OanX )fd.While OanX and Aan are not in the category Coh(OanX )fd, for any Fan inCoh(OanX )fd, the ring HomOanX (Fan,Fan) is naturally an R-algebra. By defini-tion, an Aan-module is given by an analytic coherent sheaf Fan with a morphismA→ HomOanX (Fan,Fan)of R-algebras, which must be algebraic. Therefore the analytificationCoh(A)fd ∼= Coh(Aan)fdis an equivalence of categories.Proposition 6.1.6. Let U be an analytic open subset of X . Then analytificationof Hilbn(A)U is the analytic moduli space parameterizing analytic cyclic Aan|U -modules.Proof. We first note that such analytic moduli space exists. Since Aan is locallyfree, there is a Quot space Q parameterizing quotients of Aan, and we takeMto the closed subspace of Q consisting of points [Aan → Fan] such that thekernel is Aan-invariant.Since any family of algebraic A-modules is analytic, there is a canonicalmorphismHilbn(A)U →M, (6.1)which is bijective from the previous lemma. Note that the previous lemma alsoworks for any family A-modules over a proper scheme (where GAGA applies). Itgives an equivalence between infinitesimal deformations of algebraic and analyticA-modules. In other words, (6.1) is étale and hence is an analytic isomorphism.Corollary 6.1.7. Let A1 and A2 be two coherent sheaves of non-commutative alge-bras on X . Suppose there is an analytic open subset U ⊂ X such that A1|U ∼= A2|U .58Then there is an analytic isomorphismHilbn(A1)U ∼= Hilbn(A2)U .6.2 Stratifications of Hilbert schemesFor any locally closed subset Z ⊂ X , we have Hilbn(A)Z , the locally closedsubscheme of Hilbn(A) parameterizingA-module quotients supported in Z withthe induced Hilbert–Chow map Hilbn(A)Z → SymnZ . LetZAZ (t) =∞∑n=0χvir(Hilbn(A),Hilbn(A)Z)tnbe the generating function.Proposition 6.2.1. Let X = X1∐X2∐. . .∐Xr be a stratification of X . ThenZA(t) =r∏i=1ZAXi(t).Proof. Write pin for the set of r-tuples (n1, . . . , nr) of non-negative integers suchthat n1 + . . .+ nr = n. The stratification X =∐iXi induces a stratificationHilbn(A) =∐(n1,...,nr)∈pinHilbn(A)(n1,...,nr),where Hilbn(A)(n1,...,nr) parameterizes quotients A → F such that F|Xi is oflength ni for all i. ThenHilbn(A)(n1,...,nr) ∼=r∏i=1Hilbni(A)Xi . (6.2)It remains to show that the isomorphism (6.2) induces an isomorphism(Hilbn(A)(n1,...,nr), νn) ∼= r∏i=1(Hilbni(A)Xi , νni)59in C, where νn is the Behrend function on Hilbn(A). Let p = [A → F ] ∈Hilbn(A) be a closed point. We choose analytic open subsets Ui ⊂ X such thatUi ∩ Uj = ∅ andsupp(F) ∩Xi ⊂ Ui.Then Hilbn(A)∐i Uiis analytic open in Hilbn(A)∐i Ui⊂ Hilbn(A), andHilbn(A)∐i Ui∼=∏iHilbni(A)Ui ⊂∏iHilbni(A).Thus(Hilbn(A)∐i Ui, νn)=(Hilbn(A)∐i Ui, νHilbn(A)∐i Ui)∼=∏i(Hilbni(A)Ui , νHilbni (A)Ui)=∏i(Hilbni(A)Ui , νni).which completes the proof.As in the classical case, Hilbn(X,A)S has a standard stratification indexedby partitions of n. The Hilbert–Chow map sends the stratum Hilbn(X,A)S,(n)to the diagonal S ↪→ Symn(S).Proposition 6.2.2. Suppose the induced morphism(Hilbn(A)S,(n), νn)→ Sis an analytic-local fibration with fibre (Fn, µn). ThenZAS (t) =( ∞∑n=0χ(Fn, µn) tn)χ(S).Proof. This is the analogue of [7, Theorem 4.11], but we use analytic open coverinstead of étale cover. The main idea is that there is a stratificationHilbn(A)S =∐α`nHilbn(A)S,α60and for each α = (α1 ≤ . . . ≤ αr) ` n,Hilbn(A)S,α ⊂r∏i=1Hilbαi(A)S,(αi).By the same argument in Proposition 6.2.1, we get that(Hilbn(A)S,α, νn) ⊂ r∏i=1(Hilbαi(A)S,(αi), ναi)is an immersion in C, and by assumption, the right hand side is an analytic-localfibration over S with fibre (Fαi , ναi). Then the formula follows from a standardcalculation.6.3 Calculation of DT invariantsLet (X,A) be the quantum Fermat quintic threefold. We apply Proposition 6.2.1to the stratification (5.5) of X and obtainZA(t) =3∏i=0ZAX(i)(t).Theorem 6.3.1. For each i,(Hilbn(A)X(i),(n), νn)→ X(i)is an analytic local fibration with fibre (Hilbn(C3,J(i))0, µn), where J(i)’s are thesheaves of algebras on C3 defined in Theorem 5.3.7, and µn is the Behrend functionof Hilbn(C3,J(i)).Proof. We only prove for the case i = 1, as the proofs of the other cases aresimilar. First, we choose an analytic open cover {Uα} of charts. Since A|Uα ∼=J(1)|Uα , we have an analytic isomorphismHilbn(A)Uα ∼= Hilbn(C3,J(1))Uα ,61by Corollary 6.1.7, which gives an isomorphism(Hilbn(A)Uα , νn) ∼= (Hilbn(C3,J(1))Uα , µn) (6.3)in C. Let Z be the locus of C3 defined by y = z = 0. Observe thatJ(1) = M5(C)⊗C(OC3 [v, w]/ (v5 − y, w5 − z) )is invariance under the translation of x, the Hilbert–Chow map(Hilbn(C3,J(1))Z,(n), µn)→ Zis a Zariski local fibration with fibre(Hilbn(C3,J(i))0, µn). Here we use thefact that the Behrend function is constant on orbits of a group action.Finally, since the chart Uα → C3 map X(1) into Z (see Remark 5.3.8), theisomorphism 6.3 induces an isomorphism(Hilbn(A)Uα∩X(1) , νn) ∼= (Hilbn(C3,J(1))Uα∩Z , µn)Then the theorem follows.Next we deal with the sheaves J(i) of algebras before we state our maintheorem. Recall that for i 6= 0, J(i) can be written as direct sums of M5(OC3)(see Remark 5.3.5).Lemma 6.3.2. If n is not a multiple of 5, then Hilbn(C3,M5(OC3))= ∅. Forn = 5k, there is a canonical isomorphismHilb5k(C3,M5(OC3)) ∼= Quotk(C3,O⊕5C3 ).Proof. The Morita equivalenceCoh(C3)→ Coh(M5(OC3))is given by C⊕5⊗C−, where C⊕5 is the canonical representation of M5(C). Thisimplies that the dimension of a M5(OC3)-module must be a multiple of 5.62The isomorphism is a direct consequence of the following fact: Let R be acommutative ring, and M be an R-module. For any n, we consider the simpleMn(R)-module R⊕n. Then a morphisms : Mn(R)→ R⊕n ⊗Mof Mn(R)-modules is surjective if and only if the induced morphismδ(s) : R⊕n ↪→Mn(R) s−→ R⊕n ⊗M →Mof R-modules is surjective, where the first map is the diagonal map, and the lastmap is defined by (r1, . . . , rn)⊗m 7→∑i rim.Theorem 6.3.3. We haveZA(t) = ZQ,W (t)10 ·M(−t5)−50,whereM(t) is the MacMahon function.Proof. We use Proposition 6.2.2 with Theorem 6.3.1 to obtainZA(t) =3∏i=0(ZC3,J(i)0 (t))χ(X(i)).For i 6= 0, we haveZC3,J(i)0 (t) = ZC3,M5(O)⊕5i−10 (t)=∞∑k=0χvir(Quotk((C3)∐5i−1 ,O⊕5),Quotk((C3)∐ 5i−1 ,O⊕5)0)t5k=( ∞∑k=0χvir(Quotk(C3,O⊕5),Quotk(C3,O⊕5)0)t5k)5i−1.For the first equality, see Remark 5.3.5. The second equality is given by theprevious lemma, and the third equality is a standard result about Hilbert schemesof a disjoint union of schemes.63Now, DT invariants of Quot schemes of points are well-known (for example,see [4]).∞∑k=0χvir(Quotk(C3,O⊕5),Quotk(C3,O⊕5)0)tk = M(−t)5.We conclude thatZA(t) = ZC3,J0 (t)χ(X(0) ·(M(−t5)5)χ(X(1))+5χ(X(2))+25χ(X(3)),with χ(X(0)) = 0 and χ(X(1)) + 5χ(X(2)) + 25χ(X(3)) = −10.Finally, recall that J(i)’s also are local models of J on strata C3(i) of C3. Werepeat all above arguments to (C3,J ) which lead toZC3,J (t) = ZC3,J0 (t)χ(C3(0)) ·(M(−t5)5)χ(C3(1))+5χ(C3(2))+25χ(C3(3))= ZC3,J0 (t).since χ(C3(3)) = χ(C3(3)) = χ(C3(3)) = 0. The sheaf J on C3 is given by theJacobi algebra Jac(Q,W ), and by definition, finite-dimensional quotients of Jare exactly framed representations of (Q,W ) with framing vector (1, 1, 1, 1, 1).Hence by our definition, ZC3,J (t) = ZQ,W (t).For the generating function ZQ,W , we have seen that (Q,W ) can be viewedas a quantization of an orbifold [C3/µ5], and DT invariants of an orbifold isknown to be related to colored plane partitions [35]. We will discuss more of thisin Chapter 7, and show that ZQ,W can be computed using some combinatorics.As a final remark, we give a possible geometric interpretation of this result.Recall that finite-dimension A-modules are of dimension 1 or 5. We denote Mdspthe moduli space of d-dimensional A-modules. Then Theorem 5.3.7 implies that(a) There is a morphism M1sp → X(0) which is a µ5-torsor.(b) M5sp is smooth, and there is a (ramified) covering M5sp → X \X(0), whichis 5i−1-to-1 on the stratum X(i).64In particular,χ(M5sp) = χ(X(1)) + 5χ(X(2)) + 25χ(X(3)).The factor M(−t5)−50 can be expressed asZM5sp,M5(O)(t) =∞∑k=0χvir(Quotk(M5sp,O⊕5))t5kOn the other hand, the M1sp consists of 50 points, and if we consider the Ext-quiver Q˜ associated to M1sp, then Q˜ is the disjoint union of 10 copies of thequiver Q. Therefore we can writeZQ,W (t)10 = ZQ˜,W˜ (t)andZA(t) = ZQ˜,W˜ (t) · ZM5sp,M5(O)(t)This strongly suggests that the DT invariants of the Calabi–Yau-3 categoryCoh(A)fd can be computed directly from the moduli space of simple objectsin Coh(A)fd and the Ext-quivers between simple objects.65Chapter 7The quiver Q and colored planepartitions7.1 Colored plane partitionsIn this section, we recall the notion of colored plane partitions and their relationwith orbifolds. These results are taken from [35], [9] (see also [14]).Let G = µr be the finite group of r-th roots of unity in C. We consider theµr-action on C3 with weights (a, b, c). We will denote this action µ5(a, b, c). Weidentify Gˆ, the abelian group of characters, with Z/rZ.Definition 7.1.1. A µr(a, b, c)-colored plane partition is a plane partition pi ∈ Pwith the coloring K : pi → Z/rZ defined byK(i, j, k) = ai+ bj + ck.For each color i, let |pi|i be the number of boxes in pi with color i.We define the generating function of µr(a, b, c)-colored plane partitionsZµr(a,b,c)PL (t0, . . . , tr−1) =∑pi∈Pt|pi|00 · · · t|pi|r−1r−1 .For the µr(a, b, c)-action, we consider the McKay quiver Qr(a, b, c) whose66vertices correspond to irreducible representations of µ5. Thus the setQr(a, b, c)0of vertices is identified with µˆr ∼= Z/rZ. Arrows of Qr(a, b, c) arexi : i→ i+ a, yi : i→ i+ b, zi : i+ cfor all vertex i. There is a natural potentialW = yxz − yzx,where x =∑i xi, y =∑i yi, and z =∑i zi.Given any plane partition pi, the µr(a, b, c)-coloring defines a dimensionvector|pi| := (|pi|0, . . . , |pi|r−1) ∈ Z⊕Qr(a,b,c)0 .On the other hand, the µr(a, b, c)-action defines an orbifold X = [C3/µr].For any ρ ∈ K0(Rep(µr)), we consider the Hilbert scheme Hilbρ(X ) parameter-izing µ5-invariant subschemes Z ⊂ C3 such that the induced µ5-representationon OZ is in the class ρ. The group K0(Rep(µr)) is canonically identified withZ⊕Qr(a,b,c)0 . Furthermore, it is well-known that the Hilbert scheme Hilbd(X )is isomorphic to the fine moduli space Md,e0(Qr(a, b, c),W ) of framed repre-sentations of the quiver Qr(a, b, c) with potential W , with the framing vectore0 = (1, 0, . . . , 0). We define the generating functionZX (t0, . . . , tr−1) =∑d=(d0,...,dr−1)χvir(Hilbd(X )) td00 · · · tdr−1r−1of DT invariants of the orbifold X , which is equal to the generating functionZQr(a,b,c),W,e0 in our notation.Proposition 7.1.2 ([9]). The generating function ZX is, up to signs, given by thegenerating function Zµr(a,b,c)PL of colored plane partitions. More specifically, we haveZX (t0, . . . , tr−1) =∑pi∈P(−1)|pi|0+〈|pi|,|pi|〉 t|pi|00 · · · t|pi|r−1r−1 ,where 〈−,−〉 is the bilinear form associated to the quiver Qr(a, b, c).677.2 Multi-colored plane partitionsLet Q = (Q0, Q1) be a quiver with a labeling of arrows ` : Q1 → {x, y, z}. Wedenote S(Q0) be the set of non-empty subsets of Q0.Definition 7.2.1. A Q-multi-colored plane partition consists of a plane partitionpi ∈ P with a multi-coloringK : pi → S(Q0)such that for any arrow a : v → w labeled with x, if w ∈ K(i, j, k) for some(i, j, k) ∈ pi, then v ∈ K(i − 1, j, k), and similar conditions hold for arrowslabeled with y and z. Note that there are many different Q-multi-colorings onone plane partition pi.Given a Q-multi-colored plane partition pi = (pi,K), it associates a dimen-sion vector w(pi) :=(wv(pi))v∈Q0 defined bywv(pi) = number of boxes (i, j, k) ∈ pi such that the color v ∈ K(i, j, k).For any dimension vector d ∈ Z⊕Q0 , we denote by nQ(d) the number of Q-multi-colored plane partitions with dimension vector d. We define the generatingfunctionZQPL(t) =∑dnQ(d) tdof Q-multi-colored plane partitions, where we write t for (tv)v∈Q0 , d = (dv)v∈Q0and td =∏d∈Q0 tdvv .For simplicity, from now on we only consider the quiver (Q,W ) (5.3) withpotential from the quantum Fermat quintic threefold. We will rearrange the68vertices and arrows, and write13 240   ooiiiiTTOOOO ;;4444##with outer arrows xi, yi : i → i + 1 and inner arrows zi : i + 3. The inducedpotential isW = yxz − qyzx,where x =∑xi, y =∑yi, and z = z0 + qz1 + q2z2 + q3z3 + q4z4.Our main theorem of the section is to associate DT invariants of (Q,W )with Q-multi-colored plane partitions.Theorem 7.2.2. The generating function ZQ,W is, up to signs, given by the gener-ating function ZQPL of Q-multi-colored plane partitions. More specifically, we haveZQ,W (t) =∑d(−1)|d|+〈|d|,|d|〉 nQ(d) td,where |d| = ∑i di and 〈−,−〉 is the bilinear form associated to the quiver Q.We leave the proof of Theorem 7.2.2 to the next section.Observe that this quiver is the same as the McKay quiver of µ5(1, 1, 3).Recall that the vertices of the McKay quiver correspond to the irreducible repre-sentations of µ5, in which there is a distinguished one, the trivial representation.However, the vertices of Q are simple A-modules. There are exactly 5 waysto identify the quiver Q with the McKay quiver of µ5(1, 1, 3), depending on achoice of a vertex in Q0. In other words, for each v ∈ Q0, there is a uniquebijection αv : Q0 → Z/5Z with αv(v) = 0 identifying the quiver Q with theMcKay quiver of µ5(1, 1, 3).Definition 7.2.3. Let v ∈ Q0 be a vertex. A (Q, v)-colored plane partition is a69plane partition pi with the coloring Kv : pi → Q0 making αv ◦Kv : pi → Z/5Zthe µ5(1, 1, 3)-coloring of pi.Given five plane partitions piv indexed by Q0. We may define a Q-multi-colored plane partition by taking pi to be the union of piv ’s with multi-coloringK(i, j, k) = {Kv(i, j, k) : (i, j, k) ∈ piv} ⊂ Q0.Lemma 7.2.4. Any Q-multi-colored plane partition is uniquely determined by the(Q, v)-colored plane partitions piv for v ∈ Q0.Proof. This follows from the fact that Q can be identified with a McKay quiverof a group action on C3. To be more specific, the quiver Q satisfies the followingproperties:(a) For each vertex i, there are exactly 3 arrows starting at i which are labeledwith x, y, z; and there are exactly 3 arrows ending at i, also labeled withx, y, z.(b) For each vertex i, the target of any non-trivial composition of arrows start-ing at i depends only on the numbers of arrows labeled with x, y, z.Thus given any plane partition pi with a Q-multi-coloring K , we can define pivto bepiv ={(i, j, k) ∈ pi : Kv(i, j, k) ∈ K(i, j, k)}.It is easy to check that piv is a plane partition and the union of piv ’s is pi.Corollary 7.2.5. We haveZQPL(t0, t1, t2, t3, t4) =∏i∈Z/5ZZµ5(1,1,3)PL (ti, ti+1, ti+2, ti+3, ti+4).This reduces the computation of numbers of Q-multi-colored plane parti-tions to the ones for µ5(1, 1, 3)-colored plane partitions.Remark 7.2.6. Unfortunately, the signs in the DT invariants ZQ,W and Z [C3/µ5]70do not agree. That is,ZQ,W (t0, t1, t2, t3, t4) 6=∏i∈Z/5ZZ [C3/µ5](ti, ti+1, ti+2, ti+3, ti+4),and there is no obvious modification (changes of variables or taking the productin some Poisson algebra) to equalize them.Remark 7.2.7. Both ZQ,W and Z [C3/µ5] are DT invariants of the quiver (Q,W )with potential, with different framing vectors. To put it another way, they are DTinvariants of the same Calabi–Yau-3 category with different stability conditions.Thus there should be a formula (“wall-crossing”) connecting two series ZQ,Wand Z [C3/µ5]. This can be achieved by, for example, Joyce–Song’s generalizedDT invariants [19]. However, it does not reduce to a simple formula (at least notobvious to us) due to the fact that the fact that the Euler pairing χQ(−,−) ofthe quiver Q is not trivial. To be more precise, the formula in [19, Corollary 7.24]does not hold.Here we write down the series ZQ,W up to degree 5, where we denotet(a0,...,a4) =∑i ta0i · · · ta4i+4:ZQ,W (t) = 1 + t(1,0,0,0,0) + 3t(1,1,0,0,0) − 2t(1,0,1,0,0)+ 3t(1,2,0,0,0) + t(2,0,1,0,0) − 8t(1,1,1,0,0) + 8t(1,1,0,1,0)+ t(1,3,0,0,0) + 3t(2,1,1,0,0) − 12t(1,2,1,0,0) + 7t(1,1,2,0,0)− 12t(1,2,0,1,0) + 5t(1,1,0,2,0) − 34t(1,1,1,1,0)− 3t(2,2,1,0,0) − 4t(2,1,2,0,0) − 6t(1,3,1,0,0) + 18t(1,2,2,0,0) − 2t(1,1,3,0,0)+ 8t(1,3,0,1,0) + 10t(1,2,0,2,0) + 20t(2,1,1,1,0) + 56t(1,2,1,1,0) + 35t(1,1,2,1,0)− 54t(1,1,1,2,0) − 171t0t1t2t3t4 +O((t0, t1, t2, t3, t4)6)Take t = t0 = t1 = t2 = t3 = t4, we obtainZQ,W (t) = 1 + 5t+ 5t2 + 20t3 − 210t4 − 131t5 +O(t6).71We conclude thatZA(t) =(ZQ,W (t))10 ·M(−t5)−50= 1 + 50t+ 1175t2 + 17450t3 + 184275t4 + 1450740t5 +O(t6).7.3 Proof of Theorem 7.2.2We mainly follow the same method in the computation of DT invariants on C3[7]. The path algebra CQ isCQ = C〈e, u, v, w〉/(e5 − 1, eu− que, ev − qve, ew − q3we).There is a standard T = (C∗)3-action on CQ given by(λ1, λ2, λ3) · (u, v, w) = (λ1u, λ2v, λ3w).Let T0 ⊂ T be the sub-torus defined by λ1λ2λ3 = 1, then T0 fixes the potentialW . Thus it gives a T0-action on Jac(Q,W ).The Hilbert schemes Hilb•(Q,W ) parameterize quotients of Jac(Q,W ),which are equivalent to left ideals of Jac(Q,W ). Therefore the T0-action onJac(Q,W ) induces a T0-action on Hilb•(Q,W ). The following is a generaliza-tion of [7, Lemma 4.1]Lemma 7.3.1. For each dimension vector d, there is a one-to-one correspondencebetween T0-fixed points of Hilbd(Q,W ) and Q-multi-colored plane partitions withdimension vector d.Proof. Since ei’s are idempotent, any ideal in Jac(Q,W ) is generated by poly-nomials of the form eif(x, y, z) for some i and f(u, v, w). We want toshow that any T0-invariant ideal can be generated by monomials. We re-mark that the proof of [7, Lemma 4.1] uses Hilbert’s Nullstellensatz, hence itdoes not apply directly here. However, Jac(Q,W ) contains a T0-invariant sub-ring C[u5, v5, w5]. We take I0 = I ∩ C[u5, v5, w5] is a T0-invariant ideal inC[u5, v5, w5]. Therefore I0 is a monomial ideal, and particularly, there exists nsuch that u5n, v5n, w5n ∈ I0 ⊂ I .72Now, I is generated by eigenvectors of T0, which are polynomials of the formm(u, v, w)g(uvw)ei for some monomial m, g ∈ C[t] with g(0) 6= 0. Supposem(u, v, w)g(uvw)ei ∈ I . We writeg(uvw) = a0 + ar(uvw)r + . . .Then ar(uvw)rg(uvw)(g(uvw)− ara0(uvw)rg(uvw))m(u, v, w)ei=(a0 + a˜2r(uvw)2r + . . .)m(u, v, w)ei ∈ IRepeating this process, we get (a0 + c(uvw)N + . . .)m(u, v, w)ei ∈ I for someN ≥ 5n, then m(u, v, w)ei ∈ I . This shows that I is a monomial ideal.For a monomial ideal I in Jac(Q,W ), we associate a Q-multi-colored planepartition pi as follows: we definepi ={(i, j, k) : e`uivjwk /∈ I for some `}and a Q-multi-coloringK(i, j, k) ={` : e`uivjwk /∈ I}.We left the details to the reader to check that this indeed defines a Q-multi-colored plane partitions. Also, we can associate any Q-multi-colored plane par-tition to a monomial ideal in the obvious way. To compare the dimension vectors,for any (monomial) ideal I , there is a natural decompositionI = e0I ⊕ e1I ⊕ e2I ⊕ e3I ⊕ e4Ias vector spaces, and the dimension vector d of the module induced by I isgiven bydi = dimC(eiJac(Q,W ))/(eiI),which agrees with the dimension vector of the associated multi-colored plane73partition.Corollary 7.3.2. For any dimension vector d, we haveχ(Hilbd(Q,W ))= nQ(d).Now we are ready to finalize the proof.Proof of Theorem 7.2.2. Recall that Hilbd(Q,W ) is the critical locus of the func-tion Tr(W ) on the smooth scheme Hilbd(CQ). Since T0 acts on CQ andW ∈ CQ is T0-invariant, the torus T0 acts on Hilbd(CQ) and the functionTr(W ) is T0-invariant. By [7, Proposition 3.3], the Behrend function ν ofHilbd(Q,W ) is equal to (−1)m on T0-fixed points, where m is the dimensionof Hilbd(CQ), and henceχvir(Hilbd(Q,W ))= (−1)mχ(Hilbd(Q,W )) = (−1)m nQ(d).The Hilbert scheme Hilbd(CQ) is the fine moduli space of framed represen-tations of Q with framing vector (1, . . . , 1), that is,Hilbd(CQ) =( ∏(a:i→j)∈Q1Hom(Cdi ,Cdj )×∏iCdi)//∏i∈Q0GLdi(C)which has dimensionm = dim Hilbd(CQ) =∑a:i→jdidj +∑idi −∑id2i = |d| − 〈d,d〉Q.74Chapter 8Related problems and futuredirectionsIn this chapter, we discuss several subjects related to our works and proposefuture research directions.8.1 Topological invariantsOur method also gives a computation of topological Euler characteristics ofHilbert schemes of points on the quantum Fermat quintic threefold. We denotethe generating function byZAtop(t) =∞∑n=0χ(Hilbn(A)) tn.Then Theorem 6.3.3 shows thatZAtop(t) = ZQ,Wtop (t)10 ·( ∞∑n=0χ(Quotn(C3)0)t5n)−10= ZQ,Wtop (t)10 ·M(t5)−50.75For the quiver (Q,W ) with potential, we use Theorem 7.2.2 and obtainZQ,Wtop (t0, . . . , t4) = ZQPL(t0, . . . , t4),Then Corollary 7.2.5 states thatZQPL(t0, t1, t2, t3, t4) =∏i∈Z/5ZZµ5(1,1,3)PL (ti, ti+1, ti+2, ti+3, ti+4).If we take t = t0 = . . . = t4, then Zµ5(1,1,3)PL simply becomes M(t) as it forgetsthe coloring. This yields a nice formulaZAtop(t) = M(t)χ(M1sp)(M(t5)5)χ(M5sp).One interpretation is that the topological Euler characteristics of moduli spacesof cyclic modules are completely determined by the topological Euler character-istics of moduli spaces of simple modules.Such statement is certainly true for any commutative smooth algebra A, sinceall simple modules are 1-dimensional and its moduli space is the affine schemeSpec(A), and moduli spaces of cyclic modules are precisely Hilbert schemes ofpoints on Spec(A). On the other hands, it is clearly false in some extreme cases(for example, free algebras).8.2 Cohomology theories of (X,A)In the study of DT theory, one often fixes some discrete data, such as Cherncharacters for compact Calabi–Yau 3-folds, or dimension vectors for quiverswith potential. One important property for these discrete data is the presence ofEuler pairing, which is required in, for example, wall-crossing formula.Throughout this thesis, we use Hilbert polynomials as the discretedata. However, it is not the “correct” one, as the Euler pairing χ onK0(Coh(A))fd) is non-trivial and does not descend through the induced mor-phism K0(Coh(A)fd)→ Z.Question. Find a (co)homology theory H•(X,A) for general (X,A) with a mor-76phism K0(Coh(A))→ H•(X,A) so that the Euler pairing descends.Natural candidates are some (relative) versions of cyclic (co)homology orHochschild (co)homology, in which an analogue of Chern character might bedefined.In this section, we will define a specific one that works for Coh(A)fd, and wewill use them later. The idea is that we will define an abelian group N generatedby components of the coarse moduli scheme M ss,5(A).Recall that M1sp consists of 50 points which can be regarded as the verticesof 10 copies of the quiver Q, and M5sp is a non-compact connected smoothmanifold of dimension 3. The moduli M5sp is compactified in Mss,5(A) byadding the representation of each quiver Q with dimension vector (1, 1, 1, 1, 1).We denote by eki , i = 0, . . . , 4, the vertices of the k-th quiver Q, k = 1, . . . , 10,and e the class of 5-dimensional simple A-modules. ThenN =(free abelian group generated by eki ’s and e)/(e =4∑i=0eki for all k).The morphism K0(Coh(A)fd) → N is defined by the semisimplification. TheEuler pairing descents to N since the dimension vector (1, 1, 1, 1, 1) is in thekernel of χQ. Any element in N can be written uniquely in the formne + d1 + . . .+ d10,where n ∈ Z, and dk is a dimension vector of the quiver Q, with dki = 0 forsome vertex i. The Euler pairing χ is given by χ(e,−) = 0 andχ(eki , e`j) =χQ(ei, ej), k = `,0, k 6= `.All of our results hold when we use N as the discrete data, but the formula willbecome more complicated.778.3 Motivic DT invariantsSince our computation is based on a stratification of Hilbert schemes of points,we may follow the idea from [8] to define motivic DT invariants.We define the motivic DT invariantsZAmot(t) =( ∞∑n=0[Hilbn(Q,W )0]vir tn)[X(0)]( ∞∑n=0[Quotn(C3,O⊕5)0]vir t5n)[M5sp].The virtual motivic class [Hilbn(Q,W )0]vir is defined via Hilbn(Q,W )0 ⊂Hilbn(Q,W ) which is a critical locus, see [8, Section 2.4] for details.On the other hand, motivic DT invariants of (Q,W ) is defined and is expectedto satisfy the equationZQ,Wmot (t) =( ∞∑n=0[Hilbn(Q,W )0]vir tn)[C(0)]( ∞∑n=0[Quotn(C3,O⊕5)0]vir t5n)L3−1.Combining these with the known result for [Quotn(C3,O⊕5)0]vir (see forexample, [4]), we obtainZAmot(t) = ZQ,Wmot (t)10 · EXP(L52 − L− 52L12 − L− 12· −([M5sp]− 10L3 + 10) t5(1 + L52 t5)(1 + L−52 t5)),Of course one should use the discrete data N described in the previous section,and a similar result still holds.While the potential W has a linear factor so we can apply dimension reduc-tion, the full computation of ZQ,Wmot (t) seems very difficult. Here we record the78motivic DT invariants of (Q,W ) up to degree 4:ZQ,Wmot (t) = 1 + L0t(1,0,0,0,0) + L−1(L2 + L+ 1)t(1,1,0,0,0) + L−12 (L+ 1)t(1,0,1,0,0)+ L−1(L2 + L+ 1)t(1,2,0,0,0) + L0t(2,0,1,0,0) + L−32 (L+ 1)3t(1,1,1,0,0)+ L−2(L2 + 1)(L+ 1)2t(1,1,0,1,0) + L0t(1,3,0,0,0) + L−1(L2 + L+ 1)t(2,1,1,0,0)+ L−52 (L2 + L+ 1)(L2 + 1)(L+ 1)t(1,2,1,0,0) + L−1(2L2 + 3L+ 2)t(1,1,2,0,0)+ L−52 (L2 + L+ 1)(L2 + 1)(L+ 1)t(1,2,0,1,0) + L−2(L4 + L3 + L2 + L+ 1)t(1,1,0,2,0)+ L−52 (2L4 + 4L3 + 5L2 + 4L+ 2)(L+ 1)t(1,1,1,1,0) +O(t5).We want to mention a conjecture of Cazzaniga–Morrison–Pym–Szendro˝i [11],which suggests that motivic DT invariants of (Q,W ) should in some sense de-termined by its simple modules. To be more precise, letZQ,W (t) =∑d[Md(Q,W )]vir tdbe the generating function of motivic DT invariants of (Q,W ) (without framing).Since all simple modules of Jac(Q,W ) have dimension 1 or 5 (with dimensionvector (1, 1, 1, 1, 1), the conjecture suggests that ZQ,W (t) should be of the formEXP(1L12 − L− 12t0 + t1 + t2 + t3 + t41− t0t1t2t3t4 +ML12 − L− 12t0t1t2t3t41− t0t1t2t3t4)for some M . This is clearly not true for one simple reason: ZQ,W (t) is not asymmetric function, the coefficient of t0t1 is L2(L− 1)−2 and coefficient of t0t2is L32 (L− 1)−2.The main issue is the nontrivialness of the Euler pairing χQ. It seems tous that the formula should be taken in some non-commutative products (forexample, take td1 ∗ td2 = Lχ(d1,d2)td1+d2 ). But in that case it is problematic todefine the exponential. One naive (and unsuccessful) attempt is to simply takeEXP(1L12 − L− 12(t0 + t1 + t2 + t3 + t4))=∞∑n=01[GLn]vir(t0+t1+t2+t3+t4)nand expand the right hand side using non-commutative products.79To the best of our knowledge, all quivers with potential whose motivic DTinvariants have been successfully computed have trivial Euler pairings, or equiv-alently, they are symmetric quivers.8.4 Joyce–Song’s generalized DT invariantsTo define Joyce–Song’s generalized DT invariants, one key ingredient is theBehrend function identities [19, Theorem 5.11]. Recall that for any indecomposableA-modules must support at a single point of X . Then our local models (Theo-rem 5.3.7) implies that it is sufficient to check the Behrend function identities formoduli stacks of quivers with potential, which has been verified in [19, Theorem7.11].We will use N (defined in earlier) for discrete data. For any γ ∈ N , we writeDTγ(A) for the generalized DT invariants. For any class γ, the moduli schemePIγ,n of stable pairs is just the Hilbert scheme Hilbγ(A) of points. Then thewall-crossing formula [19, Theorem 5.27] states thatχvir(Hilbγ(A)) = ∑γ1+...+γ`=γ(−1)``!∏`i=1(−1)χ(A−γ1−...−γi−1,γi)χ(A−γ1−. . .−γi−1, γi) DTγi(A).For any class γ, χ(A, γ) = |γ| is the dimension of a finite-dimensional A-module. We will use this formula to compute some invariants DTγ(A).Recall that any element in N can be written uniquely in the form ne+d1 +. . .+ d10, where dk is a dimension vector of the quiver Q with dki = 0 for somevertex i. In the case n = 0, we haveDTd1+...+d10(A) =DTdi(Q,W ), if dj = 0 for all j 6= i0, otherwise,where DT(Q,W ) is the generalized DT invariant of the quiver (Q,W ) withpotential.For the class e ∈ N , we compare the wall-crossing formula to Hilbe(A) and80to Hilb(1,1,1,1,1)(Q,W ), which yieldsχvir(Hilbe(A))−10·χvir(Hilb(1,1,1,1,1)(Q,W )) = 5(DTe(A)−10·DT(1,1,1,1,1)(Q,W ))For the left hand side, we have a stratification of Hilbe(A) that givesχvir(Hilbe(A)) = 10 · χvir(Hilb(1,1,1,1,1)(Q,W ))+ χvir(Quot1(M5sp,O⊕5)).If we pretend the non-compact 3-fold M5sp is Calabi–Yau, then the wall-crossingformula saysχvir(Quot1(M5sp,O⊕5))= −(−1)χ([O5],1)χ([O5], 1)DT1(M5sp).We conclude thatDTe(A) = 10 ·DT(1,1,1,1,1)(Q,W ) + DT1(M5sp).This result might seem natural from our computation of ZA(t), which is givenby 10 copies of (Q,W ) and M5sp. However, due to the nontrivialness of Eulerpairing χ, the wall-crossing formula is really complicated so that does not givethis result directly. It is not clear to us what would DTne(A) be for n > 1.8.5 Future directionsGeneralization to arbitrary non-commutative projective schemesThis thesis only considers non-commutative projective schemes that are finiteover their centers, which is a strong restriction in non-commutative geometry.Assuming good moduli spaces with respect to certain stability condition, theconstruction of symmetric obstruction theories seems rather formal, and we aresurprised that it is not known in the literature already.Particularly, let C be any abelian category. It is well-known that thedeformation-obstruction theory for an object F in C is governed by Ext1C(F, F )and Ext2C(F, F ). However this is not enough to construct an obstruction theoryfor moduli spaces.81Question. Does Theorem 4.1.2 (existence of the Atiyah class) hold for any abeliancategory?This would allow us to define numerical DT invariants for CY3 abelian cat-egories, which would be deformation invariants for non-commutative projectiveschemes.Deformations of the quantum Fermat quintic threefoldOur quantum Fermat quintic threefold is the Fermat quintic that lies in quantumprojective space. Because the Fermat quintic equation needs to be in the center,this puts strong restrictions on the quantum parameters qij and also on thequintic equations. There is no obvious (non-trivial) deformation of the quantumFermat quintic threefold.Unlike commutative projective spaces which are rigid, there are many non-trivial deformations of non-commutative projective spaces (see, for example, [1]for non-commutative projective planes).Question. Does there exist non-trivial deformations of the quantum Fermat quinticthreefold which lie in other non-commutative projective 4-spaces?Enumerative geometry for the quantum Fermat quintic threefoldNon-commutative projective schemes should also serve as interesting examplesin enumerative geometry. Compared to the usual (commutative) quintic three-folds, the quantum Fermat quintic threefolds seem to have “smaller” modulispaces with more complicated structures. On the DT side alone, there are al-ready many problems that can be explored. For example,Question. Is the moduli space of sheaves on the quantum Fermat threefold a d-critical locus? More generally, there is a natural derived enhancement for any non-commutative projective scheme. Does the derived moduli stack of sheaves on the quan-tum Fermat threefold carry a (−1)-shifted symplectic structure?Other directions for study include Gromov–Witten invariants,Pandharipande–Thomas invariants and homological mirror symmetry.82Generalization to non-Calabi–Yau casesClassically, DT invariants are defined for any projective 3-fold. We only managedto construct a perfect obstruction theory in the Calabi–Yau case due to the lackof notions of determinant and trace. Nevertheless, a perfect obstruction theoryshould exist more generally, for example for Fano pairs (X,A). In particular ifA is of global dimension 2, then the obstruction theory is automatically perfect.This yields an analogue of Donaldson invariants for non-commutative surfaces,which might have applications in the study of non-commutative surfaces.83Bibliography[1] M. Artin and W. F. Schelter. Graded algebras of global dimension 3. Math., 66(2):171–216, 1987. ISSN 0001-8708.doi:10.1016/0001-8708(87)90034-X. 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