Science, Faculty of
Mathematics, Department of
DSpace
UBCV
Korde, Atharva Dilip
Doctor of Philosophy - PhD
University of British Columbia
In this dissertation, the main objects of study are sheaves with a 'twisted endomorphism' on surfaces with an automorphism. The results are divided into three main chapters: In Chapter 2, we work generally and define these objects on varieties with an automorphism. The moduli stacks of these objects are shown to be algebraic. When the automorphism is of finite order, we show that the category of such objects is equivalent to a category of modules over a certain sheaf of noncommutative algebras. After fixing a stability condition, we study Hilbert schemes of points of this sheaf of noncommutative algebras. In Chapter 3, we specialize to the case of surfaces. The topological Euler characteristics of the Hilbert schemes in Chapter 2 are packaged in a generating series, which we compute. The answer generalizes the situation when the automorphism is the identity. Chapter 4 concerns virtual counts, or Donaldson-Thomas (DT) type invariants arising from these objects. We define DT-type invariants when the structure sheaf of the surface has no higher cohomology and the quotient map is a cyclic covering of smooth surfaces. This gives new examples of invariants for automorphisms of some del Pezzo surfaces, which also may be thought of as invariants of noncommutative Fano threefolds.
Some invariants for surfaces with anautomorphismbyAtharva Dilip KordeA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)October 2025© Atharva Dilip Korde 2025The following individuals certify that they have read, and recommend tothe Faculty of Graduate and Postdoctoral Studies for acceptance, the thesisentitled:Some invariants for surfaces with an automorphismsubmitted by Atharva Dilip Korde in partial fulfillment of the require-ments for the degree of Doctor of Philosophy in MathematicsExamining Committee:Kai Behrend, Professor, Mathematics, UBCSupervisorJim Bryan, Professor, Mathematics, UBCSupervisory Committee MemberKalle Karu, Professor, Mathematics, UBCSupervisory Committee MemberSabin Cautis, Professor, Mathematics, UBCUniversity ExaminerMark van Raamsdonk, Professor, Physics and Astronomy, UBCUniversity ExaminerJørgen Vold Rennemo, Professor, Mathematics, University of OsloExternal ExamineriiAbstractIn this dissertation, the main objects of study are sheaves with a âtwistedendomorphismâ on surfaces with an automorphism. The results are dividedinto three main chapters:In Chapter 2, we work generally and define these objects on varietieswith an automorphism. The moduli stacks of these objects are shown tobe algebraic. When the automorphism is of finite order, we show that thecategory of such objects is equivalent to a category of modules over a certainsheaf of noncommutative algebras. After fixing a stability condition, westudy Hilbert schemes of points of this sheaf of noncommutative algebras.In Chapter 3, we specialize to the case of surfaces. The topologicalEuler characteristics of the Hilbert schemes in Chapter 2 are packaged in agenerating series, which we compute. The answer generalizes the situationwhen the automorphism is the identity.Chapter 4 concerns virtual counts, or Donaldson-Thomas (DT) typeinvariants arising from these objects. We define DT-type invariants whenthe structure sheaf of the surface has no higher cohomology and the quotientmap is a cyclic covering of smooth surfaces. This gives new examples ofinvariants for automorphisms of some del Pezzo surfaces, which also may bethought of as invariants of noncommutative Fano threefolds.iiiLay SummaryCounting geometric objects satisfying certain constraints is a field of studydating back to the Greek geometers - Apollonius showed that there areexactly eight circles tangent to three given circles in general position. Inmodern times, geometers have discovered new ways to count more compli-cated objects objects (curves, sheaves, ..) on certain three-dimensional realand complex manifolds.A complex surface with a symmetry may be viewed as being somewhatintermediate between three real and three complex dimensions. In this the-sis, we develop a theory of counting objects on surfaces with symmetries.Since symmetries are potentially non-trivial, we naturally have to deal withnoncommutative spaces, for instance, rules for multiplying two polynomialsare described differently than usual. If the symmetry of the surface is trivial,our theory reduces to the known theory for complex threefolds.ivPrefaceThis dissertation is independent and original work by the author, AtharvaDilip Korde, with the help of mathematical discussions with his advisor,Kai Behrend. Although unpublished currently, a paper is being preparedfor publication.No Artificial Intelligence tools or Large Language Models were used inwriting any part of this thesis, including the Abstract and Lay Summary.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 A Survey on Enumerative Invariants . . . . . . . . . . . . . . 11.2 Motivation and Main Results . . . . . . . . . . . . . . . . . . 52 Moduli Spaces of Twisted Endomorphisms . . . . . . . . . . 122.1 The Stack of 0-dimensional Coherent Sheaves . . . . . . . . . 122.2 Sheaves with an Endomorphism . . . . . . . . . . . . . . . . 152.3 Sheaves with a Twisted Endomorphism . . . . . . . . . . . . 172.4 The Moduli Problem via Noncommutative Sheaves of Alge-bras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.5 Stability Conditions . . . . . . . . . . . . . . . . . . . . . . . 293 Euler Characteristics of the Moduli Spaces . . . . . . . . . 353.1 Basics on Quivers . . . . . . . . . . . . . . . . . . . . . . . . 353.2 Hilbert Schemes of Points for Associative Algebras . . . . . . 363.3 Cayley Quivers . . . . . . . . . . . . . . . . . . . . . . . . . . 37viTable of Contents3.4 Local Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.5 Computation of the Euler Characteristic . . . . . . . . . . . 504 Obstruction Theory on the Moduli Space . . . . . . . . . . 544.1 Preliminaries on Sheaves of Noncommutative Algebras . . . 544.2 Moduli Spaces of Coherent Modules . . . . . . . . . . . . . . 624.3 An Open Embedding of the Hilbert Scheme . . . . . . . . . . 634.4 Construction of the Obstruction Theory . . . . . . . . . . . . 694.5 DT-Type Invariants and Examples . . . . . . . . . . . . . . . 72Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80viiAcknowledgmentsFirst and foremost, I would like to thank my advisor, Kai Behrend, for hiscontinued patience, support, wisdom and for agreeing to take me as hisstudent.Along with Kai, I would like to thank Jim Bryan and Kalle Karu forteaching me a lot of algebraic geometry in Vancouver.I thank Jørgen Vold Rennemo for reading this dissertation carefully andfor several helpful comments which improved this work.In high school, Pawan Babel and Varun Dubey gave me some very diffi-cult and interesting problems in chemistry and physics respectively, but toldme to study mathematics. Their lessons helped me a lot, intellectually.At IIT Bombay, I chose to study mathematics after enrolling there asa student in engineering. Thank you to all my professors in Mumbai forgiving me a chance at this subject.I am grateful to Devin, RJ, Taylor and Zach for a ton of hystericaljokes throughout the years, and of course, for innumerable discussions onmathematics.Hursha and Rita Mavshi are like a second set of parents to me. Theyhave treated me like their own son and I am grateful for their support.To my parents and my late grandparents, I am indebted for all your loveand sacrifices. It is simply not possible to express in words how much youmean to me.Finally, I would like to say thank you to Priyal. I am very lucky that Iget to share life with you.viiiDedicationTo Aai and DadixChapter 1Introduction1.1 A Survey on Enumerative InvariantsAn enumerative invariant counts objects of a certain kind. In algebraic ge-ometry, one has moduli spaces of objects. If a moduli space is a finite setof points, then counting is âobviousâ. The simplest and most well-known ex-ample of this is the fact that there are exactly 27 lines on any smooth cubicsurface in P3. It is rarely the case that naturally occurring moduli spacesare finite. However, in certain settings, some moduli spaces are âexpectedâto be zero dimensional, even though they might not be and in fact, mighteven have components of various dimensions. Near the end of the 20th cen-tury, such a phenomenon was observed for moduli spaces of stable maps,giving the enumerative invariants called Gromov-Witten invariants - theseinvariants count curves on a smooth projective variety. Soon after, a similarphenomenon was observed for moduli spaces of stable sheaves on Calabi-Yauthreefolds. The resulting invariants called Donaldson-Thomas (DT) invari-ants were first defined in [50] by Thomas. These are just two of many re-lated invariants - other enumerative invariants such as Gopakumar-Vafa in-variants coming from physics, Pandharipande-Thomas invariants, and BPSinvariants are conjectured to be related via several conjectures.Since the topic of this thesis relates to DT-type invariants, we describehow DT-invariants are obtained, without the technicalities. Let X be asmooth projective Calabi-Yau (CY) threefold. After fixing a polarization onX, one has projective moduli spaces of (semi)stable sheaves with specifiedinvariants (Chern classes). For simplicity, we also put ourselves in the situa-tion where stable and semistable sheaves coincide. Then the DT-invariants11.1. A Survey on Enumerative Invariantsare enumerative invariants of such moduli spaces - these are integer-valuedvirtual counts of stable sheaves. The key word here is âvirtualâ as thesecounts are not true counts. The technical machinery (developed in [7] and[33]) allowing for such a count is that of a perfect obstruction theory givingrise to a virtual fundamental class of dimension zero, in this case. In fact,this obstruction theory is quite special - it is a symmetric obstruction theory,first discussed in [8] by Behrend and Fantechi.In [37] it was shown that DT-invariants are well-defined whenever theanticanonical bundle âKX has a nonzero section: this extended the defini-tion of DT-invariants to a broader class of threefolds than Calabi-Yau. Letus fix Chern classes (c0, c1, c2, c3) = (1, 0,âÎ², n) and consider the associatedmoduli space Mn of stable sheaves - these are rank 1 sheaves with trivialdeterminant. It turns out that this moduli space is isomorphic to a Hilbertscheme of X parametrizing those subschemes Z â X which are at most onedimensional, whose (holomorphic) Euler characteristic is n and whose one-dimensional component is in the cohomology class β. It is important to notethat the DT-invariants are sheaf-counting invariants, but in the very specialcase described above, they may also be viewed as curve-counting invariants(by counting the subschemes Z).In [38] it was shown thatMn carries a virtual class of dimension âKX ·β.Therefore, if β = 0, DT-invariants are well-defined for the Hilbert schemeHilbn(X). Let NnX :=â«[Hilbn(X)]vir 1 be the associated DT-invariant and letZDT (X, q)0 :=ânâ¥0NnXqnbe the generating series for the âdegree 0â DT-invariants (as β = 0). It wasconjectured (one of the MNOP conjectures) in [38] thatZDT (X, q)0 =M(âq)c3(TXâKX)where M(q) =ânâ¥11(1âqn)n is the Macmahon function, which also equalsânâ¥0 p3(n)qn, the generating function for space partitions of integers. This21.1. A Survey on Enumerative Invariantsconjecture was settled in various ways. It was proven by Li in [32] by a ho-motopical approach. Levine-Pandharipande used the idea of double-pointdegenerations in [31]. When X is Calabi-Yau, the right hand side of the con-jecture reduces to M(âq)e(X) (the exponent denotes the topological Eulercharacteristic). In this special case, the conjecture was proven by Behrend-Fantechi by proving that the DT-invariants are weighted Euler character-istics in [6], and computing the weighted Euler characteristics in [8]. Weremark that the series M(q) appears in topological invariants earlier: Thegenerating functionânâ¥0 e(Hilbn(X)) qn for the topological Euler charac-teristics of the Hilbert schemes of points equals M(q)e(X) for any threefold,proper or not, by Cheah first in [15] and then also in [8].A new direction of research was introduced by Szendroi in [49] whereDT-invariants were defined and computed for the noncommutative crepantresolution A (NCCR) of the conifold singularity SpecR defined by the equa-tion x1x4 â x2x3 = 0 in C4. Let Q be the quiver below.â¢ â¢a1a2b1b2The idea is that A can be described as a derivation-quotient algebra of Qwith the superpotential a1b1a2b2âa1b2a2b1 and the center of A is R. (It is aCalabi-Yau algebra, introduced by Ginzburg in [19]). The DT-invariants areobtained for moduli spaces of framed cyclic modules of this quiver, based onthe recipe that quivers with superpotential naturally lead to a symmetricobstruction theory. This was the first example of DT-type invariants arisingfrom noncommutative geometry.More recently, Kanazawa in [26] and Liu in [35], [36] have studied anexample of a ânoncommutative Calabi-Yauâ threefold - the noncommutativeFermat quintic. Roughly speaking, it is a noncommutative algebra of theformCâ¨x0, x1, x2, x3, x4â©q(x50 + x51 + x52 + x53 + x54)31.1. A Survey on Enumerative Invariantsfor a quantum polynomial ring Câ¨x0, x1, x2, x3, x4â©q with certain parametersq defining commutativity relations. The important feature of this algebra isthat it is finite over its center C[t0,t1,t2,t3,t4]t0+t1+t2+t3+t4 where ti = x5i . This allows theauthors to view the algebra as a sheaf A of noncommutative algebras overa threefold (P3 here). By analogy with Szendroiâs work, the DT-invariantsare obtained for Hilbert schemes of points of A. Like the classical casefor (commutative) CY-threefolds, the computations for both of these exam-ples exploit the existence of a symmetric obstruction theory on the relevantmoduli spaces.Enumerative invariants in the setting of a group action have been re-searched in several different directions. For example, DT-invariants for 3-dimensional CY-orbifolds (a local model is the quotient stack [A3C/G] for afinite subgroup G â SL(3,C)) have been defined in [11] by Bryan-Cadman-Young. For a finite group G, the G-fixed locus of Hilbert schemes of pointson surfaces (equivalently, Hilbert schemes of points on the stack quotients)are studied by Bryan-Gyenge in [12] for K3 surfaces, and by Pietromonaco in[41] for Abelian surfaces. They compute the generating series for the topo-logical Euler characteristics e(Hilbn(X)G) when X is a K3/Abelian surfacewith a symplectic G-action. The answers are expressed as certain modularforms.Finally we mention that other vast generalizations of DT-invariants havebeen defined, for example, Joyce-Songâs generalized DT-invariants in [25] orKontsevich-Soibelmanâs motivic DT-invariants in [27] for any 3-Calabi-Yaucategory. This is not an exhaustive list of generalizations by any means, sowe content ourselves with some of these generalizations.For purposes of motivation of this thesis, we also say a few words aboutthe Casson invariant, an enumerative invariant in the smooth category, forsmooth 3-manifolds (see [13] for more details). It may be viewed as a count offlat connections arising from an intersection of two Lagrangian submanifoldsin a moduli space of connections. The key analogy with the complex case isthat if M is a complex symplectic manifold and L1, L2 are Lagrangian sub-41.2. Motivation and Main Resultsmanifolds, the intersection L1â©L2 carries naturally, a symmetric obstructiontheory.1.2 Motivation and Main ResultsSince the DT invariants are defined for complex threefolds, we may pretendto look at them as smooth 6-manifolds. So the motivation for this thesisbegan with a rather vague question - âCan we find similar phenomena for thedimensions 4 and 5 in between?â Informally speaking, a simple example of aâmixedâ 5-manifold may be taken to be a complex surface bundle fibered overa circle. Equivalently, a complex surface X with an algebraic automorphismf : X â X produces such an object. Then, sheaves on the 5-manifoldwould be described by a sheaf F on X along with an isomorphism Ï : F âfâF . For reasons of compactification (the property of a morphism being anisomorphism is open in families), we make the next definition.Definition 1.2.1. (Also Definition 2.3.1.) Let F be a quasi-coherent sheafon X. A twisted endomorphism of F is a morphism F â fâF of sheaves.When f = idX is the identity automorphism, the 5-manifold is a trivialfibration over the circle. Algebraically, a pair (F , Ï) with Ï an isomorphismmay then be viewed as a sheaf on X Ã Gm, with the trivial fibration X ÃGm â Gm as an analogy. So one can study sheaves on the compactificationX Ã P1 and indeed, get invariants for suitable X. When f Ì¸= idX thisanalogy doesnât really hold anymore. Nevertheless, we prove that the stackPn(X, f) parametrizing pairs (F , Ï) where F is zero-dimensional of lengthn, is algebraic.The interesting case is when the automorphism f has finite order, whichwe study in depth in this thesis. The simple, but key idea is as follows. LetR be a ring with an automorphism θ : Râ R and M be an R-module. De-note by M âÎ¸ R the pullback of M via θ. A homomorphism of R-modulesz : M â M âÎ¸ R follows the rule z(rm) = Î¸â1(r)z(m) for r â R andm âM . So M naturally gets the structure of a module over a twisted poly-nomial ring R[z, Î¸â1]. The multiplication in this ring is noncommutative:51.2. Motivation and Main ResultsThe multiplication z · r is Î¸â1(r)z, that is, twisted by an automorphism.One of the reasons that the automorphism be of finite order is to ensurethat the twisted polynomial ring R[z, Î¸â1] is finite over its center, like thenoncommutative quintic.In Section 2.4, we make this idea precise to relate twisted endomorphismsto (left) modules over a certain sheaf of noncommutative algebras.Theorem 1.2.2. Let X be a smooth projective variety and f : X â Xan automorphism of finite order. Let Z = X/f be the quotient schemeand Y = X/f Ã P1. There is a coherent sheaf of noncommutative algebrasA(f) on Y satisfying the following. Let Y0 := Z Ã A1 â Y and A0(f) therestriction of A(f) to Y0. Then the stack Pn(X, f) is isomorphic to the stackCohn(Y0,A0(f)) parametrizing zero-dimensional coherent A0(f)-modules oflength n on Y0 â Y .In the theorem, the algebra A0(f) already governs twisted endomor-phisms of sheaves and A(f) may be regarded as an extension to the com-pactification Y .A more general statement is also in Theorem 2.4.8. In fact, the noncom-mutative algebra may be thought of as a noncommutative model of X Ã P1in a certain way. Unfortunately the algebra does not turn out to be Calabi-Yau (this might be âexpectedâ as X Ã P1 is never Calabi-Yau). However, itis still a new (as far as we can tell) example in noncommutative DT-theory,and of a different flavor from the example of the noncommutative Fermatquintic, as the noncommutativity is described by twisted polynomial rings.Let (F , Ï) define a twisted endomorphism. If s â Î(X,F), then Ïs iscanonically a section of F too, via the (canonical) isomorphism Î(X,F) â¼=Î(X, fâF). We say that the triple (F , Ï, s) is stable if the sections {Ïis}iâ¥0generate the sheaf F . The moduli space of stable triples is called Hn(X, f).In Section 2.5 we give equivalent formulations, expressing Hn(X, f) as theHilbert scheme of points of A0(f). The scheme Hn(X, f) is constructeddirectly as an open subscheme of the Hilbert scheme Hilbn(Y,A(f)). Thelatter is the Hilbert scheme of points of the noncommutative algebra A(f).61.2. Motivation and Main ResultsIt may be remarked that s : OX â F may not define a (closed) sub-scheme and F may not descend to the quotient [X/f ], so this situationis different from the literature on G-invariant closed subschemes/Hilbertschemes or sheaves on quotient orbifolds. It is also worth remarking thatthe schemes Hn(X, f) are rather complicated - they can be singular even forn = 1. When f = idX , these are the Hilbert schemes of points on X ÃA1.For computational purposes, we restrict ourselves to topological Eulercharacteristics of the Hilbert schemes and compute the generating functionfor these in Chapter 3.Theorem 1.2.3. Let X be a projective surface and f : X â X an auto-morphism of finite order. Then, we have a formulaânâ¥0e(Hn(X, f))qn =M(q)e(X)The result for f = idX is immediately deduced from Cheahâs resultin [15], so this generalizes the result to arbitrary f of finite order. Themethod of proof is a stratification argument which is standard for manysuch computations. The right hand side arises due to the following cut-and-paste. We can stratify X into strata Xk for each k|d. Here Xk consists ofthe points whose stabilizers have order k, so that X1 is the free locus, forexample. Writing M(q)e(X) as a productâk|dM(q)e(Xk), we show preciselythat the âcontributionâ from the stable triples (F , Ï, s) â Hn(X, f) with| suppF| in Xk is the series M(q)e(Xk).Cayley quivers provide local models for Hn(X, f). Briefly, a Cayleyquiver QG is associated to a group G along with a set S of generators. Thevertex set of the quiver is the group G and for each generator fi â S we drawa directed edge from g to gfi for every g â G. (The terminology âCayleygraphâ is used in geometric group theory but we prefer the term âCayleyquiverâ here for contextual reasons). If G is abelian, we can impose theobvious relations in the path algebra that the two paths g â gfi â gfifjand g â gfj â gfjfi agree. It turns out that such Cayley quivers withrelations describe sheaves with a twisted endomorphism (F , Ï) such that F71.2. Motivation and Main Resultsis supported in a single free orbit of f . The degenerations, i.e. cases whensheaves land on orbits of smaller size, are also described by quivers with thesame shape, but with some tweaks in the relations.As an example, the two kinds of quivers describing the case when f isof order 2 are drawn below. The quiver describing those sheaves (with atwisted endomorphism) supported in a free orbit, i.e. of size 2, isâ¢ â¢x1y1z1x2y2z2It has six relations: z1x1 = x2z1, z1y1 = y2z1, z2x2 = x1z2, z2y2 = y1z2,x1y1 = y1x1 and x2y2 = y2x2. It is the Cayley quiver associated to thegroup G = Z/2 with three generators S = {f, 1, 1} (observe that the lasttwo generators are superfluous, a possibility which we allow). It is shownin Section 3.3 that the path algebra with relations may be described rathersimply as (ke1)[x, y, z]Ã (ke2)[x, y, z] where x, y commute with ei and zei =ei+1z for i = 1, 2. Here e1, e2 are the idempotent paths at the two vertices.The quiver describing the sheaves (with a twisted endomorphism) sup-ported at an isolated fixed point isâ¢xyzIt has three relations zx = âxz, zy = âyz, xy = yx. It is a degeneration ofthe Cayley quiver drawn above, when the two points collide. The shape isthe same as that of the Cayley quiver of the trivial group with three trivialgenerators, but the relations are different. (Informally, one generator has anautomorphism of order 2). If the fixed point lies in a one-dimensional fixedlocus, one of the relations changes: zy = yz.We remark that the algebras with relations described above can be writ-81.2. Motivation and Main Resultsten as derivation-quotient algebras with respect to a twisted superpotential.It is an interesting question in its own right whether one can produce a(perfect or symmetric?) obstruction theory on the moduli space of repre-sentations of such an algebra. In literature, they are called skew or twistedCalabi-Yau algebras, see Remark 3.3.10 for details and references.In Chapter 4, we work with the compactification Y and A(f). For cycliccovers X â X/f , we are able to construct a perfect obstruction theory onH = Hilbn(Y,A).Theorem 1.2.4. Assume that X/f is smooth and the quotient X â X/f isa cyclic cover. Suppose that the cohomological conditions H0(Y,A(f)) â¼= C,H1(Y,A(f)) = H3(Y,A(f)) = 0 hold. The Hilbert scheme H has a perfectobstruction theory(Ïâ¥1RpHâRHomA(f)H (I ,I ))â¨[â1]â LHAn important preliminary result needed for this theorem is a strongerversion of Serre duality for A(f)-modules than what is described in [35].Understanding the invertibility of the dualizing bimodule ÏA(f) is a keystep in this process. Up to tensoring by a line bundle, the bimodule ÏA(f)is isomorphic to A(f) as right modules, but the action on the left is via anautomorphism (the Nakayama automorphism) of A(f).The action of ÏA(f) is not trivial on a general zero-dimensional A(f)-module but it is trivial if it is supported in a certain open subset of Y . TheseA(f)-modules correspond to the twisted endomorphisms Ï : F â fâF withF supported in the free locus of f on X and Ï an isomorphism. Since therank of a perfect complex is locally constant, we may restrict to the unionV n in Hilbn(Y,A(f)) of components which have a quotient zero-dimensionalmodule as above. As a consequence, DT-type invariants NnX,f â Z aredefined for projective surfaces X with H1(X,OX) = H2(X,OX) = 0 andautomorphisms f : X â X such that X â X/f is a cyclic cover of smoothsurfaces.The result may also be viewed as a non-commutative analogue of the9Conventionsresult of Ricolfi in [44], concerning DT invariants for the Quot scheme of asimple exceptional vector bundle F on a threefold Y satisfying H i(Y,OY ) =0 for all i > 0.ConventionsWe fix some conventions for this thesis.In Chapter 2, the base field will be an algebraically closed field k. InChapters 3 & 4, the base field will be the field of complex numbers. (Thereader might as well use complex numbers throughout).A product A Ã B of schemes A,B is always understood to be over abase field (k algebraically closed, or C). The projections A Ã B â A andA Ã B â B are denoted by pA and pB respectively. If g : A â B is amorphism and C is any scheme, the natural morphism AÃC â BÃC willbe denoted by gC .The pullback of a sheaf F on B via pB : A Ã B â B will be denotedFA. If G is a sheaf on AÃB and b â B is a point, the restriction G|Ab willsimply be denoted by Gb.For a scheme X, the notation |X| will be used to refer to its underlyingtopological space.While working with stacks in Chapter 2, we do not assume that the âtestâschemes T are Noetherian or finite-type or separated.A commutative ring or algebra will simply be called a âringâ or âalgebraâ.All algebras are assumed to have a unit. For a noncommutative ring (orsheaf of rings) A, by an A-module we always mean a left A-module. Wewrite âAop-moduleâ for a right A-module. If R is a commutative ring, wedenote the category of R-modules by R-mod.By definition, the topological Euler characteristic of a scheme X of finitetype over the complex numbers is e(X) :=âiâ¥0(â1)i dimH i(Xan,C). Itsatisfies the excision property: Let Z â X be a closed subscheme withcomplement U . Then e(X) = e(Z) + e(U).The operation hi applied to a complex in an abelian category will meantaking the cohomology at position i.10ConventionsWhen talking about obstruction theories on a scheme H, we use the(small) etale site implicitly, i.e. D(H) = D(Het).11Chapter 2Moduli Spaces of TwistedEndomorphismsLet X be a projective variety over an algebraically closed field k and let fbe a k-automorphism of X. In this chapter we will define a moduli space ofâtwisted endomorphismsâ of sheaves on (X, f).The first two sections discuss introductory constructions such as stacksof coherent sheaves and the case when f = idX . There are many referencesfor the stack of coherent sheaves on a projective variety, see for instance, [17]or [18] for a modern exposition. We do not claim any originality in Section2.1. The results of Section 2.2 may be well-known but we are unable to finda reference for these ideas and hence, detailed proofs are given which willalso be useful later.In Section 2.3 we define the stack of pairs Pn(X, f) parametrizing zero-dimensional sheaves of length n with a twisted endomorphism and provethat it is algebraic. In the last two sections, we assume that f has finiteorder. We introduce the algebras A0(f) and A(f) and describe Pn(X, f)as the stack of coherent A0(f)-modules. Finally, we introduce the modulispace Hn(X, f) and describe some properties.2.1 The Stack of 0-dimensional Coherent SheavesLet X be a quasi-projective scheme over k. A 0-dimensional sheaf F onX is a coherent sheaf F with dim(suppF) = 0. We define the stackCoh0(X) which parametrizes 0-dimensional sheaves on X. For every non-negative integer n, there is an algebraic substack Cohn(X) of Coh0(X) which122.1. The Stack of 0-dimensional Coherent Sheavesparametrizes 0-dimensional sheaves F onX of length n, i.e. dimH0(X,F) =n. Since X is Noetherian, any 0-dimensional closed subscheme has finitelength. Therefore we also have a stratification Coh0(X) =ânâ¥0 Cohn(X)of the full stack. Whenever X is quasi-projective, we prove that the stackCohn(X) is algebraic.Let T be a k-scheme.Definition 2.1.1. A family of 0-dimensional sheaves on X parametrizedby T is a coherent sheaf F on X Ã T such that (1) F is T -flat. (2) Thesupport suppF is finite over T . We call such a family F a T -family of0-dimensional sheaves on X.Definition 2.1.2. For a non-negative integer n, a family of 0-dimensionalsheaves of length n on X parametrized by T is a T -familyF of 0-dimensionalsheaves with the additional condition (3) pT âF is a vector bundle of rank non T . We call such a family F a T -family of length n sheaves on X.Define the category Coh0(X) whose objects are pairs (T,F ) where Tis a k-scheme and F is a T -family of 0-dimensional sheaves on X. Themorphisms from (T â²,F â²) to (T,F ) are pairs (h, Ï) where h : T â² â T is amorphism of schemes and Ï : hâXF â F â² is an isomorphism of sheaves onX Ã T â². The category Cohn(X) is defined by the extra condition (3) andthe same morphisms between objects as in Coh0(X). The obvious forgetfulmorphisms from Coh0(X),Cohn(X) to Schk make either of these categoriesfibered in groupoids over Schk.By Grothendieckâs theory of descent for quasi-coherent sheaves, Coh0(X)becomes a stack. By the semicontinuity theorem, for a T -family F of 0-dimensional sheaves, the function tâ dimH0(X,Ft) is locally constant onT since F is T -flat. That is, the rank of pT âF is continuous on T where Nhas the discrete topology. So Cohn(X) becomes an open and closed substackof Coh0(X).The algebraicity of the stack of coherent sheaves is a well-known resultproven in much more general contexts (see [34], Theorem 2.1.1 for example).132.1. The Stack of 0-dimensional Coherent SheavesWe content ourselves with the basic result for X projective, with a sketchof the idea.Theorem 2.1.3. The stack Cohn(X) is algebraic.Proof. (Sketch) Consider the quot scheme Quot(OânX , n) which exists afterfixing a polarization on X. This parametrizes quotients OânX â E â 0where E is a 0-dimensional coherent sheaf of length n on X. There isthe obvious forgetful morphism F : Quot(OânX , n) â Cohn(X) which, onpoints, is the map p = [OânX â E â 0] â E . It is surjective as any 0-dimensional sheaf on X is generated by its global sections. The morphismF is smooth: the relative deformations and obstructions of F at the point plie in Hom(OânX , E) = H0(X, E)ân and Ext1(OânX , E) = H1(X, E)ân respec-tively but the obstruction space vanishes as E is 0-dimensional.The locus of points p for which the induced mapH0(X,OânX )â H0(X, E)is surjective, is open in the quot scheme. Let U be this open locus. The re-striction f |U : U â Cohn(X) is still surjective, so U becomes a smooth atlas.Moreover, GLn acts on U in the natural way by acting on OânX and becausewe restricted to U , the fiber of f |U over E â Cohn(X) is precisely a GLn-torsor corresponding to all possible ordered bases of H0(X, E). Therefore,Cohn(X) â¼= [U/GLn] is a global quotient stack.Let V be an open subscheme of X with i : V âªâ X the inclusion. Itdetermines an open substack CV of Cohn(X) because for a T -family F oflength n sheaves on X, the subset {t â T | supp(Ft) â Vt} is open in T .This is the open substack parametrizing the length n sheaves on X whosesupport lies in V .It can be proven that there is a morphism of stacks Cohn(V )â Cohn(X)which sends a T -familyF of length n sheaves on V to iT âF . Infact Cohn(V )is canonically isomorphic to the substack CV defined above, hence Cohn(V )is an open substack of Cohn(X) and is algebraic. It follows that Cohn(X) isalgebraic even if X is quasi-projective.142.2. Sheaves with an Endomorphism2.2 Sheaves with an EndomorphismFor an affine morphism, the next equivalence is well-known.Proposition 2.2.1. Let X be a scheme and A a quasi-coherent (commuta-tive) OX-algebra. Let Z = SpecXA be the relative spectrum and Ï : Z â Xthe structure morphism. Then the pushforward G â ÏâG induces an equiv-alence between the category of quasi-coherent OZ-modules and the categoryof quasi-coherent A-modules.Suppose A = OX [t] so that Z = X Ã A1 and Ï = pX is the projec-tion to X. The category QCohX(OX [t]) of quasi-coherent OX [t]-modules isequivalent to the category C whose objects are pairs (F , Ï) with F a quasi-coherent sheaf on X and Ï : F â F is an endomorphism. The morphismsin this latter category from (F , Ï) to (F â², Ïâ²) are morphisms h : F â F â²such that Ïâ²h = hÏ. By the proposition, this category is also equivalent tothe category QCohXÃA1 of quasi-coherent sheaves on X Ã A1.Let CohXÃA1/X be the full subcategory of QCohXÃA1 whose objects arethe coherent sheaves on X Ã A1 with proper support over X.Lemma 2.2.2. Under the correspondence of 2.2.1, CohXÃA1/X is equivalentto the full subcategory of C consisting of pairs (F , Ï) with F coherent.Proof. Let G â CohXÃA1/X with j : suppG âªâ X Ã A1 the inclusion of thesupport. Then pX â¦ j is proper and pXâG = ÏâjâjâG = (Ï â¦ j)â(jâG) iscoherent, being the pushforward of a coherent sheaf via a proper morphism.Conversely, if (F , Ï) â C is a pair with F a coherent OX -module, cer-tainly F is coherent when viewed as a sheaf onXÃA1 under the equivalence.To show that its support is proper over X, it suffices to work locally: LetU = SpecR be an open affine in X and F|U = MË for a finitely generated R-module. Let Ï|U be the endomorphism T :M âM . The determinant trickimplies that T satisfies an equation of the form T d+ rdâ1T dâ1+ . . .+ r0 = 0for some coefficients ri â R. The polynomial q(t) = td + rdâ1tdâ1 + . . .+ r0annihilates M , therefore suppR[t]M is contained in the closed subschemeSpecR[t]/q(t). This closed subscheme is finite over SpecR as q(t) is monic,hence suppR[t]M is finite (and also proper, equivalently) over SpecR.152.2. Sheaves with an EndomorphismThe relation between the supports is clear by the next lemma.Lemma 2.2.3. In the equivalence above, the scheme theoretic image ofsuppG is supp(pXâG).Proof. Once again, this is a local question so we may assume X = SpecR isaffine with Ï corresponding to the natural morphism R â R[t] and G = NËfor a finitely generated R[t]-module N . Let I â R[t] be the annihilatorideal defining suppG. The scheme theoretic image of suppG is defined bythe kernel ideal of OX â pXâOsuppG or equivalently, by the kernel of thecomposition R â R[t] â R[t]/I. The kernel of this composition is clearlyannR(N), proving the lemma.Remark 2.2.4. If X is Noetherian, the proof of Lemma 2.2.2 also showsthat the equivalence restricts to 0-dimensional sheaves on X Ã A1 and 0-dimensional sheaves on X. Since H0(XÃA1,F) = H0(X, pXâF), the lengthof a 0-dimensional sheaf is also preserved.Let X be a projective scheme. We define the category Pn(X). Theobjects are triples (T,F ,Φ) where (T,F ) â Cohn(X) and Φ : F â F isan endomorphism. The morphisms from (T â²,F â²,Î¦â²) to (T,F ,Φ) are themorphisms (h, Ï) from (T â²,F â²) to (T,F ) in Cohn(X) such that Î¦â² â¦ Ï =Ï â¦ (hâXÎ¦).hâXF Fâ²hâXF Fâ²ÏhâXÎ¦ Î¦â²ÏBy the equivalence in Lemma 2.2.2, for a triple (T,F ,Φ), the sheafF gives a coherent sheaf G on X Ã T Ã A1 = (X Ã A1) Ã T such thatpXÃT âG = F and suppG finite over X Ã T . Here pXÃT is the projectionX Ã T Ã A1 â X Ã T . Momentarily, we set qT to be the projection X ÃA1 Ã T â T so that qT = pT â¦ pXÃT .Lemma 2.2.5. (T,G ) is an object of Cohn(X Ã A1)162.3. Sheaves with a Twisted EndomorphismProof. We need to check that G defines a T -family of length n sheaves onX Ã A1.1. By Lemmas 2.2.2 and 2.2.3, suppG maps to suppF under pXÃT â andsuppG â suppF is finite. Since suppF is finite over T , suppG â Tis also finite, being a composition of finite morphisms.2. Since qT âG = pT âpXÃT âG = pT âF , it is a rank n vector bundle on T .3. Since pXÃT is an affine morphism, G is flat over T iff pXÃT âG is flatover T (follows from [48, Tag 0FLM]). But pXÃT âG equals F whichis flat over T by assumption.This proves the lemma.The equivalence in Lemma 2.2.2 is functorial under pullbacks by mor-phisms T â² â T . Therefore Lemma 2.2.5 shows that Pn(X) is an algebraicstack which is canonically isomorphic to the stack Cohn(X Ã A1).2.3 Sheaves with a Twisted EndomorphismLet X be a projective scheme with a k-automorphism f : X â X.Definition 2.3.1. Let F be a quasi-coherent sheaf on X. A twisted endo-morphism of F is a morphism Ï : F â fâF .In this section, we construct an algebraic stack Pn(X, f) which willparametrize pairs (F , Ï), where F is a coherent sheaf of length n on Xand Ï is a twisted endomorphism of F . When f = idX is the identityautomorphism, Pn(X, idX) = Pn(X).We define the category Pn(X, f) in a manner similar to Pn(X). The ob-jects are triples (T,F ,Φ) where (T,F ) â Cohn(X) and Φ : F â (fT )âF isa morphism. The morphisms from (T â²,F â²,Î¦â²) to (T,F ,Φ) are the mor-phisms (h, Ï) from (T â²,F â²) to (T,F ) in Cohn(X) such that Î¦â² â¦ Ï =Ï â¦ (hâXÎ¦).172.3. Sheaves with a Twisted EndomorphismhâXF Fâ²hâX(fT )âF (fT â²)âF â²ÏhâXÎ¦ Î¦â²ÏThe morphism Φ is not a morphism of the sheaf F but instead can beviewed as âlying overâ the automorphism fT . To prove that Pn(X, f) is analgebraic stack, we use a different strategy.There is a forgetful morphism F : Pn(X, f) â Cohn(X) of categoriesfibered in groupoids, which forgets the twisted endomorphism. That is,(T,F ,Φ) maps to (T,F ).The next theorem is the main result of this section.Theorem 2.3.2. The morphism F is representable by schemes. HencePn(X, f) is an algebraic stack.The second sentence of the theorem follows immediately by the generalproposition for stacks explained in [48, Tag 05XY].Proposition 2.3.3. Let J : X â Y be a morphism of categories fibered ingroupoids over Schk. Assume that J is representable by schemes. Then if Yis an algebraic stack, so is X .It remains to prove the first sentence in Theorem 2.3.2. We prove twointermediary lemmas in 2.3.4 and 2.3.5 first and complete the proof of therepresentability of F after these lemmas.For (T,F ) â Cohn(X), let VT = HomOT ((pT )âF , (pT )âF ). This is alocally free sheaf on T . The geometric vector bundle associated to VT willbe denoted by αT : VT â T .Lemma 2.3.4. For (T,F ) â Cohn(X), the sheafET := (pT )âHomOXÃT (F , (fT )âF )is isomorphic to a subsheaf of VT .182.3. Sheaves with a Twisted EndomorphismProof. Denote by Z the support of (fT )âF . The scheme Z is the inverseimage fâ1T (suppF ) and is finite over T . Denote by g the map (fT )|Z : Z âsuppF so that g induces an isomorphism of schemes. If i : suppF â XÃTdenotes the closed embedding of the support ofF , then j = iâ¦g : Z â XÃTis the closed embedding of the support of (fT )âF . By adjunction for thesheaves F on X Ã T and jâ(fT )âF on Z we get,jâHomOZ (jâF , jâ(fT )âF ) â¼= HomOXÃT (F , jâjâ(fT )âF )The canonical map (fT )âF â jâjâ(fT )âF is an isomorphism, so the righthand side is HomOXÃT (F , (fT )âF ). Applying (pT )â we getET â¼= (pT )âjâHomOZ (jâF , jâ(fT )âF )Since q = pT â¦j is a finite morphism, A := qâOZ is a coherentOT -algebra.ThereforeET â¼= HomqâOZ (qâjâF , qâjâ(fT )âF )We compute easily thatqâjâ(fT )âF = (pT )â(jâjâ(fT )âF ) = (pT )â(fT )âF = (pT )âFandqâjâF = (pT )â(jâjâF ) = (pT )â(iâgâgâiâF ) = (pT )â(iâiâF ) = (pT )âFhence ET â¼= HomA((pT )âF , (pT )âF ). As the A-linear homomorphisms arealso OT -linear, ET is a subsheaf of HomOT ((pT )âF , (pT )âF ) = VT .In the computation of ET , we passed to the support Z of (fT )âF . How-ever, the proof holds whenever Z is any closed subscheme of X Ã T (1)containing the support of (fT )âF and (2) finite over T , as these are theonly two properties of Z that were used.Lemma 2.3.5. The inclusion ET â VT described above is compatible underany morphism (h, Ï) : (T â²,F â²)â (T,F ) in Cohn(X).192.3. Sheaves with a Twisted EndomorphismProof. The condition hâXF â¼= F â² implies that suppF â² = supphâXF is con-tained in the closed subscheme (hX)â1(suppF ) of XÃT â² which is finite overT . Therefore, the support of (fT â²)âF â² is contained in the closed subscheme(fT â²)â1[(hX)â1(suppF )] = (hX)â1 supp(fT )âF . Using the remark above,computations for the sheaf ET â² may be done on the bigger closed subscheme(hX)â1 supp(fT )âF .Keeping the notation Z and A from the lemma above, let Z â² = (hX)â1Zwhich by the remark, may be assumed to be the support of F â². Let qâ² bethe map Z â² â T â² and Aâ² = qâ²âOZâ² . Since the morphisms q, qâ² are finite, weget Aâ² â¼= hâA.Checking that the inclusion ET â² â VT â² is the base change by h of theinclusion ET â VT is a local question on T â². Thus, we reduce to the caseT = SpecA, T â² = SpecAâ², Z = SpecB, Z â² = SpecBâ² with Bâ² = B âA Aâ²and B,Bâ² finitely generated modules over A,Aâ² respectively. The sheaf Fis given by a B-module M which is A-flat and F â² is given by the Bâ² moduleM â² =M âB Bâ². Since (pT )âF , (pT â²)âF â² are rank n vector bundles we mayalso assume that SpecA and SpecAâ² are chosen such that M â¼= An andM â² â¼= (Aâ²)n as modules over A and Aâ² respectively.In this case the inclusions ET â VT and ET â² â VT â² are simply the inclu-sionsHomB(An, An)â HomA(An, An) =Mn(A)HomBâ²((Aâ²)n, (Aâ²)n)â HomAâ²((Aâ²)n, (Aâ²)n) =Mn(Aâ²)Therefore, after base-change by Aâ², it suffices to show that the natural mapHomB(An, An)âA Aâ² â HomBâ²((Aâ²)n, (Aâ²)n) (2.3.1)is an isomorphism of Aâ²-modules.Let {b1, · · · , br} be a finite generating set for B as an A-module. Then{b1 â 1, · · · br â 1} is a generating set for Bâ² as an Aâ²-module. Each bkacts A-linearly on An hence gives a matrix Bi in Mn(A). The B-linearhomomorphisms An â An are precisely the matrices X = (Xij) which202.3. Sheaves with a Twisted Endomorphismcommute with the finite generating set {b1, · · · , br}, that is X â Mn(A)with BkX âXBk = 0 for all k. So HomB(An, An) is the closed subschemeSpecA[Xij ]/I of Mn(A) = SpecA[Xij ] where I is the ideal generated by{BkX âXBk = 0 | k = 1, · · · , r}in A[Xij ]. Similarly, the Bâ²-linear homomorphisms (Aâ²)n â (Aâ²)n are thematrices X â Mn(Aâ²) with Bâ²iX â XBâ²i = 0 for all i where Bâ²i is the im-age of Bi in Mn(Aâ²). Write HomBâ²((Aâ²)n, (Aâ²)n) as the closed subschemeSpecAâ²[Xij ]/I â² of Mn(Aâ²) = SpecAâ²[Xij ] where I â² is the ideal generated by{Bâ²kX âXBâ²k = 0 | k = 1, · · · , r} in Aâ²[Xij ].We see that A[Xij ]/I âA Aâ² â¼= Aâ²[Xij ]/I â² and hence the natural map(2.3.1) is an isomorphism.The lemma above also shows the existence of closed subschemes WT âVT for every T , with the following property. A global section of VT (equiva-lently a section of αT : VT â T ) is a section of ET iff it factors through WT .The closed subschemeWT â VT is described locally by the closed immersionSpecA[Xij ]/I â SpecA[Xij ] over SpecA â T and is well-defined on T . Theisomorphism in (2.3.1) shows that the closed subschemesWT are compatibleunder base change.We can now finish the proof of the main theorem.Proof. (of Theorem 2.3.2) We show that F is represented by the schemesWT â T , i.e. for a morphism T â Cohn(X) given by the T -family F , thediagramWT Pn(X, f)T Cohn(X)FFis cartesian. Let h : T â² â T and T â² â Pn(X, f) be morphisms such that212.3. Sheaves with a Twisted Endomorphismthe outer quadrilateral commutes.T â²WT Pn(X, f)T Cohn(X)hFFWe need to show that the dotted arrow exists and is unique. The morphismT â² â Cohn(X) is given by the T â²-family hâXF therefore the morphism T â² âPn(X, f) is given by a lift of hâXF , i.e., (hâXF ,Î¦â²). Since Î¦â² is a globalsection of ET â² , the Lemma 2.3.5 implies that it gives a unique a section ofWT â² which gives the dotted arrow after composition with WT â² â WT . Thedotted arrow is unique as the lifts ofWT â T to T â² are equivalently, sectionsof WT ÃT T â² â¼=WT â² â T â².This proof works without any assumption on the order of the automor-phism. In the next section, we restrict to f having finite order and give asimpler description.We now describe certain open and closed loci in the stack Pn(X, f).Lemma 2.3.6. Let (T,F ,Φ) â Pn(X, f). The locus of points t â T suchthat Φt is an isomorphism is open in T .Proof. Let L be the cokernel of Φ. Its support is proper over T , so theimage pT (suppL ) â T is closed. For any t in the open complement, wehave Lt = 0 by definition, so Φt is surjective. Conversely, if Φt is surjective,then Lt = 0 so t /â pT (suppL ). By restricting to this open complement,we may also assume that Φ is surjective. Let K be the kernel of Φ. SinceF and fâTF are T -flat, the restriction Kt is isomorphic to the kernel of Φtfor every t â T (Lemma 2.1.4 in [24]). Since suppK is also proper, theargument for L applies to K too. Thus, there is an open subset of T suchthat t is in this open subset iff Φt is injective. The proof is complete as Φtis an isomorphism iff Φt is injective and surjective.222.4. The Moduli Problem via Noncommutative Sheaves of AlgebrasLemma 2.3.7. Let (T,F ,Φ) â Pn(X, f). The locus of points t â T suchthat Φt = 0 is closed in T .Proof. This follows directly from Lemma 2.3.5 which shows that the locus oft â T with Φt = 0 is an intersection of two sections of VT â T : the sectioncorresponding to Φ and the zero section.2.4 The Moduli Problem via NoncommutativeSheaves of AlgebrasLet f : X â X be an automorphism of a k-scheme. Assume that (1) theautomorphism f is of finite order d. (2) X is separated. (3) Every pointof X has an affine open neighborhood invariant under f . For instance,a projective (or even quasi-projective) scheme with an automorphism offinite order satisfies these conditions. To see that the last condition holds,embed X in projective space PN . Any (finite) orbit in X is contained in thecomplement X âH of a hyperplane H â PN . Then (X âH)â© f(X âH)â©Â· · · â© fdâ1(X âH) is an open affine which contains the orbit.Definition 2.4.1. (Twisted Polynomial and Laurent Rings) Let R be a ringand θ : Râ R be a ring automorphism. The twisted polynomial ring R[z, θ]has the same elements and additive structure as the polynomial ring R[z]but with âtwistedâ multiplication given by zd.r = θd(r)zd for all r â R andd â¥ 0. The twisted Laurent ring R[z, zâ1, θ] is defined similarly with themultiplication rule holding for all d â Z, or equivalently zâ1r = Î¸â1(r)zâ1for all r â R.When θ = idR is the identity automorphism of R, these are the usualpolynomial/Laurent rings. For θ Ì¸= idR, the twisted analogues are noncom-mutative but share many of the desired algebraic and homological propertiesthat polynomial rings have.Proposition 2.4.2. If R is Noetherian, then R[z, θ] is left-Noetherian (andright-Noetherian too).232.4. The Moduli Problem via Noncommutative Sheaves of AlgebrasProof. A reference is in Chapter 1, Section 2, 2.9 of [39].Lemma 2.4.3. Let R be a domain. Suppose that θ has finite order d. Thenthe center of R[z, θ] is the polynomial subring Rθ[zd].Proof. It is clear that zd is a central element as zdr = θd(r)z = rz. Theinvariants Rθ = {r â R | θ(r) = r} are also central hence Rθ[zd] is apolynomial subring of R[z, θ] and central in it.Conversely, if p(z) =âni=0 rizi â R[z, θ] is central, the equality p(z) Â·z =z · p(z) shows that each ri â Rθ. Also, the equation p(z) · r = r · p(z) holdsfor every r â R. This impliesθi(r)ri = rri =â (Î¸i(r)â r)ri = 0 (2.4.1)for every i â¥ 0 and every r â R. When d divides i, the equation (2.4.1) istautological. Otherwise, θi Ì¸= idR and choosing r such that θi(r) Ì¸= r impliesthat ri = 0. So p(z) =âiâ¥0 rdi(zd)i â Rθ[zd].Remark 2.4.4. Let R be a finitely generated k-algebra which is a domainand let θ have finite order. Then R is finitely generated as an Rθ-module.Indeed, this is because R is integral over Rθ. Then R[z, θ] is finitely generatedas an Rθ[zd]-module via the chain of inclusions Rθ[zd] â R[zd] â R[z, θ].So RË[z, θ] may be viewed as a sheaf of coherent noncommutative algebras onSpecRθ[zd].Remark 2.4.5. If α : R â Râ² is a morphism of rings and Î¸â² is an auto-morphism of Râ² of order d such that Î¸â²Î± = αθ, then α naturally induces aring homomorphism of twisted polynomial algebras R[z, Î¸]â Râ²[z, Î¸â²] whichmaps the central elements Rθ[zd] into (Râ²)Î¸[zd].We want to globalize this construction. Assume that {Ui = SpecRi}i isa cover of X by open affine subschemes which are f -invariant. The intersec-tions Ui â© Uj = Uij := SpecRij and Ui â© Uj â© Uk := SpecRijk are affine asX is separated and are also f -invariant. The automorphism f restricts to242.4. The Moduli Problem via Noncommutative Sheaves of AlgebrasUi and hence gives a ring automorphism θi : Ri â Ri. Similarly, we get au-tomorphisms θij and θijk of Rij and Rijk respectively which are compatibleunder the ring homomorphisms Ri â Rij â Rijk.Ri Rij RijkRi Rij Rijkθi θij θijkWe recall the construction of the scheme X/f first. The invariantsRθii â Ri give categorical quotients Ïi : Ui â Vi = SpecRθii which arefinite surjective maps. Since Uij is an invariant open subset of Ui, the im-age Ïi(Uij) â Vi is also open. Thus Uij â Ïi(Uij) is also a categoricalquotient, i.e. the image is precisely SpecRθijij â Vi. Similarly the imageÏj(Uij) in Vj is SpecRθijij â Vj . So we can glue Vi and Vj along Vij = Uij/f .By the same reasoning, the image of Uijk under the quotient Uij â Vij isSpecRθijkijk â Vij and analogously for Ujk â Vjk and Uik â Vik. Thereforethe cocycle condition for gluing holds and the schemes {Vi}i glue to give thequotient X/f .Let Ï : X â X/f be the quotient map. Consider the quasi-coherentsheaf B0(f) = Ïâ(OX [z]) on X/f . Equivalently on each Vi we consider thequasi-coherent sheaf RËi[z]. An algebra structure on B0(f) is an OX/f -linearmultiplication map m : B0(f)âOX/f B0(f)â B0(f), or equivalently, a Rθii -algebra structure on each Ri[z] which is compatible under restriction. Setthe multiplication mi : Ri[z]âRÎ¸ii Ri[z]â Ri[z] as twisted multiplication viaÎ¸â1i . Then it restricts to Vij as(Ri[z]âRÎ¸ii Ri[z])âRÎ¸ii Rθijij Ri[z]âRÎ¸ii RθijijRij [z]âRÎ¸ijijRij [z] Rij [z]miâ1â¼= â¼=where restriction is by the tensor product RiâRÎ¸ii RÎ¸ijijâ¼= Rij . Since the au-tomorphisms are compatible under Ri â Rij , the restriction is just twisted252.4. The Moduli Problem via Noncommutative Sheaves of Algebrasmultiplication by Î¸â1ij . So m is well-defined. Since Rθii [zd] is central, B0(f)is the pushforward of an algebra A0(f) on Y0 := X/f Ã A1zd .We can perform the same construction with the sheaf Bâ(f) := ÏâOX [zâ1]with the multiplication Ri[zâ1]âRÎ¸iiRi[zâ1]â Ri[zâ1] given by twisted mul-tiplication by θi. Then Bâ(f) is the pushforward of an algebra Aâ(f) onYâ := X/f Ã A1zâdThe algebras A0(f) and Aâ(f) glue on the intersection X/fÃGm whereGm = Spec k[zd, zâd]. This is clear as the multiplication on both algebrasrestricts locally on SpecRθii [zd, zâd] as the multiplication on the twistedLaurent ring R[z, zâ1, Î¸â1]. Let Y = X/f Ã P1 where P1 is obtained bygluing the affine lines A1zdand A1zâd (we drop the subscripts henceforth).The discussion above proves the next theoremTheorem 2.4.6. Let Y be the quotient of XÃP1 by the group Z/dZÃZ/dZwith the first factor acting on X by f and the second factor acting on P1 bya primitive d-th root of unity. Let q : X Ã P1 â Y be the quotient. ThenqâOXÃP1 has a noncommutative algebra structure A(f) as described above.Let Cf be the category whose objects are the pairs (F , Ï) with F a quasi-coherent sheaf on X and Ï : F â fâF a morphism. The morphisms aredefined in the same way as in Section 1.2.Lemma 2.4.7. The functor Ïâ induces an equivalence of categories Cf andQCohX/f (B0(f)).Proof. For (F , Ï) in Cf , since fâF = (fâ1)âF we have ÏâfâF = ÏâF .Applying Ïâ to Ï gives amorphism ÏâÏ : ÏâF â ÏâF . Checking that ÏâÏdefines a B0(f)-module structure on ÏâF is a local question, so assume thatF = MË on SpecR which is an f -invariant affine open inX and f correspondsto a ring automorphism θ of R. Then fâF is the sheaf induced by the moduleM Î¸â1which is M as an abelian group but with R-action via fâ1. Further,ÏâF = RËÎ¸M and Ï :M âM Î¸â1 satisfies Ï(rm) = Î¸â1(r)Ï(m) for all r â Rand m âM . HenceÏâÏ :Rθ M âRÎ¸ (M Î¸â1) =Rθ M262.4. The Moduli Problem via Noncommutative Sheaves of Algebrasdefines a left R[z, Î¸â1]-module structure on RθM .Conversely, if NË is a quasi-coherent R[z, Î¸â1] module on SpecRθ then Nhas a structure of an R-module and z : N â N satisfies z(rn) = Î¸â1(r)z(n)so z is a map N â N Î¸â1 of R-modules. This completes the proof.Lemma 2.4.8. The category QCohX/f (B0(f)) is naturally equivalent toQCohY0(A0(f)). The equivalence restricts to an equivalence between co-herent B0(f)-modules and coherent A0(f)-modules with proper support overX/f .Proof. This is the same proof as in Section 2.2. Just note that Y0 isX/fÃA1where the affine line has coordinate zd. Thus we consider the endomorphism(ÏâÏ)d in the local setting above to get the result.It is clear that all the constructions above are compatible under base-change. That is, if T is any k-scheme, then carrying the construction forthe automorphism fT of X Ã T gives the sheaf of algebras B0(f)T on (X ÃT )/fT â¼= X/f Ã T and similarly, the algebras A0(f)T (resp. A(f)T ) onY0 Ã T (resp. Y Ã T ).Let Cohn(Y0,A0(f)) be the category fibered in groupoids whose ob-jects are (T,F ) â Cohn(Y0) with F a coherent A0(f)T -module. Themorphisms (T â²,F â²) â (T,F ) are the morphisms (h, Ï) in Cohn(Y0) withÏ : hâY0F â F â² an isomorphism of A0(f)Tâ²-modules. Define the categoryCohn(X/f,B0(f)) similarly.Theorem 2.4.9. The stack Pn(X, f) is canonically isomorphic to the cat-egory fibered in groupoids Cohn(X/f,B0(f)). The latter is canonically iso-morphic to Cohn(Y0,A0(f)).Proof. This is the same proof as in Section 2.2 using the Lemmas 2.4.6and 2.4.7 instead. Namely, the isomorphism takes an object (T,F ,Φ) âPn(X, f) to (T, ÏâF ) â Cohn(X/f,B0(f)).Remark 2.4.10. Let us use the natural notation YÃ := X/f Ã Gm andAÃ(f) := A(f)|YÃ. Suppose that Ï : F â fâF is an isomorphism. Let272.4. The Moduli Problem via Noncommutative Sheaves of AlgebrasQ be the corresponding A0(f)-module on Y0. Then Q is supported in YÃ.Indeed, this is because ÏâF , and equivalently (ÏâF)d, are isomorphisms.The converse is clearly true as well. It follows that the open substack (seeLemma 2.3.6) of Pn(X, f) parametrizing pairs (F , Ï) with Ï an isomorphismis naturally isomorphic to Cohn(YÃ,AÃ(f)).The twisted polynomial/Laurent rings are standard and have appearedin similar contexts in the literature, see for example [2]. To the best of ourknowledge, the global construction of the algebra A(f), although standard,has not been spelled out explicitly in the literature. Noncommutative pro-jective schemes have been introduced by Artin and Zhang in their seminalpaper [4]. If A is a locally finite N-graded algebra over an algebraicallyclosed field, we have the categories Gr(A) of finitely generated graded A-modules and T (A) â Gr(A) of torsion A-modules. Abstractly, the noncom-mutative projective scheme associated to A is really the quotient categoryGr(A)/T (A) with the shift functor shifting the grading of modules. Now,for a projective scheme S with a polarization OS(1) and a coherent sheaf Aof noncommutative OS-algebras, form the homogenous coordinate ringA =âmâ¥0H0(X,A(m))which is an N-graded locally finite noncommutative algebra. For F a coher-ent A-module, form the associated graded A-modulegrA(F) :=âmâ¥0H0(X,F(m))It is proven that the association F â grA(F) induces a graded equivalenceof the category CohX(A) of A-modules and the noncommutative projectivescheme associated to A abstractly.In our case, although Y will be projective, we do not have a naturalpolarization on X or Y (and we do not need to mention one) therefore wedo not pursue this line of approach.282.5. Stability Conditions2.5 Stability ConditionsWe continue with the assumptions on X and f from the last section. Whenworking in this setup, we will fix the notation Z := X/f and Ï : X â Z forthe quotient map. Let p = pZ : Y0 = Z Ã A1 â Z be the projection.Let F be a zero-dimensional sheaf of length n. For a twisted endomor-phism Ï : F â fâF and a section s â Î(X,F), the stability condition wewant to impose is: The sections Ïks; k â¥ 0 generate the sheaf F . A littleelaboration is needed for this statement. First, Ïk is shorthand for the k-foldcomposition((fâ)kâ1Ï) â¦ · · · â¦ (fâÏ) â¦ Ï : F â (fâ)kFThen Ïks â Î(X, (fâ)kF) and this vector space is canonically isomorphic toÎ(X,F). In this way, we view all sections as sections of F . This is a naturalstability condition to consider (see Remark 2.5.7 later).Lemma 2.5.1. For (F , Ï, s) as above, the following notions of stability areall equivalent:1. The sections {Ïks; k â¥ 0} generate F .2. There is a zero-dimensional sheaf G of length n on Z with ÏâF = Gand a surjection B0(f)â G of B0(f)-modules.3. There is a zero-dimensional sheaf Q of length n on Y0 with pâQ = Gand a surjection A0(f)â Q of A0(f)-modules.Proof. If (1) holds then ÏâF gets the structure of a left B0(f)-module byLemma 2.4.7 and s â Î(X,F) = Î(Z, ÏâF) defines a morphism B0(f) âF of B0(f)-modules. In this formulation, the stability condition is thatB0(f) â F is surjective. So (1) and (2) are equivalent. The equivalence of(2) and (3) follows straightaway from Lemma 2.4.8.Definition 2.5.2. Define the contravariant functor Hn(X, f) : Schk â Setas follows. For a k-scheme T , Hn(X, f)(T ) consists of equivalence classesof triples (F ,Φ, s) where F is a T -family of length n sheaves on X, Φ :292.5. Stability ConditionsF â fâF and s â Î(X,F ) is a section such that the sections Φis, i â¥ 0generate F . The equivalence relation is: (F ,Φ, s) â¼ (F â²,Î¦â², sâ²) iff there isan isomorphism Ψ (as in Section 2.3) with Ψs = sâ².For a morphism h : T â² â T , the induced morphism Hn(X, f)(T ) âHn(X, f)(T â²) is obtained by pullback.The goal of this section is to show that Hn(X, f) is representable by aquasi-projective scheme.Remark 2.5.3. By Lemma 2.5.1, the functor Hn(X, f) is isomorphic tothe (contravariant) functor sending T to equivalence classes of quotientsA0(f)T â Q â 0 of A0(f)T -modules, where Q is a T -family of length nsheaves on Y0. It is also isomorphic to the (contravariant) functor sendingT to equivalence classes of quotients B0(f)T â G â 0 of B0(f)T -modules,where G is a T -family of length n sheaves on Z. Therefore we identifyHn(X, f) with either of these functors.The representability of the Hilbert functor for sheaves of noncommuta-tive algebras is addressed next. Although this is âwell-knownâ by the resultsin [47], we give a direct proof which might be illuminating.Theorem 2.5.4. Let (Y,OY (1)) be a projective scheme with a polarizationand let A be a sheaf of coherent OY -algebras. Let h be a polynomial withinteger coefficients. There exists a projective scheme Hilbh(Y,A) which rep-resents the functor sending a scheme T to equivalence classes of quotientsAT â E â 0 of A-modules where for any geometric point t â T , Et on Ythas the Hilbert polynomial h with respect to the polarization.In particular, if h â¡ n is a constant polynomial, the Hilbert schemeHilbn(Y,A) exists and is independent of the polarization.Proof. Consider the Quot scheme Q = Quoth(Y,A) which is projective. LetAQ â E â 0 be the universal quotient on Y ÃQ and let K be the kernel.For a point q â Q corresponding to the quotient A â Eq â 0, its kernel isKq by Lemma 2.1.4 of [24]. The quotient Eq has an A-module structure iffKq is A-invariant, i.e. the compositionKq âOY A â AâOY A â Aâ Eq302.5. Stability Conditionsvanishes (the middle map is the multiplication of A). Therefore the desiredHilbert scheme will be the âzero-locusâ of the natural mapα : K âOYÃQ AQ â AQ âOYÃQ AQ â AQ â ESince pQ : Y ÃQâ Q is a projective morphism and E is Q-flat, the resultsin Corollaire 7.7.8 and Remark 7.7.9 in [20] imply that there is a coherentsheaf N on Q such thatHomOY (N ,M) â¼= pQâHomOYÃQ(K âOYÃQ AQ,E âOYÃQ pâQM)for every coherent sheaf M on Q. Taking M = OQ, we get a morphismN â OQ corresponding to α. The image of N â OQ defines a closedsubscheme H â Q satisfying the representability property: a map T â Q,equivalently a T -family of quotients AT â E â 0 with kernel K, factorsthrough H iff KâOYÃT AT â E is zero. Thus H is exactly Hilbh(Y,A) andis projective as well.We remark that the sheaf N is only coherent, so Hilbh(Y,A) is notnecessarily the zero locus of a section of a vector bundle on Quoth(Y,A).The scheme Quoth(Y,A) may not be smooth even in the case h â¡ n. Sothis does not necessarily provide a global Kuranishi model for Hilbh(Y,A).In order to pass to an open subscheme U of Z, the Hilbert functorHilbn(U,AU ) parametrizing sheaves with support in U makes sense and isdescribed by the next lemma.Lemma 2.5.5. For a quotient AT â E â 0 in Hilbn(Y,A)(T ) the set ofpoints {t â T | supp Et â U} is open in T . Hence Hilbn(U,AU ) is an opensubscheme of Hilbn(Y,A).Proof. At the level of topological spaces we have an equality | supp Et| =| supp E â© Yt|. Since supp E is finite over T , the image of supp E â© (Y âU) is closed in T . The complement, which is the desired set, is open inT . Applying this to the universal family over Hilbn(Y,A) gives an opensubscheme Hilbn(Y,A)U which we claim represents Hilbn(U,AU ).312.5. Stability ConditionsLet i : U â Y denote the open embedding. For a quotient (AU )T âE â 0 in Hilbn(U,AU )(T ), we may apply iT â and compose with the naturalmap A(f)T â iT âiâTAT = iT â(AU )T to get a morphism AT â iT âE ofleft AT -modules. This is surjective because we can restrict to supp E =supp(iT âE) for the calculation and (AU )T |supp E = AT |supp E â E|supp E issurjective. Hence we get a map Hilbn(U,AU ) â Hilbn(Y,A). Clearly, thismap factors through Hilbn(Y,A)U . Furthermore, the map Hilbn(Y,A)U âHilbn(U,AU ) is given by restricting the universal quotient over Hilbn(Y,A)Uto the subscheme U ÃHilbn(Y,A)U . These maps are clearly inverses of eachother, which proves the proposition.Since Y0 â Y is open, the Hilbert scheme Hilbn(Y0,A0(f)) also existsand is an open subscheme of Hilbn(Y,A(f)). We have provedTheorem 2.5.6. The functor Hn(X, f) is represented by Hilbn(Y0,A0(f)).Remark 2.5.7. For f = idX , the stability condition gives the Hilbert schemeof points on X Ã A1, i.e. Hn(X, idX) is represented by Hilbn(X Ã A1).Remark 2.5.8. The Hilbert scheme Hilbn(X) of n points on X is a closedsubscheme of Hn(X, f) because it is described by the locus of points (F , Ï, s)such that Ï = 0 (see Lemma 2.3.7). In general, Hn(X, f) is much largerthan Hilbn(X), as the examples below will show.Remark 2.5.9. The locus of points (F , Ï, s) such that Ï is an isomorphismis open in Hn(X, f) but this does not mean that F descends to the quo-tient stack [X/f ] as the isomorphism Ï is not required to satisfy the cocyclecondition.Example 2.5.10. As an example, we discuss (without giving much impor-tance to complete details) how H1(X, f) should look like. Let Ï : F â fâFbe a twisted endomorphism. The sheaf F must be a skyscraper sheaf, i.e.supported at a single point. If F is supported on a non-fixed point of f , thenÏ = 0. If F is supported on a fixed point of f , then there is a one-dimensionalspace of morphisms F â fâF corresponding to scalars. If (F , Ï, s) is apoint in Hn(X, f), then s Ì¸= 0. After normalizing s = 1, it follows that322.5. Stability Conditions(F , Î»1Ï, 1) and (F , Î»2Ï, 1) are not isomorphic if λ1 Ì¸= λ2 and Ï Ì¸= 0. Insummary, H1(X, f) has an open stratum isomorphic to the complement ofthe fixed locus and closed strata isomorphic to the trivial line bundle overevery component of the fixed locus. The zero locus Hilb1(X) = X inter-sects transversally with the trivial line bundle over the fixed locus. (Withslightly more work, it is easy to make this argument rigorous by workingwith families).Example 2.5.11. Even in the next simplest case n = 2, the situation be-comes vastly complicated. We list the various cases when X = P2 with theautomorphism f acting by the diagonal matrix diag(â1,â1, 1). Let (F , Ï, s)be a stable triple in H2(X, f).Case I: Let F be a sheaf of length 2 supported on a single point. Ifthe point is not a fixed point, then Ï = 0 and hence (F , s) defines a closedsubscheme of X of length 2. If the point is a fixed point, then there are extratriples.Case II: Let F be a direct sum of two skyscraper sheaves. The possibilitiesfor the support are: two fixed points, one fixed and one non-fixed point, twonon-fixed points which are not in the same orbit and two points in the sameorbit. The first three cases are described in the same way as in the previousexample, however, the last case is more interesting. If {p, q} is the orbit, thenF = Cp â Cq and hence a morphism Ï : F â fâF is described by a pair oflinear maps z1 : Cp â Cq and z2 : Cq â Cp. Let s = (sp, sq) â CpâCq. Thestability condition implies that one of sp, z2sq is nonzero and one of sq, z1spis nonzero. Let sp = 1, sq = 0, z1 an isomorphism and z2 be the zero map.The points corresponding to such triples (F , Ï, s) are not in Hilb2(X) andÏ is not an isomorphism either.It is a tedious task to try to describe the moduli spaces Hn(X, f) thisway. In the next chapter, we will use a different strategy to give a descriptionof the strata of this moduli space via quiver representations.Remark 2.5.12. Let X be a connected variety. We conjecture that Hn(X, f)is a connected scheme for every n â¥ 1. The rough idea is to degenerate everysheaf to one supported on a single point - similar to Hartshorneâs proof in332.5. Stability Conditions[22] for the classical Hilbert schemes. The analogous statement is no longertrue for the compactification Hilbn(Y,A(f)) as can be seen by consideringHilb1(Y,A(f)) for a fixed-point free f . However, we do think that the casen = 1 is the only case where this fails.34Chapter 3Euler Characteristics of theModuli SpacesIn this chapter, we compute the Euler characteristics of the spacesHn(X, f).The first two sections consist of preparatory material on quiver representa-tions and Hilbert schemes of associative algebras. In Section 3.3 we applythese to study Cayley quivers. In Section 3.4, we determine local modelsfor Hn(X, f) via a Hilbert-Chow morphism Hn(X, f) â Symn(X/f). Thestrategy is to count contributions from sheaves whose support lies in a fixedorbit, and hence we use the quotient scheme X/f . The fibers over the deep-est stratum are Hilbert schemes of various associative algebras arising fromthe quivers in 3.3. The calculations are put together in the last section.3.1 Basics on QuiversA quiver Q = (Q0, Q1) consists of a finite set Q0 of vertices and a finite setQ1 of directed edges between the vertices of Q. A representation V of Q isthe data of a finite dimensional vector space Vi for each vertex i â Q0 anda linear map of vector spaces Tji : Vi â Vj for each edge (iâ j) â Q1. Thehead of the edge (iâ j) is i and the tail is j. The dimension vector of V is(dimVi)iâQ0 â N|Q0|.A path in Q is a sequence of edges ananâ1 . . . a1 such that the tail of aiequals the head of ai+1 for all i. Therefore the convention we adopt is thatthe path ananâ1 . . . a1 is read from right to left, i.e. the first edge is a1. Thepath algebra kQ of the quiver is defined as follows: As a vector space, it hasa basis consisting of all paths in the quiver, including a trivial path ei looping353.2. Hilbert Schemes of Points for Associative Algebrasat i for every i â Q0. The multiplication p2 · p1 is given by concatenationof the path p2 with p1 if the ending vertex of p1 coincides with the startingvertex of p2, and is 0 otherwise. In kQ the elements ei are orthogonalidempotents: e2i = ei for every i, eiej = 0 for i Ì¸= j andâiâQ0 ei = 1. Inthis way kQ becomes a unital associative graded k-algebra, graded by pathlength. As an algebra, kQ is generated over (kQ)0 by (kQ)1, the degree 1paths which are the edges.It is well known that a representation of Q is equivalent to a left kQ-module. Indeed, the correspondence takes a representation V to the leftkQ-module M = âiâQ0Vi. Since Homk(M,M) = âi,jâQ0 Hom(Vi, Vj), themodule structure is as follows: A trivial path ei acts as the identity onVi and zero otherwise. An edge (i â j) acts by the linear map Tji âHom(Vi, Vj) and zero otherwise. Conversely, if M is a left kQ-module weget a representation V with Vi = eiM . It is indeed a representation asan edge (i â j) equals ej(i â j)ei and therefore acts as a linear mapeiM â ejM . This correspondence induces an equivalence of the categoryof representations of Q and the category of left kQ-modules.3.2 Hilbert Schemes of Points for AssociativeAlgebrasA framing of the representation V is the data of a vector vi â Vi for eachi â Q0. Equivalently, it is an element m âM or a map kQâM of left kQmodules. The framing is said to generate V if the following condition holds:For every j â Q, the vectors pvi â Vj for paths from i to j as i ranges overQ0 generate Vj . In equivalent terms, the map kQâM is surjective.In [29], Larsen and Lunts have studied the Hilbert scheme of points forany finitely generated associative k-algebra. We give a quick description oftheir moduli space.Let A be a finitely generated associative algebra over k. For T â Schkdenote by AT the sheaf of OT -algebras associated to the presheaf OT âk A.363.3. Cayley QuiversDefinition 3.2.1. (The Hilbert Functor for A) Let MnA : Schopk â Set be thecontravariant functor defined as follows: For a scheme T , the set MnA(T )consists of equivalence classes of pairs (M,v) where M is a left AT -modulewhich is locally free of rank n as an OT -module and v â Î(T,M) generatesM as an AX-module. The equivalence relation is (M,v) â¼ (M â², vâ²) iff thereis an isomorphism Ï :M âM â² of AT -modules such that Ï(v) = vâ².A morphism h : T â² â T of sheaves gives an isomorphism of algebrashâAT â AT â² and hence gives the map of sets MnA(T )â MnA(T â²) by pullback.Theorem 3.2.2. The functor MnA is representable by a k-scheme HnA offinite type, called the Hilbert scheme of n points of A. The k-points of HnAparametrize the left ideals in A of codimension n.In the next section, we will compute the Euler characteristics e(HnA) forsome special associative algebras A. These Hilbert schemes will occur asstrata in Hn(X, f).3.3 Cayley QuiversDefinition 3.3.1. Let G be a finite abelian group with a set S = (f1, f2, . . . , fr)of generators (this set is NOT assumed to be minimal - some of the gen-erators can be superfluous). The Cayley quiver associated to this data is aquiver with relations QG = (Q0, Q1) where the vertex set Q0 = G and theedge set Q1 consists of the edges pgi := (g â gfi) for each g â Q0 and eachi = 1, 2, . . . , r.In the path algebra kQG, we will impose the âobviousâ relations: thepaths g â gfi â gfifj and g â gfj â gfjfi = gfifj must agree forevery g â G and every pair of generators fi, fj . Therefore the algebra withrelations is AG := kQG/R where R is the two-sided ideal generated by(pgfji pgj â pgfij pgi ) for all g â Q0 and i, j â {1, 2, . . . , r}.Let pi be the sum of edgesâgâG pgi . Let {eg, g â G} be the idempotentsat each vertex of QG. Then pieg = pggfi= egfipi for all g â G and i =1, 2, . . . , r. Therefore kQG and AG are generated by the elements pi and373.3. Cayley Quiversthe idempotents eg. Because of the imposed relations, in AG we have anequalitypipj =ï£«ï£­âgâGpgiï£¶ï£¸ï£«ï£­âgâGpgjï£¶ï£¸ =âgâGpgfji pgj =âgâGpgfij pgi = pjpiHence we have the lemmaLemma 3.3.2. Any element of AG can be written uniquely as a sumâgâGeghg(p)where hg(p) = hg(p1, p2, . . . , pr) is a polynomial in the commuting indeter-minates pi.Proof. For any a â AG, let a(g) be the sum of all monomials in a whichgive paths ending at g â QG, so that a = eg(âgâG a(g)). Then it suffices toprove the claim for a single path ending at g â G. Such a path looks likeg(fâ1i1 fâ1i2· · · fâ1im )â · · · â gfâ1i1 â gwhere i1, i2, · · · , im are (possibly repeated) indices from {1, 2, · · · , r}. Thenthis path is clearly eg times the monomial pi1 · · · pim . The latter âmonomialâis actually a well-defined monomial in the indeterminates pi since thesecommute. So the claim is proved. The uniqueness of the representation isclear.Consider the ring D =âgâG k â¼=âgâG(keg). Every element h â Ginduces an automorphism θh of D by the permutation eg â egh. The auto-morphisms θh, h â G commute pairwise (because G is abelian). Therefore,we proved the next theoremTheorem 3.3.3. The algebra with relations AG is the twisted polynomialringD[p1, · · · , pr; θf1 , · · · , θfr ]383.3. Cayley Quiverswhere the twist on each pi is via θfi.We note that the algebra is extremely simple but we are not able to findit referenced in the literature. Cayley graphs have been studied extensively,albeit without the imposed relations.Since AG is a finitely generated associative k-algebra, the âHilbert schemeof pointsâ of AG exists by the previous section. The Euler characteristic ofHnAG can be computed by utilizing torus actions. For simplicity, we prove itonly for r = 3 because this is the case we need.We use the results of Bialynicki-Birula in [9]. For an algebraic scheme Xwith a Gm-action, the Euler characteristic e(X) is the Euler characteristic ofthe fixed locus e(XGm). We also use the fact that if T â¼= Grm is an algebraictorus acting on an algebraic scheme, the T -fixed points are the Gm-fixedpoints induced by general weights Gm â T . So we will use T -actions for alarger torus T .Theorem 3.3.4. Let G be generated by a set of generators {f1, f2, f3}. Thene(HnAG) =ââgâG ng=n,ngâ¥0âgâGp3(ng)Proof. The torus T = G3m acts on the ring AG by scaling the indeterminatesp1, p2, p3 and hence acts on the C-points ofHnAG . We characterize the T -fixedideals, so that e(HnAG) is the count of these.Let I â AG be a left ideal which is fixed under the action. Let h =âgâG eg · hg(p1, p2, p3) â I. Write hg(p1, p2, p3) =âmâ¥0 hg,m(p1, p2)pm3 .Thenh =âmâ¥0ï£«ï£­âgâGeg · hg,m(p1, p2)pm3ï£¶ï£¸Let us act by (1, 1, λ) to getâmâ¥0ï£«ï£­âgâGeg · hg,m(p1, p2)(Î»p3)mï£¶ï£¸ = âmâ¥0Î»mï£«ï£­âgâGeg · hg,m(p1, p2)pm3ï£¶ï£¸ â I393.3. Cayley Quiversfor every λ â Gm. In the outer sumâmâ¥0 there are only a finite num-ber of terms m = 0, 1, · · · , N . Using N + 1 distinct values λ0, λ1, · · ·λN ,the associated Vandermonde matrix being invertible implies that the sumâgâG eg · hg,m(p1, p2)pm3 â I for every m. Repeat the same process withhg,m(p1, p2). Continuing this way, we get that for all k, l,m â¥ 0, the sumâgâGeg · hg,klmpk1pl2pm3 â Ibut since I is a left ideal, multiplying by a fixed but arbitrary idempotenteg on the left shows that each term of the sum eg · hg,klmpk1pl2pm3 is also inI. Thus, every monomial term of h is in I, so I is a monomial left-ideal.A finite generating setM of monomials for I may be written as a disjointunionâgâGMg where Mg consists of the monomials corresponding to pathsending at g. As a k-vector space, AG is isomorphic to âgâG(eg) Â·k[p1, p2, p3].Therefore, the vector space quotient A/I is isomorphic toâgâG(A/I)g :=âgâG(eg) · k[p1, p2, p3]/â¨Mgâ©where â¨Mgâ© denotes the vector subspace generated by the monomials Mg.Finally, the number of fixed points is obtained by counting the monomialleft-ideals I such that dim(A/I)g = ng â¥ 0 withâgâG ng = n, and summingover all such tuples (ng). For a fixed tuple (ng), the number of such I isâgâG p3(ng), proving the formula.We will also need the Euler characteristic of the Hilbert schemes associ-ated to a noncommutative polynomial ring in three variables. Let kâ¨x, y, zâ©be a free polynomial algebra. Fix α, β, γ â kâ and let J be the two-sidedideal with generators xy â γyx, yz â αzy, zxâ βxz.Theorem 3.3.5. Let A = kâ¨x, y, zâ©/J . Then e(HnA) = p3(n).Proof. It suffices to observe that because of the relations in J , any elementh â A may be written uniquely as âhklmxkylzm; then the proof of 3.3.4applies in showing that any G3m-invariant left ideal of A is generated by403.3. Cayley Quiversmonomials.Definition 3.3.6. (Quiver representations and endomorphisms) Let Q =(Q0, Q1) be a quiver, with relations R. The endomorphism quiver of (Q,R)is the quiver Q+ whose vertex set is Q0 and the edge set is Q1 â¨¿ x wherex = (xv)vâQ0 is set of self-loops, one at each vertex of Q. The relations R+are: Along with R, we impose that for every edge (i â j) â Q1 we imposethe equality of pathsxj · (j â i) = (j â i) · xiThe definition of (Q+,R+) implies that a representation of (Q+,R+) isa representation of (Q,R) (equivalently, a kQ/R-module V ) with an endo-morphism x : V â V of kQ-modules.Remark 3.3.7. For a Cayley quiver associated to G and (f1, f2, · · · , fr),the description of Q+G is rather simple: It is the Cayley quiver associated toG and (f1, f2, · · · , fr+1) where fr+1 = 0G, the identity in G, is added as asuperfluous generator.Example 3.3.8. As an example, we show the quivers QG and Q++G (withoutwriting down all the relations) for (G,S) where G is cyclic of order 3 andS consists of one element, a generator of G.The quiver QGâ¢â¢ â¢z0z2z1The quiver Q++Gâ¢â¢ â¢y0x0y2x2x1y1z0z2z1413.3. Cayley QuiversA representation of Q++G is simply the data of a representation V of QGalong with two commuting endomorphisms.The next corollary is now a direct consequence of Theorem 3.3.4 andRemark 3.3.7 applied repeatedly.Corollary 3.3.9. Let G = Z/d. Then, kQ++G is isomorphic to a noncom-mutative twisted polynomial ring (âdâ1i=0 kei)â¨x, y, zâ© with rule zei = ei+1zfor all i. If A = kQ++G /R++ is the algebra with relations, then [x, y] =[y, z] = [z, x] = 0 in A ande(HnA) =ââdâ1i=0 ni=n,niâ¥0dâ1âi=0p3(ni)Remark 3.3.10. (Twisted CY-algebras and Twisted Superpotentials) Let Abe a N-graded, locally finite k-algebra with A0 = k. Let Ae := A âk Aop bethe enveloping algebra. The algebra A is called a twisted Calabi-Yau algebraof dimension d if the following two conditions hold (1) As an Ae-module,A has a finite length resolution by projective Ae-modules. (2) For i Ì¸= d,ExtiAe(A,Ae) = 0 and ExtdAe(A,Ae) = U as Ae-modules, for some invertibleAe-module U . We focus on the case when A is of the form kQ/R for aquiver with relations (Q,R). In this case, U is of the form 1Aµ (meaningthat A acts on the right by µ) for an automorphism µ of A. The algebra Ais Calabi-Yau of dimension d if µ is an inner automorphism, equivalently,U â¼= A as Ae-modules. In fact, the twisted CY-algebras coincide with theArtin-Schelter regular algebras, by [42], Lemma 1.2.Let Q be a (finite) quiver and µ an automorphism of kQ. A µ-twistedsuperpotential is a homogenous element Ï of degree n in the vector spacekQ/[kQ, kQ] which is invariant under the k-linear mapananâ1 . . . a1 â µ(a1)an . . . a2The corresponding derivation quotient is kQ/I where I is the two-sided idealgenerated by the derivatives âaÏ for a â Q1 (The derivative acts by cancel-lation: âa(ananâ1 . . . a1) = anâ1 . . . a1 if an = a and is zero otherwise). The423.4. Local Modelsexample in Corollary 3.3.9 is of this type. Observe thatÏ = (xyz + zxy + yzx)â (xzy + yxz + zyx)is invariant under the cyclic permutation automorphism of the quiver Q++Gand it is easy to compute that the ideal of derivatives equals xy â yx, yz âzy, zxâ xz. Therefore A is twisted CY of dimension 3, according to [43].3.4 Local ModelsLet f : X â X be an automorphism of order d, of a smooth surface.Assume that the order of f is a prime number. This assumption is only tosimplify the exposition, as in this case there are only two kinds of stabilizersof points: corresponding to a free locus and a fixed locus. We compute theEuler characteristics of the moduli spaces represented by Hn = Hn(X, f).In computations, it will be easier to work with the algebra B0(f). Thatis, we take the definition of Hn(T ) as a flat T -family of quotients B0(f)T âG â 0 with G on X/f Ã T = Z Ã T .If G on Z is zero-dimensional with a B0(f)-action, then the decom-position G = âzâsuppG Gz as coherent sheaves is also a decomposition ofB0(f)-modules, as the action is local. Thus, a Hilbert-Chow morphism cn :Hn(X, f)â Symn Z exists: On points, it takes a quotient [B0(f)â G â 0]to the coherent sheaf G (forgetting the action) and then applying the âusualâHilbert-Chow morphism for coherent sheaves,G ââzâsuppG(lengthz Gz) · zas in [45].To understand the local behavior of an automorphism near a fixed point,the lemma of H. Cartan (the original formulation is in [14]) will be useful.Lemma 3.4.1. Let G be a finite group acting on a complex manifold X ofdimension n. If p â X is a fixed point, the action of G can be linearized433.4. Local Modelsaround p, that is, there exist coordinates (x1, · · · , xn) on an analytic openneighborhood of p on which G acts linearly.Applying this to the situation at hand, we find that every component ofthe fixed locus of f is smooth: Because fd = idX , one can find coordinatesx, y around any fixed point such that the action of f is diagonal, even.Therefore the fixed locus is analytically locally a one-dimensional linearspace or an isolated fixed point. Let R â X be the fixed locus of f , clearlyR is a closed subscheme. Let W = X â R be the open subset on which facts freely: This is because the order of f is assumed to be a prime. LetÏ : X â Z be the quotient map. Restricted toW , the map Ï|W :W âW/fis a Galois cover.To understand the local structure of the moduli spaceHn(X, f), we mustfirst analyze the simplest case of the affine plane.Lemma 3.4.2. Consider the scheme X which is a disjoint union of d copiesof A2. Let f be the automorphism of X which is a cyclic permutation of thecomponents, so that Ï : X â A2 = Z = X/f is a trivial covering. Let Hn(n)be the pre-image of the deepest stratum Z â Symn Z under the Hilbert-Chowmorphism cn. Let Hn0 be the fiber over 0 â Z. Then Hn(n) is a trivial productbundle Z ÃHn0 .Proof. First note that even though X is not projective, the moduli spaceHn(X, f) still exists, say by Theorem 3.2.2. as X is the spectrum of the ringR = (ke0 Ã · · · Ã kedâ1)[x, y] and twisted endomorphisms Ï : F â fâF arethen equivalent to modules over a finitely generated associative algebra. Sothe Hilbert-Chow morphism exists too. Further, there is an obvious actionof Z on X: for any a â A2 = Z, translation on every component of X by a isan automorphism of X. Denote this translation by tXa . Then Ï â¦ tXa = ta â¦Ïwhere ta is just the translation on Z by a. Let 0 be the origin in Z and 0ithe origins in X for i = 0, 1, . . . , dâ 1.Now for a scheme T , an element in Hn(n)(T ) is given by a T -family ofquotients B0(f)T â G â 0 in H(T ) and a map a : T â Z, which isequivalently a section of Z Ã T â T with image the (topological space ofthe) support of G . Equivalently, let (F ,Φ : F â fâF , s) on X specify the443.4. Local Modelsdata of G . On X Ã T , we may translate the sheaf F and the data Φ, s viathe algebraic section a on each component. More precisely, consider the mapXÃT idXÃaâââââ XÃZ â X where the second map is the action of Z on X. Itinduces an isomorphismXÃT â XÃT such that in the pullback of the data(F ,Φ : F â fâF , s), the pullback sheaf is supported only on the sections0i Ã T . Said differently, we have translated G to 0Ã T â Z Ã T . Thereforethere is a map Hn(n) â Hn0 described on T -points as above, hence also amap Hn(n) â Z ÃHn0 . It is now clear that an inverse map Z ÃHn0 â Hn(n)is described by the inverse translation, which proves the lemma.The Euler characteristic of Hn has been computed in Corollary 3.3.9.Further, the proof of Theorem 3.3.4 shows that the Euler characteristicof the fiber Hn0 in the lemma above is equal to the Euler characteris-tic of Hn. Indeed, the T -fixed points of the scheme Hn are all in Hn0 .For notational clarity, we denote the fiber Hn0 by Fn,â where the â inthe superscript is because the fixed point set of f is empty. So we gete(Fn,â ) =ââdâ1i=0 ni=n,niâ¥0âdâ1i=0 p3(ni).Lemma 3.4.3. Consider X = A2 with coordinates x, y and let f be theautomorphism f : x â x, f : y â µy for a primitive d-th root of unity µ.The fixed locus R is the x-axis and the quotient Z is A2 with coordinatesx, yd. Identify R with its image in Z. Keeping the notation Hn(n) and Hn0from Lemma 3.4.2 we have câ1n (R) = RÃHn0 â Hn(n).Proof. The same proof as 3.4.2 goes through, except that we apply transla-tions only along the fixed locus.Once again, we observe that e(Hn0 ) = e(Hn). Here the algebra B0(f) isdescribed by the noncommutative polynomial ring kâ¨x, y, zâ©/(xy â yx, yz âµzy, zxâ xz) on Z. The T -fixed points of Hn are all in Hn0 . We denote thefiber Hn0 in this case by Fn,1µ where the 1 in the superscript indicates thatthe fixed locus is one-dimensional. So e(Fn,1µ ) = p3(n).To analyze the map cn for general X, we will reduce to the trivial casesstudied above. We follow the ideas in Appendix A of [5]. By abuse of453.4. Local Modelsnotation, we just use Ï for Ï|W : W â W/f and B0(f) for B0(f)|W/f sincerestriction to an open doesnât really change anything, courtesy Lemma 2.5.5Since W âW/f is a Galois cover, the base change W ÃW/f W is a disjointunion of d copies of W . By flat base change in the diagramWâ. . .âW WW W/fÏÏwe see that on ÏâB0(f) = OâdW [z], the noncommutative algebra structurecomes from the cyclic permutation automorphism Ï of Wâ. . .âW . Notethat the Hilbert scheme Hn(W â . . .âW,Ï) = Hilbn(W,OâdW [z]) also existsas the disjoint union is quasi-projective. (Although Ï = Ï, we use a differentletter in order to emphasize that we will be working with the base changemap).Lemma 3.4.4. Let [OâdW [z]â Qâ 0] be a point in Hilbn(W,OâdW [z]) suchthat Ï is injective on | suppQ|. Then the compositionB0(f)â ÏâÏâB0(f) â¼= ÏâOâdW [z]â ÏâQis a surjection, hence defines a point in Hn(W, f) â Hn(X, f).Proof. First, by suitably shrinking, we may replaceW by an open neighbor-hood U of | suppQ| with the property that for any w â | suppQ| we haveÏâ1(Ï(w)) = {w}. To check surjectivity, note that ÏâQ is supported only atthe points in Ï(| suppQ|) so we may replace W/f by the image Ï(U) (whichis open as Ï is etale, and therefore, can assume that Ï is faithfully flat.By faithful flatness for quasi-coherent sheaves, it now suffices to check thatÏâB0(f)â ÏâÏâQ is surjective. But Ï is etale and bijective on | suppQ|, sothe canonical map ÏâÏâQ â Q is an isomorphism. Thus ÏâB0(f) â ÏâÏâQmay be identified with the original surjection OâdW [z] â Q, proving thelemma.The rule above associating to P = [OâdW [z]â Qâ 0] the âpushforwardâ463.4. Local Modelsquotient [B0(f) â ÏâQ â 0] may be denoted by Ïâ again by abuse ofnotation. It is well-defined for P but we show in fact, that it is well-definedon an open neighborhood of P .Lemma 3.4.5. There is an open neighborhood V â Hilbn(W,OâdW [z]) of thepoint P and a map V â Hn(W, f) extending the rule Ïâ to V .Proof. Consider the commutative diagramW Ã P W ÃHilbn(W,OâdW [z]) WW/f Ã P W/f ÃHilbn(W,OâdW [z]) W/fiÏpWÏË Ïj pW/fIn the right square, the base change map pâW/fÏâ â ÏËâpâW is an isomorphismbecause Ï is affine. Let pâWOâdW [z] â Q â 0 be the universal family ofquotients on W ÃHilbn(W,OâdW [z]).From the canonical map B0(f)â ÏâÏâB0(f) = ÏâOâdW [z] we get a canon-ical compositionpâW/fB0(f)â pâW/fÏâOâdW [z] â¼= ÏËâpâWOâdW [z]â ÏËâQ (3.4.1)onW/fÃHilbn(W,OâdW [z]). We know by Lemma 3.4.4 that (3.4.1) restrictedto W/f Ã P in the left square is a surjection. To extend the rule Ïâ to anopen neighborhood of P , first observe that ÏËâQ is coherent (as ÏË is properand Q has proper support) so the cokernel K of (3.4.1) is coherent as well.Then the image of suppK under the projection pW/f is also closed (strictlyspeaking, we need to use properness of W/f which it isnât, but we cancompactify to Z to justify this assertion). If V is the open complement, weclaim that the rule Ïâ extends on V .It is immediate from the construction that on W/f ÃV , (3.4.1) restrictsto a surjection. Also, V is nonempty as it contains P . It only remains tocheck the flatness of ÏËâQ over V , which follows from [48, Tag 0FLM] asboth base-changes in the diagram are affine. In this way we have obtainedthe map V â Hn(W, f).473.4. Local ModelsRemark 3.4.6. The map V â Hn(W, f) is etale too, by an application ofthe formal criterion for etaleness. This fact is not needed later.Let O â Hn(W â . . .âW,Ï) be the open subscheme satisfying the fol-lowing property: If [OâdW [z]â Qâ 0] is a point in O, then Ï is injective on| suppQ|. As a consequence of the two Lemmas 3.4.4 and 3.4.5 there is an(etale) morphism O â Hn(W, f). By the subscript (n), we will mean thepreimage of the deepest stratum, as in 3.4.2.Corollary 3.4.7. There is a Cartesian squareHn(n)(Wâ. . .âW,Ï) Hn(n)(W, f)W W/fcn cnÏwhere the vertical maps are the âHilbert-Chowâ maps on the deepest strata.Proof. The fact that the square is Cartesian is immediate after we knowthe horizontal map Hn(n)(Wâ. . .âW,Ï) â Hn(n)(W, f) exists. It does, asHn(n)(Wâ. . .âW,Ï) â O.Corollary 3.4.8. The induced map Hn(n)(W, f) = câ1n (W/f) â W/f is alocally trivial fiber bundle with fiber Fn,â in the analytic topology.Proof. The scheme W has a (finite) cover by Zariski open neighborhoods{Wi}i with etale maps Wi â A2. Applying the previous theory to the etalemaps Wi â A2 and the Cartesian squareWiâ. . .âWi A2â. . .âA2Wi A2we see from Lemma 3.4.2 that Hn(n)(Wâ. . .âW,Ï)âW is Zariski locallytrivial on the cover, with fiber Fn,â . Then Hn(n)(W, f) â W/f is analyticlocally trivial with the same fiber as Ï : W â W/f is analytic locally, anisomorphism.483.4. Local ModelsRemark 3.4.9. We remark that there is no hope for the fibrations above tobe Zariski locally trivial. Even in the simplest example n = 1, the fibrationcâ11 (W/f) = W â W/f is not Zariski locally trivial, so we are forced tostate the result in the analytic category.Corollary 3.4.10. Let R1 in R be any 1-dimensional component. Theinduced maps câ1n (R1) â R1 are analytically locally trivial fiber bundleswith fiber Fn,1µ for µ a primitive dâ th root of unity intrinsic to f .Proof. The proof is identical to Corollary 3.4.8 but in the analytic category:By the Cartan Lemma, R1 â X has a finite cover by analytic open f -invariant neighborhoods {Ui} â A2 such that Ui â Ui/f is equivariantto the automorphism in Lemma 3.4.3. It only remains to note that µ isconstant on R1. The natural isomorphism fâÏX â ÏX is a fixed nonzeroscalar on all of X, which is precisely µ.Remark 3.4.11. For sheaves supported on an isolated fixed point R0 â X,the analysis is the easiest. By the Cartan lemma, we may choose analyticlocal coordinates x, y on which the action of f is diagonal. The completionof the local ring at R0 is isomorphic to S = Cx, y with the induced au-tomorphism f# on S acting diagonally. It is well-known (see, for instance,[18]) that the punctual stack of coherent sheaves on X supported at R0 canbe understood by passing to S via an isomorphism. Thus, a coherent sheafF of length n supported at R0 is equivalent to a vector space V of dimensionn with an action of S. By the Nakayama Lemma, if mS â S is the maximalideal, then mnS Â·V = 0 and we may also restrict to an action of C[x, y] on V .Finally, a twisted endomorphism Ï : F â fâF is a linear map z : V â Vsuch that zx = f#(x)z = µxz and zy = f#(y)z = νyz for some d-th roots ofunity µ, ν. We conclude that the stratum Hn(X, f)R0 of sheaves supportedon R0 is (non-canonically) isomorphic to the Hilbert scheme HnA of the al-gebra A = kâ¨x, y, zâ©/(xyâyx, yzâÎ½â1zy, zxâµxz). By Theorem 3.3.5, theEuler characteristic of Hn(X, f)R0 is p3(n).Denote by HnW = Hn(X, f)W the inverse image câ1n (Symn(W/f)). It isopen in Hn and consists of the quotients supported in W/f . Let EW (q) :=493.5. Computation of the Euler Characteristicânâ¥0 e(HnW )qn be the generating function for the Euler characteristics ofHnW . For any one-dimensional component of R, define the series ER1(q) anal-ogously. The computation of these series is similar to that in [8], Proposition4.11 by using a stratification of Symn(W/f).Lemma 3.4.12. We have the equalities EW (q) = M(q)e(W ) and ER1(q) =M(q)e(R1).Proof. In Corollary 3.4.5 it was shown that over the deepest stratumW/f âSymn(W/f), the scheme Hn(X, f)|W restricts as an analytically locallytrivial bundle with fiber Fn,Ï. Now using Proposition 5.9 in [36], we getEW (q) = (ânâ¥0 e(Fn,Ï)qn)e(W/f). Substituting the expression for e(Fn,Ï)shows immediately thatEW (q) =ï£«ï£¬ï£­ï£«ï£­ânâ¥0p3(n)qnï£¶ï£¸dï£¶ï£·ï£¸e(W/f)=M(q)e(W )where we used the fact that e(W ) = d · e(W/f). The proof of the secondequality is identical, using the analytic local fibration câ1n (R1)â R1.3.5 Computation of the Euler CharacteristicWe can now prove the main theorem of this chapter.Theorem 3.5.1. Let X be a smooth projective surface and f : X â X bean automorphism of prime order. Let E(q) :=ânâ¥0 e(Hn(X, f))qn be thegenerating function for the topological Euler characteristics. Then we havethe equality E(q) =M(q)e(X).Proof. There is a stratificationX =Wâ(âiR1i)â(âiR0j)where R1i and R0j denote the one-dimensional and zero-dimensional com-ponents (isolated fixed points) of the fixed locus of f . For fixed n, let503.5. Computation of the Euler CharacteristicCn be the set of non-negative integer tuples m,n1i and n0j such that n =m+âi n1i +âj n0j . The stratification induces a stratificationHn(X, f) =âCnHn(X, f)(m,n1i ,n0j )where Hn(X, f)(m,n1i ,n0j ) is the stratum parametrizing quotients B0(f) âG â 0 such that G|W/f is of length m, G|R1i is of length n1i for every i, andG|R0j is of length n0j for every j. Thus,Hn(X, f)(m,n1i ,n0j ) = HmW Ã(âiHn1iR1i)Ãï£«ï£­âjHn0jR0jï£¶ï£¸and E(q) = EW (q) · (âiER1i(q)) · (âiER0j (q)). Substituting for all the serieson the right hand side yieldsE(q) =M(q)e(W ) Â·M(q)âi e(R1i ) Â·M(q)âj 1 =M(q)e(X)completing the proof.The proof of the case when the automorphism f is not of prime orderis entirely similar, but with more strata to handle. Indeed, let Râ² be acomponent of X which is fixed by fk for some k dividing d (In the casewhen d has prime order, k is necessarily 1 or d giving the fixed and freelocus respectively). According to Lemma 3.4.12, the key step is to prove thatERâ²(q) =M(q)e(Râ²) and this is enough to prove Theorem 3.5.1 generally. Weproceed to show this by a trick.Let p â Râ² and let l = d/k. The orbit of p consists of k points p =p0, . . . , pkâ1. Applying the Cartan Lemma to fk, we have analytic localcoordinates x0, y0 at p such that fk acts on x0, y0 diagonally by µ and νrespectively, for some l-th roots of unity µ, ν. (If Râ² is a one-dimensionalcomponent, µ may be chosen to be 1). Let xi, yi be the coordinates at piinduced by x0, y0 at p. Let Ï : F â fâF be a twisted endomorphism withF supported in the orbit of p. It follows that Ï : F â fâF is described by513.5. Computation of the Euler Characteristicrepresentations of the quiver in Example 3.3.8 (with size k) but with twodifferent relationszkâ1xkâ1 = µx0zkâ1andzkâ1ykâ1 = Î½y0zkâ1These two different relations arise because of the âlastâ step in completingthe cycle in the quiver.To get a nice description of the algebra with relations of this quiver asin Section 3.3, let α, β be numbers satisfying αk = µâ1 and βk = Î½â1. Thetrick is to set x =âkâ1i=0 αixi and y =âkâ1i=0 βixi instead. Let z =âkâ1i=0 zias in Section 3.3. We computezx =kâ1âi=0Î±izixiandxz =kâ2âi=0Î±i+1xi+1zi + x0zkâ1sozxâ Î±â1xz = Î±kâ1zkâ1xkâ1 â Î±â1x0zkâ1 = (Î±kâ1µâ Î±â1)x0zkâ1 = 0and similarly zy â Î²â1yz = 0. It follows that the algebra with relations isdescribed by(kâ1âi=0kei)â¨x, y, zâ©/(xy â yx, yz â βzy, zxâ Î±â1xz)Hence, the Euler characteristic of the Hilbert scheme of points of this asso-ciative algebra is described by Theorem 3.3.4 again.Lemma 3.5.2. If Râ² is a component fixed by fk, then ERâ²(q) =M(q)e(Râ²)Proof. The same arguments imply that câ1n (Râ²) â Râ² is an analytic localfibration. The fiber Fn is the moduli space of codimension n left ideals of523.5. Computation of the Euler Characteristicthe algebra(kâ1âi=0kei)â¨x, y, zâ©/(xy â yx, yz â βzy, zxâ Î±â1xz, xn, yn)Therefore, by Lemma 3.4.12,ERâ²(q) = (ânâ¥0e(Fn)qn)e(Râ²/f) =ï£«ï£­ânâ¥0ï£«ï£­ ââni=n,niâ¥0kâ1âi=0p3(ni)ï£¶ï£¸ qnï£¶ï£¸e(Râ²/f)The last series is simply M(q)k·e(Râ²/f) and since Râ² is fixed by fk we gete(Râ²) = k · e(Râ²/f). This completes the proof.So we have proved the more general theorem.Theorem 3.5.3. Let X be a smooth projective surface and f : X â X bean automorphism of finite order. Then the series E(q) described in Theorem3.5.1 equals M(q)e(X).53Chapter 4Obstruction Theory on theModuli SpaceLet Y be a Gorenstein projective threefold and A a locally free sheaf onY , of global dimension 3. In this chapter, a perfect obstruction theory onthe Hilbert scheme Hilbn(Y,A) is constructed under certain assumptionson A. We apply this to noncommutative algebras A(f) on the threefoldY = X/f Ã P1 arising from the original problem of the moduli of twistedendomorphisms. In this case, Donaldson-Thomas type invariants are definedfor a class of surfaces with automorphisms where the surface is a cycliccover of the quotient surface. This includes some del Pezzo surfaces withautomorphisms.4.1 Preliminaries on Sheaves of NoncommutativeAlgebrasLet Y be a variety and A be a sheaf of coherent OY -algebras (Therefore,there is a structure map OY â A and OY is central in A). Denote byCoh(A) the category of coherentA-modules. The objects of this category arethe coherent OY -modules with an A-module structure (which is a morphismA â HomOY (F ,F) of sheaves ofOY -algebras). The morphisms areA-linearmorphisms of coherent sheaves.We will use several algebraic constructions for A-modules, listed below.â¢ (Duals) For F â Coh(A), the dual Fâ¨ is the sheaf HomA(F ,A). Ithas the structure of a right A-module (equivalently an Aop-module)by the following rule. For γ â Fâ¨(U) and a â A(U), γ · a â Fâ¨(U)544.1. Preliminaries on Sheaves of Noncommutative Algebrasis defined by (γ · a)(m) := γ(m)a â A(U) for m â F(U). Repeatingthis for the dual, the double dual Fâ¨â¨ := HomAop(Fâ¨,A) gets thestructure of a left A-module. The natural evaluation map F â Fâ¨â¨is well defined and it is easy to check that it is A-linear.â¢ (Tensor Products) Let F â Coh(A). If G â Coh(OY ), the tensorproduct F âOY G has the structure of a left A-module in the obviousway. The functor â âOY â : Coh(A) Ã Coh(OY ) â Coh(A) is rightexact in either argument. Furthermore, if G â Coh(Aop), we can formthe tensor product G âA F in the obvious way again and it is an OY -module. The functor â âA â : Coh(Aop) Ã Coh(A) â Coh(OY ) isright exact in either argument.â¢ (Hom) For F â Coh(A) and H â Coh(A) the sheaf of A-modulehomomorphismsHomA(F ,H) is anOY -module (becauseOY is centralin A). The functor HomA(F ,â) : Coh(A)â Coh(OY ) is right exact.Hence the functor HomA(F ,â) : Coh(A) â OY (Y )-mod is also rightexact as it is the composition Î(Y,HomA(F ,â)).â¢ (Hom-Tensor Adjunction I) For F ,H â Coh(A) and G â Coh(OY ) thenatural adjunctionHomA(F âOY G,H) â¼= HomOY (G,HomA(F ,H))continues to hold.â¢ (Hom-Tensor Adjunction II) For F an A-module, G an Aop-moduleand H an OY -module, the sheaf HomOY (G,H) is naturally an A-module by the following rule. For γ â HomOY (G,H)(U) and a âA(U), a · γ â HomOY (G,H)(U) is defined by (a · γ)(m) := γ(ma).There is a natural isomorphismHomOY (G âA F ,H) â¼= HomA(F ,HomOY (G,H))We also discuss some standard facts in homological algebra related tothese functors. The category mod(A) of A-modules has enough injectives.554.1. Preliminaries on Sheaves of Noncommutative AlgebrasTherefore, for F â mod(A), we have right derived functorsExtiA(F ,â) : mod(A)â mod(OY (Y ))andExtiA(F ,â) : mod(A)â mod(OY )for the functors HomA(F ,â) and HomA(F ,â) respectively. We make theeasy but useful remark that the functor ExtiA(A,â) coincides withH i(X,â).SinceHomA(A,â) â¼= HomOY (OY ,HomA(A,â)) = Î(Y,HomA(A,â))by taking F = A and G = OY in the Hom-Tensor adjunction I, we getRHomA(A,â) = RÎ â¦ RHomA(A,â) but HomA(A,H) = H for everyH â mod(A), i.e. it is an exact functor. So the functors RHomA(A,â) andRÎ(Y,â) coincide: for every A-module F we have ExtiA(A,F) = H i(Y,F)where on the right hand side, F is viewed as an OY -module.For a short exact sequence 0 â F â² â F â F â²â² â 0 of A-modules, weget an associated long exact sequences· · · â ExtiA(G,F â²)â ExtiA(G,F)â ExtiA(G,F â²â²)â Exti+1A (G,F â²)â · · Â·and· · · â ExtiA(F â²â²,G)â ExtiA(F ,G)â ExtiA(F â²,G)â Exti+1A (F â²â²,G)â · · Â·for any A-module G.Finally, we may also derive the tensor productââA â : mod(Aop)Ãmod(A)â mod(OY )in either factor. The derived versions of the adjunction formulae continueto remain true with the derived tensor product ââLA â and derived Hom.564.1. Preliminaries on Sheaves of Noncommutative Algebrasâ¢ (Derived Hom-Tensor Adjunction I) For complexes Fâ¢,Hâ¢ ofA-modulesand Gâ¢ of OY -modules, there is a natural adjunctionRHomA(Fâ¢ âLOY Gâ¢,Hâ¢) â¼= RHomOY (Gâ¢,RHomA(Fâ¢,Hâ¢))â¢ (Derived Hom-Tensor Adjunction II) For complexes Fâ¢ of A-modules,Gâ¢ of Aop-modules and H of OY -modules, there is a natural adjunctionRHomOY (Gâ¢ âLA Fâ¢,Hâ¢) â¼= RHomA(Fâ¢,RHomOY (Gâ¢,Hâ¢))After applying h0 â¦RÎ on both sides, it gives an adjunction betweenthe group homomorphisms on either side.Assume that Y is a proper Gorenstein variety with a dualizing line bundleÏY . We want a version of Serre duality for coherent A-modules.Definition 4.1.1. The dualizing complex Ïâ¢A is RHomOY (A, ÏY ). This is acomplex of (A,A)-bimodules with action coming from the rule (a·γ Â·aâ²)(v) :=Î³(aâ²va) for γ â ÏA(U) and a, aâ², v â A(U).Recall that an A-module E is locally projective if there is a cover {Ui}I âI of Y by open affine subsets such that E(Ui) is finitely generated projectiveas an A(Ui)-module for every i â I. Therefore, E(Ui) is a direct summandof a finitely generated free A(Ui)-module.We now recall perfect complexes of A-modules and some of their proper-ties. A strictly perfect complex Eâ¢ of A-modules on Y is a bounded complexwhose terms are all locally projective. A perfect complex Eâ¢ of A-moduleson Y is one for which there exists an open covering {Ui}iâI of Y such thatfor each i, Eâ¢|Ui can be represented by a strictly perfect complex Eâ¢i . Allthe results for perfect complexes in the commutative case continue to holdhere. For example, the following results will be useful.â¢ If Eâ¢ is a perfect complex ofA-modules, then the dual complex (Eâ¢)â¨ :=RHomA(Eâ¢,A) is a perfect complex of Aop-modules and the canonicalmorphism Eâ¢ â (Eâ¢)â¨â¨ is an isomorphism.574.1. Preliminaries on Sheaves of Noncommutative Algebrasâ¢ If Eâ¢ is a perfect complex of A-modules, then we have a canonicalisomorphism(Eâ¢)â¨ âLA Fâ¢ â¼= RHomA(Eâ¢,Fâ¢)for any complex Fâ¢ of A-modules.â¢ Suppose that A is locally free as an OY -module. If Eâ¢ and Fâ¢ areperfect complexes of A and Aop-modules respectively, then the tensorproduct Fâ¢ âLA Eâ¢ is a perfect complex of OY -modules. In particular,RHomA(Eâ¢, Eâ¢) is a perfect complex of OY -modules.All of the assertions follow from local computations for a finitely gener-ated projective module over a noncommutative ring, see [48, Tag 0FP7] forinstance.We remark that the assumption that A be locally free in the last pointis both necessary and sufficient. Indeed, this reduces to the following lo-cal problem. Let A be a noncommutative ring finitely generated over itscenter R. If P and Q are finitely generated projective A and Aop-modulesrespectively, must QâA P be projective as an R-module?Taking P = A and Q = A shows that the condition is necessary asQ âA P = HomA(A,A) = A as an R-module. In general, Q âA P is adirect summand of HomA(A,A) = A (again viewed as an R-module) so itis projective if A is projective over R.We can now discuss duality for A-modules.Proposition 4.1.2. (Serre Duality in the noncommutative setup) AssumethatRHomOY (A, ÏY ) = HomOY (A, ÏY ) (4.1.1)in which case the dualizing complex is a dualizing sheaf, denoted ÏA. Then,for a perfect complex Fâ¢ of A-modules and any complex Gâ¢ â D(Coh(A)),we have a canonical isomorphismHomA(Fâ¢,Gâ¢) = HomA(Gâ¢, ÏA âLA Fâ¢[n])â¨584.1. Preliminaries on Sheaves of Noncommutative Algebraswhere n = dimY .Proof. By Serre duality forOY -modules and using the fact that Fâ¢ is perfect,HomA(Fâ¢,Gâ¢) = HomOY (OY ,RHomA(Fâ¢,Gâ¢))= HomOY (RHomA(Fâ¢,Gâ¢), ÏY [n])â¨= HomOY ((Fâ¢)â¨ âLA Gâ¢, ÏY [n])â¨(4.1.2)where the dual (Fâ¢)â¨ is RHomA(Fâ¢,A). Using the derived Hom-Tensoradjunction II, (4.1.2) continues asHomOY ((Fâ¢)â¨ âLA Gâ¢, ÏY [n])â¨ = HomA(Gâ¢,RHomOY ((Fâ¢)â¨, ÏY [n]))â¨Since Fâ¢ is perfect, there is a natural isomorphism of A-modulesRHomOY ((Fâ¢)â¨, ÏY ) â¼= RHomOY (A, ÏY )âLA Fâ¢To see that the last isomorphism holds, it suffices to work locally and con-struct a natural isomorphism as follows. If A is a noncommutative ring finiteover its center R, and P is a finitely generated projective A-module, thenthere is a canonical isomorphismHomR(A,R)âA Pâ¼=ââ HomR(HomA(P,A), R)of A-modules given by the rule: a tensor Î·â p maps to the R-linear map onthe right sending h to η(h(p)).Now by our assumption, the right hand side is ÏA âLA Fâ¢. This provesthe proposition.Remark 4.1.3. The assumption (4.1.1) is obtained automatically if A islocally free over OY because HomOY (A,â) is exact in this case. It does holdmore generally for some cases even when A is not locally free. For example,if R = C[x, y] with the automorphism θ acting by x â âx and y â ây,the quotient Rθ is Gorenstein with trivial canonical bundle. Let A = R[z, θ]which is finite over its center B = Rθ[z2]. Then A is not locally free over B594.1. Preliminaries on Sheaves of Noncommutative Algebrasyet RHomB(A,B) = HomB(A,B) â¼= A.Remark 4.1.4. If ÏA is an invertible (A,A)-bimodule, then the derivedtensor product ÏA âLA â is the underived product ÏA âA â. We will provelater that this holds for our considerations.In practice, we want to apply Serre duality to coherent A-modules, there-fore it is necessary to have resolutions of coherent A-modules by (finitelength) complexes of locally projective A-modules. This is naturally a ho-mological question relating to the global dimension of A.Definition 4.1.5. The global dimension dimA of A is the largest integer m(possibly infinite) such that ExtmA(F ,G) Ì¸= 0 for some F ,G â Coh(A). Wecall A smooth if dimA <â.As remarked before, one can compute the Ext functors by choosing aresolution of the first argument by locally projective A-modules. Any F âCoh(A) has such a resolution (but it may not be of finite length). Define theprojective dimension of F to be the infimum of the largest integer n (again,possibly infinite) such that Eân Ì¸= 0, where the infimum is taken over allresolutions Eâ¢, the terms of which are all locally projective. It is standardto check that dimA is also the supremum of the projective dimensions of Franging over all F â Coh(A).A word of warning: Unlike the commutative case, it is important to usethe notion of âlocally projectiveâ rather than âlocally freeâ here, as the twonotions are not equivalent for noncommutative sheaves of algebras.Let f : X â X be an automorphism of finite order of a smooth projectivevariety. The algebra A(f) was defined on Y = X/f Ã P1 in Chapter 2. Wework with such a Y for the rest of this section.We need the next result from homological algebra, which says that theglobal dimension of twisted polynomial rings behaves in the expected fash-ion.Proposition 4.1.6. Let R be a commutative ring of finite global dimension604.1. Preliminaries on Sheaves of Noncommutative Algebrasm. For any automorphism θ of R, the global dimension of the ring R[z, θ]is m+ 1.Proof. This is Theorem 5.3, (iii) in Ch. 7 of [39].Theorem 4.1.7. The algebra A(f) defined in Section 2.4 is smooth.Proof. The construction of the sheaf of algebras A(f) shows that any point yhas a basis of affine neighborhoods U with A(f)|U the algebra described by atwisted polynomial ring R[z, θ] with gl.dimR = dimX. For any such U , wehave ExtiA(F ,G)|U = 0 as ExtiA(F ,G)|U â¼= ExtiA(f)|U (F|U ,G|U ) and the righthand side is the sheaf associated to the module ExtiA(f)(U)(F(U),G(U)). ByProposition 4.1.6, ExtiA(f)(U)(F(U),G(U)) vanishes for all i > dimX+1 andhence the stalks ExtiA(f)(F ,G)y vanish for all y â Y and all i > dimX + 1.Therefore ExtiA(f)(F ,G) = 0 for i > dimX+1. On the E2-page of the local-to-global spectral sequence, the termsHp(Y, ExtqA(f)(F ,G)) vanish wheneverp+q > 2(dimX+1) because if p > dimX+1, the sheaf cohomology vanishesand if q > dimX+1, the local Ext sheaves vanish. So ExtiA(f)(F ,G) vanisheswhenever i > 2(dimX + 1) and the proof is complete.We remark that smoothness is intrinsic to the algebra: A could besmooth even when Y is not smooth. For rather simple examples of thiskind, one could take an affine morphism g : Y â² â Y with Y â² smooth but Ynot smooth. Then gâOY â² is smooth even though the underlying scheme onwhich it lives is singular.Corollary 4.1.8. Let Y be smooth and assume that ÏA(f) is invertible.Then Serre duality holds for coherent sheaves of A(f)-modules and more-over, ExtiA(f)(F ,G) = 0 for i > n = dimY . Therefore, the global dimensionof A(f) equals dimY .Proof. Recall that the underlying sheaf of A(f) is qâOXÃP1z where q : X ÃP1z â Y = X/f Ã P1zd . Since Y is smooth, q is flat by the Miracle Flatnesslemma (see [48, Tag 00R4]) and hence A(f) is locally free over OY .By Theorem 4.1.7, A(f) is smooth. So any coherent sheaf is quasi-isomorphic to a strictly perfect complex of A(f)-modules and hence Serre614.2. Moduli Spaces of Coherent Modulesduality applies to any two coherent A(f)-modules F ,G. We getExtiA(f)(F ,G) = ExtnâiA(f)(G, ÏA(f) âA(f) F)â¨We emphasize that on the RHS, the arguments are actual sheaves (andnot complexes). If i > n, ExtnâiA(f)(â,â) vanishes for any two objects inCoh(A(f)), proving the claim. The last sentence is clear from the fact thatExtnA(f)(A(f), ÏA(f)) is nonzero.4.2 Moduli Spaces of Coherent ModulesAssume that (Y,OY (1)) is a polarized projective scheme. Let A be a co-herent sheaf of OY -algebras. We give a quick overview of stability for A-modules. For the details, we refer to Simpsonâs constructions in [47].For a coherent A-module F , its Hilbert polynomial p(F) and rank r(F)are defined to be the same as the underlying coherent sheaf. The A-moduleF is pure of dimension d if suppF â² has dimension d for any A-submodulesheaf 0 Ì¸= F â² â F .Definition 4.2.1. A coherent A-module is (semi)stable if it is pure and forany nontrivial subsheaf F â² â F with 0 < r(F â²) â¤ r(F), we havep(F â²)(m)r(F â²) (â¤)p(F)(m)r(F)for m >> 0.In [47] it is also proven that all the standard facts for (semi)stablesheaves also hold for (semi)stable A-modules. In particular, the naturalmap Câ HomA(F ,F) is an isomorphism for any stable A-module F . Themoduli stacks M(s)s,h(Y,A) of (semi)stable A-modules having fixed Hilbertpolynomial exist as algebraic stacks of finite type. The coarse moduli spaceM ss,h(Y,A) exists and is projective - to establish properness of the modulispace, a similar argument as in Langtonâs theorem (see [28]) applies here.624.3. An Open Embedding of the Hilbert SchemeThe stable locus M s,h(Y,A) âM ss,h(Y,A) is open. The natural morphismM(s)s,h(Y,A)âM ss,h(Y,A) is a Câ-gerbe, i.e. a locally trivial BCâ-bundle.We also have the Hilbert scheme Hilbh(Y,A) as in Chapter 1. Thedeformation-obstruction theory of Hilbh(Y,A) is worked out in the sameway as for the usual Hilbert/Quot schemes.Proposition 4.2.2. Let P = [A â F â 0] be a point of Hilbh(Y,A)and I = ker(A â F). The tangent space at P is naturally isomorphic toHomA(I,F). Let T â T a square-zero extension of fat points with sheafof ideals J . If T â Hilbh(Y,A) is a morphism whose image is P , theobstruction to extending it to T lies in Ext1A(I,F).Proof. The proof is the same as that for the Hilbert scheme of closed sub-schemes of a scheme. Infact, a more general statement appears in the contextof abstract Hilbert schemes in [3], Proposition E2.4 and Lemma E2.6.4.3 An Open Embedding of the Hilbert SchemeNext we relate the Hilbert scheme of points to the moduli of stable sheaves.In the classical case of Donaldson-Thomas theory for the Hilbert schemes ofthreefolds, there is an isomorphism of the Hilbert scheme of one dimensionalsubschemes and the (coarse) moduli space of stable sheaves with trivial de-terminant. In the noncommutative setting, we have an open embeddinginstead, under certain conditions. We remark that the results in Lemma4.3.1 and Theorem 4.3.7 have also appeared in [35] but the crucial assump-tion that H0(Y,A) â¼= C is missing, without which the claim is false.We assume that Y is a projective Gorenstein threefold and A has globaldimension 3, is locally free over OY with ÏA invertible.Lemma 4.3.1. Let ξ â Y be the generic point. If A is a domain (thatis, A(U) does not have nontrivial zero divisors for any open U), there is amorphismj : Hilbn(Y,A)âM s,p(A)ân(Y,A)634.3. An Open Embedding of the Hilbert Schemewhich on points, is the map taking a quotient [A â F â 0] to [I = ker(A âF)].Proof. First note thatAξ is a division algebra overOY,ξ = k(Y ), the functionfield. The stalk Aξ is a finite dimensional kξ-algebra which is also a domain.A standard algebra argument shows that every nonzero x â Aξ satisfies apolynomial with coefficients in kξ and non-zero constant coefficient, so x hasa two-sided inverse.For such a quotient [A â F â 0], we first prove that [I] âM s,p(A)ân(Y,A).For any A-submodule 0 Ì¸= I â² â I, the left ideal I â²Î¾ â Aξ is either 0 or Aξ.By the hypothesis, it cannot be 0 else I â² would be a torsion submodule of A,a contradiction. We get that I is pure as r(I â²) = r(I) = r(A). For I â² Ì¸= I,the quotient I/I â² Ì¸= 0 is supported on a proper closed subset of S sop(I)(m)â p(I â²)(m) = p(I â²/I)(m) > 0for m >> 0 proving the claim. Then constructing the morphism is routinein families: On T -points construct the morphismHilbn(Y,A)(T )âMs,p(A)ân(Y,A)(T )sending a family AT â F â 0 to I = ker(AT â F ). Note that I isT -flat since AT ,F both are and I â Ms,p(A)ân(Y,A)(T ) because It =ker(A â Ft) by Lemma 2.1.4. of [24]. Composing the map above withMs,p(A)ân(Y,A)âM s,p(A)ân(Y,A) yields the desired map.For the rest of this section we make the following assumptions: Y is asmooth threefold and A is a smooth algebra of global dimension 3 with ÏAinvertible. Under the extra assumptions H0(Y,A) â¼= C and H1(Y,A) = 0,we alter the deformation-obstruction theory of the Hilbert scheme to thatof the moduli of stable sheaves.Let P = [A â F â 0] â Hilbn(Y,A) and I be as defined earlier.Lemma 4.3.2. Assume that H0(Y,A) = C and H1(Y,A) = 0. There thereis an isomorphism HomA(I,A) = C and moreover, Ext1A(I,A) = 0644.3. An Open Embedding of the Hilbert SchemeProof. Applying HomA(â,A) to the exact sequence of P yields portionsExtiA(F ,A)â ExtiA(A,A)â ExtiA(I,A)â Exti+1A (F ,A)butExtiA(F ,A) â¼= Ext3âiA (A, ÏA âA F)â¨ â¼= H3âi(Y, ÏA âA F)â¨ = 0for i = 0, 1, 2 as F has zero-dimensional support. This gives an isomorphismC = HomA(A,A) â¼= HomA(I,A)by the assumption on H0(S,A). We also get an isomorphismExt1A(A,A) â¼= Ext1A(I,A)giving the vanishing Ext1A(I,A) = 0 from the assumption on H1(Y,A).Lemma 4.3.3. Continuing with the same assumptions, there is an isomor-phism HomA(I,F) â¼= Ext1A(I, I) and an inclusion Ext1A(I,F) âªâ Ext2A(I, I).Proof. Applying HomA(I,â) to the same exact sequence of P yields0â HomA(I, I)â HomA(I,A)â HomA(I,F)â Ext1A(I, I)The map HomA(I,F) â Ext1A(I, I) is surjective because Ext1A(I,A) = 0.It is also injective as HomA(I, I)â HomA(I,A) is an isomorphism Câ C(since I is stable). Therefore we have an isomorphismHomA(I,F) â¼= Ext1A(I, I)The long exact sequence continues to give an injective map0 = Ext1A(I,A)â Ext1A(I,F)â Ext2A(I, I)proving the lemma.654.3. An Open Embedding of the Hilbert SchemeFor later use we also prove the next lemma.Lemma 4.3.4. If H3(Y,A) = 0 we also have Ext3A(I, I) = 0Proof. Consider the long exact sequence of cohomology associated to 0 âI â A â F â 0. It has a portionH2(Y,F)â H3(Y, I)â H3(Y,A)Since F has zero-dimensional support, H2(Y,F) = 0. Then H3(Y, I) = 0as both ends vanish. Applying HomA(â, I) gives the sequence0 = H3(Y, I) = Ext3A(A, I)â Ext3A(I, I)â Ext4A(F , I)Since A has global dimension 3, Ext4A(F , I) = 0. Hence Ext3A(I, I) = 0 aswell.Let DefAâF be the deformation functor associated to the quotient A âF representing P â Hilbn(Y,A). That is, for an Artin local algebra B, theset DefAâF (B) consists of pairs (AB â F â², θ) where the first argumentis a flat deformation of the surjection A â F and θ : iâF â² â F is anisomorphism such that the diagramiâAB iâF â² 0A F 0= θcommutes. Here i : Y â Y ÃC B is the closed embedding associated tothe closed point of B. Define similarly the deformation functor DefI forthe kernel I of A â F and observe that there is a natural morphism offunctors η : DefAâF â DefI since if I â² is the kernel of a deformation(AB â F , θ) â DefAâF (B) then in the diagram0 IâI â² IâAB F â² 00 I A F 0= θ664.3. An Open Embedding of the Hilbert Schemethe top row is exact as F â² is B-flat, hence the dotted arrow exists uniquelyand is an isomorphism. Both deformation functors are pro-representable:For the target, this crucially relies on the fact that HomA(I, I) â¼= C (see,for example, [24], Theorem 4.5.1 or [40], Theorem 29).Lemma 4.3.5. The map η is naturally an isomorphism of deformationfunctorsProof. The induced map on the tangent spaces HomA(I,F) â Ext1A(I, I)is an isomorphism and on the obstruction spaces Ext1A(I,F)â Ext2A(I, I)is an inclusion. The claim now follows from Remark 2.3.8 of [46], see alsoLemma 2.7 of [44].We can now finish the proof of the main theorem.Theorem 4.3.6. Under the assumptions on Y and A, assume that theextra hypotheses (1) H0(Y,A) = C (2) H1(Y,A) = 0 also hold. Then, themorphism j is an open embedding.Proof. We show j is injective and etale. To show injectivity on points,suppose that P1 = [0 â I1 â A â F1] and P2 = [0 â I2 â A â F2]are two points in Hilbn(Y,A) such that their images in M s,p(A)ân(Y,A)are equal, i.e. let α : I1 â I2 be an isomorphism. Consider the twoinclusions I1 â A and I1 Î±ââ I2 â A in HomA(I1,A). By Lemma 4.3.2.HomA(I1,A) = C. Thus, there is a scalar λ Ì¸= 0 making the left squarecommute in the diagram below0 I1 A F1 00 I2 A F2 0α λ βStandard diagram-chasing implies the existence of the dotted arrow β :F1 â F2 and the five-lemma implies that it is an isomorphism. Then theisomorphism Î»â1Î² does induce an isomorphism of the quotients A â F1 andA â F2, proving injectivity. To show that j is etale, the same argument inProposition 2.8 of [44] applies, using the formal criterion. For a square-zero674.3. An Open Embedding of the Hilbert Schemeextension T â T of fat points, we require a unique dotted arrow as in thesquareT Hilbn(Y,A)T M s,p(A)ân(Y,A)jgor equivalently in the diagramHomP (T ,Hilbn(Y,A)) HomP (T,Hilbn(Y,A))HomQ(T ,Ms,p(A)ân(Y,A)) HomQ(T,M s,p(A)ân(Y,A))Î·T ηTwe require a lift of g to HomP (T ,Hilbn(Y,A)) (Here P and Q are the imagesof T and T respectively). The lift exists uniquely as the vertical arrows arenatural isomorphisms by Lemma 4.3.6.Remark 4.3.7. The morphism j is also a closed embedding as Hilbn(Y,A)is proper. So it is the inclusion of a union of connected components.Remark 4.3.8. The morphism is not an embedding if dimH0(Y,A) > 1(Observe that C â H0(Y,A) always for a domain A). Indeed for P =[0 â I â A â F ] a point in Hilbn(Y,A) as before, the map of tangentspaces jP : HomA(I,F) â Ext1A(F ,F) would be an isomorphism if j wasan embedding. However the proof of Lemma 3.3.2 shows that H0(Y,A)/Cis in the kernel of jP .Remark 4.3.9. It is natural to ask if the embedding holds for a Hilbertscheme with a fixed linear polynomial as datum, i.e. for quotients withone-dimensional support. Unfortunately, the conclusion Ext1A(I,A) = 0 ofLemma 4.3.2 is no longer valid because the assumption that the quotientshave zero-dimensional supports was necessary. It remains to investigate ifthe result could be proven true with some extra hypotheses.684.4. Construction of the Obstruction Theory4.4 Construction of the Obstruction TheoryTo construct an obstruction theory on Hilbn(Y,A), we will need the existenceof the Atiyah class for the deformation-obstruction theory of sheaves of A-modules. Let T be a scheme and T a square-zero extension of T . Let Jbe the ideal sheaf of T â T . Let the truncated cotangent complex of T bedenoted by LT and κ(T/T ) â Ext1T (LT ,J ) be the Kodaira-Spencer class ofthe extension. The obstruction theory of A-modules on Y is made explicitdue to the following theorem in [35].Theorem 4.4.1. Let F be a flat family of coherent AT -modules on Y ÃT .There is a natural classatA(F ) â Ext1AT (F ,F âOYÃT pâTLT )satisfying the following: There exists a flat familyF of coherent AT -moduleson Y Ã T with F |YÃT â¼= F iff the composition, denoted ob(F )F F â pâTLT [1] F â pâTJ [2]atA(F ) idFâpâT κ(T/T )vanishes in Ext2AT (F ,F â pâTJ ).Since Hilbn(Y,A) has a universal quotient (and hence a universal simplesheaf), the construction of a perfect obstruction theory is direct by followingthe arguments in [23] or [44].On H = Hilbn(Y,A), let 0â I â AH â F be the universal quotient.Since I is H-flat and A is smooth, it is a perfect object in the derivedcategory of AH -modules. LetatA(I ) â HomAH (I ,I âOYÃH LYÃH [1])be the Atiyah class. Since I is canonically isomorphic to I â¨â¨ we can use694.4. Construction of the Obstruction Theorythe Derived Hom-Tensor adjunction IIHomAH (I ,I âOYÃH LYÃH [1]) = HomAH (I , (I â¨)â¨ âOYÃH LYÃH [1])= HomAH (I ,RHomOYÃH (I â¨,LYÃH [1]))= HomOYÃH (Iâ¨ âLAH I ,LYÃH [1])= HomOYÃH (RHomAH (I ,I ),LYÃH [1])Projecting onto the factor and pâHLH of LYÃH = pâY LY â pâHLH yields aclass inHomOYÃH (RHomAH (I ,I ),LH [1]) (4.4.1)The 3-dimensional morphism pH : Y ÃH â H has a relative dualizingsheaf Ï := pâY ÏY as Y is Gorenstein. Grothendieck duality for pH readsRpHâRHomOYÃH (Kâ¢, pâHLâ¢ âOYÃH Ï[3]) = RHomOH (RpHâKâ¢,Lâ¢)Set Kâ¢ = RHomAH (I ,I )âOYÃHÏ[2] and Lâ¢ = LH . After applying h0â¦RÎto both sides, the left hand side becomes (4.4.1). Therefore the projectedAtiyah class (4.4.1) induces an element inHomOH (RpHâ(RHomAH (I ,I )âOYÃH Ï[2]),LH)For simplicity, we use notation F := RHomAH (I ,I ).Theorem 4.4.2. Let Y be a projective Gorenstein threefold. Let A a smoothalgebra of global dimension 3 which is locally free over OY and ÏA is invert-ible. Assume that (1) H0(Y,A) = C (2) H i(Y,A) = 0 for i = 1, 3.Then H carries a perfect obstruction theoryob : (Ïâ¥1RpHâF)â¨[â1]â LHProof. First, by the remarks in Section 4.1 on perfect complexes, the com-plex F is perfect as I is, and hence RpHâF is perfect too, by [48, Tag0DJQ].From the vanishing in Lemma 4.3.4, we observe that RpHâF is isomor-704.4. Construction of the Obstruction Theoryphic to a perfect complex of amplitude [0, 2] in D(OH). Indeed, this isbecause for any point P = [A â F â 0] with I = ker(A â F), we havethat the derived restriction of RpHâF to P is a perfect complex (of vec-tor spaces) with cohomology only in [0, 2] as Ext3A(I, I) = 0. So the sameargument in Lemma 4.2 of [23] applies.Now consider the mapOH â RpHâFwhich is in adjunction to the canonical map OYÃH â F. For every pointP as before, the map restricts to an isomorphism C â HomA(I, I) andhence the cone of this map is Ïâ¥1RpHâF. Therefore, (Ïâ¥1RpHâF)â¨[â1] hasamplitude [â1, 0].Set E := RpHâ(RHomAH (I ,I ) âOYÃH Ï[2]) = RpHâ(F âOYÃH Ï[2])which is a perfect complex on H. In the discussion before, the Atiyah classinduced a mapE â¼= (Eâ¨)â¨ â LHWe compute Eâ¨ by Grothendieck duality. Set Kâ¢ = F âOYÃH Ï[2] andLâ¢ = OH to getEâ¨ = RpHâRHomOYÃH (F,OYÃH)[1]but F is self dual so the right hand side simply reduces to RpHâF[1]. So wehave a map(RpHâF[1])â¨ = (RpHâF)â¨][â1]â LHDualizing the map RpHâF â Ïâ¥1RpHâF and composing with the mapabove, we get a mapob : (Ïâ¥1RpHâF)â¨[â1]â LHIt remains to check that ob is really an obstruction theory. The com-putations in Lemmas 3.3.2 and 3.3.3 together prove it is, in the sense ofDefinition 4.4 of [7] (As pointwise, for a point P â H as before, h0(ob) is714.5. DT-Type Invariants and Examplesthe isomorphism Ext1A(I, I)â¨ â HomA(I,F)â¨ and hâ1(ob) is the surjectionExt2A(I, I)â¨ â Ext1A(I,F)â¨).4.5 DT-Type Invariants and ExamplesIn this last section, we give examples of surfaces X with an automorphism ffor which the theory holds for A = A(f) and Y are satisfied. Since A(f) =qâOXÃP1 for the finite quotient map q : XÃP1 â Y , the cohomology groupsH i(Y,A(f)) equal H i(X Ã P1,OXÃP1) for every i â¥ 0. By the Kunnethformula ([48, Tag 0BEC])H i(X Ã P1,OXÃP1) â¼=âk+l=iHk(X,OX)âH l(P1,OP1) = H i(X,OX)for every i â¥ 0 as OP1 has no higher cohomology. So the conditionsH0(Y,A(f)) â¼= C and H3(Y,A(f)) = 0 are automatic. The conditionH1(Y,A(f)) = 0 holds iff the surface X has irregularity H1(X,OX) = 0.We assume this last condition for the rest of the chapter.We assume that the quotient Z = X/f is smooth. Then A(f) is locallyfree (see Corollary 4.1.8) over OY . We now address the issue of the invert-ibility of ÏA(f). Let us assume that g : X â Z = X/f is totally ramified ofdegree d, or in other words, it is a (ramified) cyclic cover of degree d in thesense of [30]. Therefore, gâOX = OZâLâ· · Â·âLâ(dâ1) for some line bundleL on Z and g is totally ramified over a smooth divisor D â Z defined bya section of Lââd. The situation on local charts is this: we have a degreed cover SpecS[x]/(xd â s)â SpecS which is totally ramified over the zerolocus of s â S. The automorphism f acts by a primitive d-th root of unityon x, which is in fact, a global automorphism of the (total space of the) linebundle Lâ1.Considering the quotient q : X Ã P1z â Y = X/f Ã P1zd , it followsimmediately that A(f) = qâOXÃP1z is the locally free sheafA(f) =â0â¤i,jâ¤dâ1Li â O(â1)j724.5. DT-Type Invariants and ExamplesLemma 4.5.1. Let R = S[x]/(xd â s) as before with θ the automorphismof R fixing S and acting on x by a primitive d-th root of unity, ξ. LetA = R[z, θ] and the center B of A is S[zd]. Then, there is a bilinear perfectpairing Ï : BâAB â A, which is not symmetric if d Ì¸= 1. There is a B-linearautomorphism η of A such that Ï(a1, a2) = Ï(a2, η(a1)) for all a1, a2 â Aand it induces an isomorphismηA1 â HomB(A,B)of (A,A)-bimodulesProof. Note that B is a free A-module of rank d2 with basis xizj with0 â¤ i, j â¤ dâ1. So it suffices to describe the pairing on these basis elements.Define Ï(xizj , xkzl) to be the coefficient of xdâ1zdâ1 in the expansion of(xizj)(xkzl) in the basis above. By construction, this pairing is bilinear andÏ(xizj , xkzl) = 0 unless i + k = d â 1 = j + l, in which case it equals ξjk.Therefore, in the matrix associated to the basis, Ï has non-zero entries onlyon the anti-diagonal, which are roots of unity. It follows that Ï is perfect asthis matrix is invertible in B. Note also, that Ï(a1, a2a3) = Ï(a1a2, a3) bydefinition, as the equality holds whenever the ai are basis elements.For a fixed a â A, the map Ï(a,â) : A â B is B-linear and hencemay also be described as Ï(â, η(a)) for a unique a â A, by perfectness ofthe pairing. The map η is bijective by perfectness and we claim that it isa B-linear automorphism of A. It is enough to check that it is B-linearand a ring homomorphism. For arbitrary c â A and fixed but arbitrarya1, a2 â A, b â B, we follow the definitionsÏ(c, η(a1 + ba2)) = Ï(a1 + ba2, c)= Ï(a1, c) + bÏ(a2, c)= Ï(c, η(a1)) + bÏ(c, η(a2))= Ï(c, η(a1) + bη(a2))734.5. DT-Type Invariants and Exampleswhich shows η(a1 + ba2) = η(a1) + bη(a2) andÏ(c, η(a1a2)) = Ï(a1a2, c)= Ï(a1, a2c)= Ï(a2c, η(a1))= Ï(a2, cη(a1))= Ï(cÎ·(a1), η(a2))= Ï(c, η(a1)η(a2))which shows η(a1a2) = η(a1)η(a2). To understand η, it suffices to know howit acts on x and z. We know that 1 = Ï(x, xdâ2zdâ1) = Ï(xdâ2zdâ1, η(x)).It follows that the coefficient of x in η(x) is ξ. Also, 0 = Ï(x, xizj) =Ï(xizj , η(x)) for any other different pair (i, j) Ì¸= (d â 2, d â 1) so it followsthat η(x) = ξx. Similarly, we find that η(z) = Î¾â1z.It remains to check that there is an isomorphism Î·â1A1 â HomB(A,B)of (A,A)-bimodules. For the natural isomorphism A â HomB(A,B) ofB-modules sending a to Ta : c â Ï(a, c), it suffices to check the desiredlinearities to upgrade it to a bimodule isomorphism. We have(a1 · Ta2)(c) = Ta2(ca1)= Ï(a2, ca1)= Ï(a2c, a1)= Ï(Î·â1(a1), a2c)= Ï(Î·â1(a1)a2, c)= TÎ·â1(a1)a2(c)and(Ta1 · a2)(c) = Ta1(a2c)= Ï(a1, a2c)= Ï(a1a2, c)= Ta1a2(c)744.5. DT-Type Invariants and Examplesand this shows Î·â1A1â¼=ââ HomB(A,B) as (A,A)-bimodules. This completesthe proof.Remark 4.5.2. The algebra A over B may be regarded as a generalizationof the concept of a Frobenius algebra over a field. In the literature, the auto-morphism η (or Î·â1) is called the Nakayama automorphism of the Frobeniusalgebra.Now we can easily globalize the construction of Lemma 4.5.1. On Y , wehave a bilinear pairing A(f)âOY A(f)â Ldâ1â O(1âd) which is describedon the local charts SpecS[x]/(xd â s) â SpecS and is well-defined on Yby the construction of the cyclic cover. The automorphism η is well-definedglobally too, because the local action xâ ξx is a global action. Therefore,the next corollary is immediate.Corollary 4.5.3. The sheaf ÏA(f) is an invertible (A(f),A(f))-bimodule.Proof. We have an isomorphism Î·â1A(f)1 â HomOY (A(f), Ldâ1â O(1âd))by globalizing Lemma 4.5.1 and hence, we getÏA(f) â¼= HomOY (A(f), Ldâ1 â O(1â d))âOY (L1âd â O(dâ 1)âOY ÏY )= (Î·â1A(f)1)âOY (L1âd â O(dâ 1)âOY ÏY )which is an isomorphism of bimodules, making ÏA(f) invertible. Note thatthe first isomorphism follows from Proposition 7.7 of [21].Example 4.5.4. As a curious example, consider Z = P1 Ã P1 and D â Za smooth divisor of type (3, 3). Then, one can construct the degree 3 cycliccover g : X â Z ramified totally over D and it is simple to see that X is a K3surface. It has a natural (necessarily non-symplectic) automorphism of order3 which we call f . In fact, X may be viewed as a degree 6 surface in P4. i.e.as an intersection of a quadric hypersurface F2(x0, x1, x2, x3, x4) = 0 and acubic hypersurface F3(x0, x1, x2, x3) + x34 = 0 where F2, F3 are homogenousof degree 2, 3 respectively. (For more details, see Remark 2.8 in [10] orRemark 4.8 in [1])754.5. DT-Type Invariants and ExamplesIn this case, keeping the notation of this section, we have L = O(â1,â1),d = 3 and ÏY = O(â2,â2)â O(â2) so the line bundle (L1âdâ O(dâ1))âOYÏY in Corollary 4.5.3 above is trivial. Therefore, in this example, we haveexhibited A(f) as a sheaf of non-symmetric Frobenius algebras. This is incontrast to the quantum Fermat quintic studied by Kanazawa and Liu, wherethe sheaf of algebras is symmetric Frobenius, see Lemma 3.8 in [35].Remark 4.5.5. Consider the algebra Î·â1A(f)1 restricted to the open subset(Z âD)ÃGm â Y . Working locally and borrowing the notation of Lemma4.5.1, we have that x and z are invertible. Then η is an inner automorphismas(xz)x(xz)â1 = x(zx)zâ1xâ1 = x(Î¾xz)zâ1xâ1 = ξx = η(x)and similarly(xz)z(xz)â1 = x(zxâ1) = x(Î¾â1xâ1z) = Î¾â1z = η(z)It follows that Î·â1A(f)1 is isomorphic to A(f) as (A(f),A(f))-bimoduleswhen restricted to the open subset (Z âD)ÃGm.Remark 4.5.6. To view the isomorphism described in Remark 4.5.5, wecan also use the more standard trace form(qâOXÃP1)âOY (qâOXÃP1)â OYinduced by the finite morphism X Ã P1 â Y . This is evidently bilinear andsymmetric. In a simple case d = 2, we exhibit why it is perfect on the openlocus (Z âD)ÃGm of Y (the general case is similar). Once again, we maywork locally as in Lemma 4.5.1 with the standard basis {1, x, z, xz} and withrespect to this basis, the matrix corresponding to the trace form isï£«ï£¬ï£¬ï£¬ï£¬ï£­4 0 0 00 4x2 0 00 0 4z2 00 0 0 4x2z2ï£¶ï£·ï£·ï£·ï£·ï£¸764.5. DT-Type Invariants and Exampleswhich is clearly invertible as x and z are. However, this form is certainlynot perfect on Y which is why we argue as in Lemma 4.5.1.We see that Theorem 4.4.2 holds when X is a surface with H1(X,OX) =0, the quotient Z = X/f is smooth, and X â Z is a cyclic cover ramifiedover a smooth divisor D. We show that certain components of the modulispace Hilbn(Y,A(f)) have a virtual cycle.Lemma 4.5.7. Let P = [A(f)â F â 0] be a point in Hilbn(Y,A(f)) andI = ker(A(f)â F). If Ï(â,â) denotes the Euler pairingÏ(â,â) :=3âi=0(â1)i dimExtiA(f)(â,â)then we haveÏ(I, I) = Ï(A(f),A(f)) + Ï(F ,F)and moreover, if suppF â (Z âD)ÃGm, then Ï(F ,F) = 0.Proof. We haveÏ(I, I) = Ï(A(f),A(f)) + Ï(F ,F)â Ï(A(f),F)â Ï(F ,A(f))by considering [I] = [A(f)]â [F ] as K-Theory classes. First,Ï(A(f),F) = dimExt0A(f)(A(f),F) = dimH0(Y,F)as F is zero-dimensional. On the other hand,Ï(F ,A(f)) = âdimExt3A(f)(F ,A(f))as the other terms vanish by Serre duality and the fact that F is zero-dimensional. ButdimExt3A(f)(F ,A(f)) = dimH0(Y, ÏA(f) âA(f) F)774.5. DT-Type Invariants and ExamplesUsing the formula in Corollary 4.5.3 shows that ÏA(f)âA(f) F â¼= (Î·â1F), soÏ(F ,A(f)) = âdimH0(Y, (Î·â1F)) = âdimH0(Y,F)Hence Ï(I, I) = Ï(A(f),A(f)) + Ï(F ,F). For the second statement, notethatÏ(F ,F) = âÏ(F , ÏA(f) âA(f) F) = âÏ(F , (Î·â1F))and if F is supported in (Z â D) Ã Gm, then Remark 4.5.5 shows thatÎ·â1F â¼= F and hence Ï(F ,F) = 0.Our results allow us to define DT-type invariants in certain cases. As-sume that H2(Y,A(f)) = H2(X,OX) = 0. Then Ï(A(f),A(f)) = 1. If Pis a point as in Lemma 4.5.7 with F supported in (Z âD)ÃGm, then thelemma shows thatdimExt1A(f)(I, I)â dimExt2A(f)(I, I) = 0Since the rank of a perfect complex is a locally constant function, it followsthat the rank of the obstruction theory in Theorem 4.4.2 equals zero onany component containing such a point P . Let V n â Hilbn(Y,A(f)) bethe union of all such components. Then V n has a virtual cycle [V n]vir ofdimension 0.Definition 4.5.8. Let X be a projective surface satisfying H1(X,OX) =H2(X,OX) = 0. Assume that f : X â X is an automorphism of finiteorder such that X/f is smooth and X â X/f is a cyclic cover.Consider the union V n of components of Hilbn(Y,A(f)) on which theobstruction theory has rank 0. The DT-type invariants for Hilbn(Y,A(f))are the integersNnX,f :=â«[V n]vir1defined by the virtual fundamental class [V n]vir.Earlier, we have conjectured that the union V n equals Hilbn(Y,A(f))except in the special case n = 1 and X â X/f non-trivial etale - so that784.5. DT-Type Invariants and Examplesthere are no isomorphisms Ï : F â fâF at all. In the case when f =idX , we have Hilbn(Y,A(f)) = Hilbn(X Ã P1) which is connected, henceV n = Hilbn(X Ã P1) as known already and the invariants reduce to theDonaldson-Thomas invariants of the threefold X Ã P1.Let X be a del Pezzo surface, i.e. by definition, âKX is ample. In-deed, the Kodaira vanishing theorem applied to âKX implies the vanishingH i(X,OX) = 0 for all i > 0. The automorphism groups of del Pezzo surfaceshave been studied in detail by Dolgachev and Iskovskikh in [16], Section 6.We give some examples where the quotient is a cyclic cover.Example 4.5.9. Let X = P1 Ã P1 with f the involution swapping the twofactors. The fixed locus of f is the diagonal P1 â X and X/f â¼= P2. Thequotient X â X/f is a cyclic cover of degree 2 ramified over a conic in P2.Example 4.5.10. (Degree 2 del Pezzo surfaces) Let X be a del Pezzo surfaceof degree 2. It is known that X arises as a double cover of P2 ramified alonga smooth quartic curve C â P2. 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Behrend, K. (Kai)
Some invariants for surfaces with an automorphism
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