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Categorification of Donaldson-Thomas Invariants Via Perverse Sheaves

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Categorification of Donaldson-Thomas Invariants Via Perverse Sheaves Categorification of Donaldson-Thomas invariants via perverse sheaves Young-Hoon Kiem Jun Li Department of Mathematics and Research Institute of Mathemat- ics, Seoul National University, Seoul 151-747, Korea E-mail address: [email protected] Department of Mathematics, Stanford University, CA 94305, USA E-mail address: [email protected] arXiv:1212.6444v5 [math.AG] 21 Mar 2016 2010 Mathematics Subject Classification. Primary 14N35, 55N33 Key words and phrases. Donaldson-Thomas invariant, critical virtual manifold, semi-perfect obstruction theory, perverse sheaf, mixed Hodge module, categorification, Gopakumar-Vafa invariant. The first named author was supported in part by Korea NRF Grant 2011-0027969. The second named author was partially supported by NSF grant DMS-1104553 and DMS-1159156. Abstract. We show that there is a perverse sheaf on a fine moduli space of stable sheaves on a smooth projective Calabi-Yau 3-fold, which is locally the perverse sheaf of vanishing cycles for a local Chern-Simons functional. This perverse sheaf lifts to a mixed Hodge module and gives us a cohomology theory which enables us to define the Gopakumar-Vafa invariants mathematically. This paper is a completely rewritten version of the authors’ 2012 preprint, arXiv:1212.6444. Contents Preface vii Part 1. Critical virtual manifolds and perverse sheaves 1 Chapter 1. Critical virtual manifolds 5 1.1. Background 5 1.2. Definition of Critical Virtual Manifolds 6 1.3. Orientability of a critical virtual manifold 10 1.4. Semi-perfectobstructiontheoryandBehrend’stheorem 11 1.5. Critical virtual manifolds and d-critical loci 20 Chapter2. Perversesheavesoncriticalvirtualmanifolds 23 2.1. Perverse sheaves of vanishing cycles 23 2.2. Gluing of perverse sheaves 27 2.3. Gluing of mixed Hodge modules 30 Chapter 3. Rigidity of perverse sheaves 35 3.1. Sebastiani-Thom isomorphism and rigidity 35 3.2. Vector fields and isotopies 37 3.3. Obstruction theory and isotopy 39 3.4. Proof of Theorem 2.12 42 Part 2. Critical virtual manifolds and moduli of sheaves 45 Chapter 4. Critical loci and moduli of simple vector bundles 49 4.1. Semiconnections and Chern-Simons functional 49 4.2. CS framings and CS charts 52 4.3. Critical virtual manifold structure 55 Chapter 5. Critical virtual manifolds and moduli of vector bundles 59 5.1. Proof of Proposition 4.16 59 5.2. Proof of Proposition 4.15 68 Chapter 6. Moduli of sheaves and orientability 71 6.1. Critical virtual manifold structure on moduli of simple sheaves 71 6.2. Orientability for moduli of sheaves 73 Chapter7. Gopakumar-Vafainvariantviaperversesheaves 79 7.1. Intersection cohomology sheaf 79 7.2. GVinvariantsfromperversesheaves 80 7.3. K3-fibered Calabi-Yau 3-folds 82 v vi CONTENTS Bibliography 87 Preface The Donaldson-Thomas invariant (DT invariant, for short) is a virtual count of stable sheaves on a smooth projective Calabi-Yau 3-fold Y over C which was de- fined as the degree of the virtual fundamental class of the moduli space X of stable sheaves on Y ([48]). Using microlocal analysis, Behrend showed that the DT in- variant is in fact the Euler number of the moduli space, weighted by a constructible function νX , called the Behrend function ([1]). Since the ordinary Euler number is the alternating sum of Betti numbers of cohomology groups, it is reasonable to ask if the DT invariant is in fact the Euler number of a cohomology theory on X. On the other hand, it is known that the moduli space is locally the critical locus of a holomorphic function, called a local Chern-Simons functional ([19]). Given a holomorphic function f on a complex manifold V , one has the perverse sheaf φf (Q[dim V 1]) of vanishing cycles supported on the critical locus. So the moduli space is covered− by charts each of which comes with the perverse sheaf of vanishing cycles φf (Q[dim V 1]). This motivated Joyce and− Song to raise the following question ([19, Question 5.7]). Let X be a moduli space of simple coherent sheaves on Y . Does there exist a natural perverse sheaf P on the underlying analytic space which is locally isomorphic to the sheaf φ (Q[dim V 1]) of vanishing cycles for (V,f) above? f − The purpose of this paper is to provide an affirmative answer. Theorem 0.1. Let X be a moduli space of simple sheaves on a smooth projective Calabi-Yau 3-fold Y . Suppose the reduced scheme Xred of X is of finite type and admits a tautological family. Then the collection of perverse sheaves φf (Q[dim V 1]) geometrically glue to a perverse sheaf P on X. The same holds for polarizable− mixed Hodge modules. As an application of Theorem 0.1, we use the hypercohomology Hi(X, P ) of P to deduce the DT (Laurent) polynomial Y i i DTt (X)= t dim Hc(X, P ) i X Y such that DT−1(X) is the ordinary DT invariant by [1]. Another application is a mathematical theory of Gopakumar-Vafa invariants (GV invariants, for short) proposed in [12]. Let X be a moduli space of stable sheaves supported on curves of homology class β H2(Y, Z). The GV invariants are integers n (β) for h Z defined by an sl sl∈ action on some cohomology of h ∈ ≥0 2 × 2 X such that n0(β) is the DT invariant of X and that they give all genus Gromov- Witten invariants Ng(β) of Y via a relation in [12] (cf. (7.1)). Applying Theorem vii viii PREFACE 0.1 to X, we obtain a perverse sheaf P on X which is locally the perverse sheaf of vanishing cycles. By the relative hard Lefschetz theorem for the morphism to the ∗ Chow scheme ([42]), we have an action of sl2 sl2 on H (X, Pˆ) where Pˆ is the gradation of P by the weight filtration of P . This× gives us a geometric theory of GV invariants which we conjecture to give all the Gromov-Witten invariants Ng(β). Our proof of Theorem 0.1 uses gauge theory. By the Seidel-Thomas twist ([19, Chapter 8]), it suffices to consider only vector bundles on Y . Let si = si/ be the space of semiconnections on a hermitian vector bundle E moduloB theA gaugeG group action and let X be a locally closed complex analytic subspace parameterizing ⊂Bsi integrable semiconnections. Let CS : si C be the holomorphic Chern-Simons functional. We call a finite dimensionalA complex→ submanifold V of a CS chart if A the critical locus of f = CS V is an open complex analytic subspace of X. Our proof of Theorem 0.1| consists of three parts: I. We show that there are (1) an open cover X = X ; ∪α α (2) Xα = (dfα = 0) for an r-dimensional CS chart (Vα,fα), with Vα and f = CS ; ⊂ A α |Vα (3) for any pair (α, β), there are open neighborhoods Vαβ Vα and Vβα Vβ of X X , and a biholomorphic map ϕ : V V ⊂with f ϕ =⊂f . α∩ β αβ αβ → βα β◦ αβ α An analytic space X equipped with the above (1)-(3) will be called a critical virtual manifold. See Definition 1.5 for the precise statement. II. We prove that given a critical virtual manifold structure, we can extract perverse sheaves Pα on Xα for all α (i.e. the perverse sheaves of vanishing cycles) and gluing isomorphisms σαβ : Pα Xαβ Pβ Xαβ of geometric origin (cf. Definition 2.9) where X = X X . The| same→ hold| for polarizable mixed Hodge modules. αβ α ∩ β III. We show that the 2-cocycle obstruction for gluing P to a global perverse { α} sheaf is Z2-valued and vanishes if there is a square root of the determinant line bundle det Rp∗R om( , ) Xred , where denotes (local) tautological family while p : Xred Y XHred isE theE | projection.E Therefore the local perverse sheaves P × → { α} glue to a global perverse sheaf if there is a square root of det Rp∗R om( , ) Xred in Pic(Xred) whose existence is guaranteed by an argument of OkounkovH whenE E X| red admits a global tautological family. This paper consists of two parts and a detailed layout of each part is provided at the beginning. This paper is a completely revised version of the authors’ preprint posted on the arXiv in December 2012. All the ideas and arguments are essentially the same. Some related results were independently obtained by Chris Brav, Vittoria Bussi, Delphine Dupont, Dominic Joyce and Balazs Szendroi in [7]. We are grateful to Dominic Joyce for his comments and suggestions. We thank Martin Olsson and Yan Soibelman for their comments. We also thank Takuro Mochizuki and Masaki Kashiwara for answering questions on mixed Hodge modules. Part 1 Critical virtual manifolds and perverse sheaves A critical virtual manifold is a (complex) analytic space which is locally the critical locus of a holomorphic function on a complex manifold of fixed dimen- sion (Definition 1.5). One may say that a critical virtual manifold is the gluing of Landau-Ginzburg models. Usual complex manifolds are special cases of critical vir- tual manifolds. As we will show in Part 2, moduli spaces of sheaves on Calabi-Yau 3-folds are critical virtual manifolds. In Part 1, we investigate several interesting structures on critical virtual manifolds, such as orientability, semi-perfect obstruc- tion theory, virtual fundamental class, Donaldson-Thomas type invariant, weighted Euler characteristic, local perverse sheaves of vanishing cycles, their gluing isomor- phisms and mixed Hodge modules. Since a critical virtual manifold X is locally the critical locus Xα of a holo- morphic function fα on a complex manifold Vα, it comes with a natural symmetric obstruction theory Eα = [TVα Xα ΩVα Xα ] LXα on each Xα.
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