Categorification of Donaldson-Thomas invariants via perverse ...

CATEGORIFICATION OF DONALDSON-THOMAS INVARIANTSVIA PERVERSE SHEAVESYOUNG-HOON KIEM AND JUN LIAbstract. We show that there is a perverse sheaf on a fine moduli spaceof stable sheaves on a smooth projective Calabi-Yau 3-fold, which is locallythe perverse sheaf of vanishing cycles for a local Chern-Simons functional,possibly after taking an étale Galois cover. This perverse sheaf lifts to a mixedHodge module and gives us a cohomology theory which enables us to definethe Gopakumar-Vafa invariants mathematically.1. IntroductionThe Donaldson-Thomas invariant (DT invariant, for short) is a virtual countof stable sheaves on a smooth projective Calabi-Yau 3-fold Y over C which wasdefined as the degree of the virtual fundamental class of the moduli space X of stablesheaves ([35]). Using microlocal analysis, Behrend showed that the DT invariant isin fact the Euler number of the moduli space, weighted by a constructible functionν X , called the Behrend function ([1]). Since the ordinary Euler number is thealternating sum of Betti numbers of cohomology groups, it is reasonable to ask ifthe DT invariant is in fact the Euler number of a cohomology theory on X. On theother hand, it has been known that the moduli space is locally the critical locusof a holomorphic function, called a local Chern-Simons functional ([12]). Givena holomorphic function f on a complex manifold V , one has the perverse sheafφ f (Q[dim V − 1]) of vanishing cycles supported on the critical locus and the Eulernumber of this perverse sheaf at a point x equals ν X (x). This motivated Joyce andSong to raise the following question ([12, Question 5.7]).Let X be the moduli space of simple coherent sheaves on Y . Does there exist anatural perverse sheaf P • on the underlying analytic variety X = X red which islocally isomorphic to the sheaf φ f (Q[dim V − 1]) of vanishing cycles for f, V above?The purpose of this paper is to provide an affirmative answer.Theorem 1.1. (Theorem 3.14 and Theorem 3.16)Let X be a quasi-projective moduli space of simple sheaves on a smooth projectiveCalabi-Yau 3-fold Y with universal family E and let X = X red be the reduced schemeof X. Then there exist an étale Galois coverρ : X † → X = X † /Gand a perverse sheaf P • on X † , which is locally isomorphic to the perverse sheafφ f (Q[dim V − 1]) of vanishing cycles for the local Chern-Simons functional f. InDate: v4; March 18, 2013.YHK was partially supported by NRF grant 2011-0027969. JL was partially supported by thegrant NSF-1104553.1

2 YOUNG-HOON KIEM AND JUN LIfact, for any étale Galois cover ρ : X † → X, there exists such a perverse sheaf P •if and only if the line bundle ρ ∗ det(Ext • π(E, E)) admits a square root on X † whereπ : X × Y → X is the projection and Ext • π(E, E) = Rπ ∗ RHom(E, E).We will also prove the same for mixed Hodge modules (Theorem 7.1), i.e. there isa mixed Hodge module M • on X † whose underlying perverse sheaf is rat(M • ) = P • .(See §7.) Note that the perverse sheaf P • may not be unique because we can alwaystwist P • by a Z 2 -local system on X.As an application of Theorem 1.1, when det Ext • π(E, E) admits a square root, thehypercohomology H i (X, P • ) of P • gives us the DT (Laurent) polynomialDT Y t(X) = ∑ it i dim H i (X, P • )such that DT−1(X) Y is the ordinary DT invariant by [1].Another application is a mathematical theory of Gopakumar-Vafa invariants(GV for short) in [9]. Let X be a moduli space of stable sheaves supported oncurves of homology class β ∈ H 2 (Y, Z). The GV invariants are integers n h (β) forh ∈ Z ≥0 defined by an sl 2 × sl 2 action on some cohomology of X such that n 0 (β) isthe DT invariant of X and that they give all genus Gromov-Witten invariants N g (β)of Y . By Theorem 1.1, when det Ext • π(E, E) admits a square root, there exists aperverse sheaf P • on X which is locally the perverse sheaf of vanishing cycles. Bythe relative hard Lefschetz theorem for the morphism to the Chow scheme ([30]),we have an action of sl 2 × sl 2 on H ∗ (X, ˆP • ) where ˆP • is the graduation of P • bythe filtration of P • which is the image of the weight filtration of the mixed Hodgemodule M • with rat(M • ) = P • . This gives us a geometric theory of GV invariantswhich we conjecture to give all the GW invariants N g (β).Our proof of Theorem 1.1 relies heavily on gauge theory. By the Seidel-Thomastwist ([12, Chapter 8]), it suffices to consider only vector bundles on Y . Let B =A/G be the space of semiconnections on a hermitian vector bundle E modulo thegauge group action and let B si be the open subset of simple points. Let cs : B → Cbe the (holomorphic) Chern-Simons functional. Let X ⊂ B si be a locally closedcomplex analytic subspace. We call a finite dimensional complex submanifold V ofB si a CS chart if the critical locus of f = cs| V is V ∩ X and is an open complexanalytic subspace of X. By [12], at each x ∈ X, we have a CS chart V withT x V = T x X, which we call the Joyce-Song chart (JS chart, for short). Thus wehave a perverse sheaf P • | V on V ∩ X. One of the difficulties in gluing the localperverse sheaves P • | V is that the dimensions of the JS charts V vary from point topoint. In this paper, we show that there are(1) a locally finite open cover X = ∪ α U α ;(2) a (continuous) family V α → U α of CS charts of constant dimension r, eachof which contains the JS chart;(3) a homotopy V αβ → U αβ × [0, 1] from V α | Uαβ = V| t=0 to V β | Uαβ = V| t=1where U αβ = U α ∩ U β . We call such a collection CS data. (See Proposition 3.12.)From the CS data, we can extract perverse sheaves P α• on U α for all α andgluing isomorpisms σ αβ : P α| • Uαβ → Pβ • | U αβ. (See Proposition 3.13.) The 2-cocycleobstruction for gluing {P α} • to a global perverse sheaf is shown to beσ αβγ = σ γα ◦ σ βγ ◦ σ αβ = ±1 ∈ Z 2

CATEGORIFICATION OF DONALDSON-THOMAS INVARIANTS 3which coincides with the 2-cocycle obstruction for gluing the determinant line bundlesof the tangent bundles of CS charts. Since the perfect obstruction theoryExt • π(E, E) for X is symmetric, the determinant of the tangent bundle is a squareroot of det Ext • π(E, E). Therefore the local perverse sheaves {P α} • glue to a globalperverse sheaf if and only if there is a square root of det Ext • π(E, E) in Pic(X). (SeeTheorem 3.14.)When X is the moduli scheme of one dimensional stable sheaves on Y , we showthat the torsion-free part of det Ext • π(E, E) has a square root by Grothendieck-Riemann-Roch. More generally, Z. Hua ([11]) proved that it is true for all sheaves.In §9, we simplify his proof and generalize his result to the case of perfect complexes.By taking a spectral cover using a torsion line bundle, we obtain a finite étale Galoiscover ρ : X † → X with a cyclic Galois group G and a perverse sheaf P • on X †which is locally the perverse sheaf of vanishing cycles of a local CS functional. (SeeTheorem 3.16.)The layout of this paper is as follows. In §2, we recall necessary facts about theperverse sheaves of vanishing cycles and their gluing. In §3, we collect the mainresults of this paper. In §4, we prove that there exists a structure which we callpreorientation data on X. In §5, we show that preorientation data induce CS datamentioned above. In §6, we show that CS data induce local perverse sheaves, gluingisomorphisms and the obstruction class for gluing. In §7, we prove an analogue ofTheorem 1.1 for mixed Hodge modules. In §8, we develop a theory of GV invariants.In §9, we discuss the existence of square root of det Ext • π(E, E).An incomplete version of this paper was posted in the arXiv on October 15,2012 (1210.3910). Some related results were independently obtained by C. Brav,V. Bussi, D. Dupont, D. Joyce and B. Szendroi in [5]. We are grateful to DominicJoyce for his comments and suggestions. We thank Martin Olsson for his comments,Zheng Hua for informing us of his paper [11] and Yan Soibelman for his comments.We also thank Takuro Mochizuki for answering questions on mixed Hodge modules.Notations. A complex analytic space is a local ringed space which is coveredby open sets, each of which is isomorphic to a ringed space defined by an idealof holomorphic functions on an analytic open subset of C n for some n > 0, andwhose transition maps preserve the sheaves of holomorphic functions. A complexanalytic variety is a reduced complex analytic space. We will denote the varietyunderlying a complex analytic space X by X. We will use smooth functions to meanC ∞ functions. In case the space is singular, with a stratification by smooth strata,smooth functions are continuous functions that are smooth along each stratum. Weuse analytic functions to mean continuous functions that locally have power seriesexpansions in the real and imaginary parts of coordinate variables. We will workwith analytic topology unless otherwise mentioned.2. Perverse sheaves of vanishing cyclesIn this section, we recall necessary facts about perverse sheaves of vanishingcycles. Let X be a complex analytic variety and Dc(X) b the bounded derivedcategory of constructible sheaves on X over Q. Perverse sheaves are sheaf complexeswhich behave like sheaves.

4 YOUNG-HOON KIEM AND JUN LIDefinition 2.1. An object P • ∈ D b c(X) is called a perverse sheaf (with respect tothe middle perversity) if(1) dim{x ∈ X | H i (ı ∗ xP • ) = H i (B ε (x); P • ) ≠ 0} ≤ −i for all i;(2) dim{x ∈ X | H i (ı ! xP • ) = H i (B ε (x), B ε (x) − {x}; P • ) ≠ 0} ≤ i for all iwhere ı x : {x} ↩→ X is the inclusion and B ε (x) is the open ball of radius ε centeredat x for ε small enough.Perverse sheaves form an abelian category P erv(X) which is the core of a t-structure ([2, §2]). An example of perverse sheaf is the sheaf of vanishing cycleswhich is the focus of this paper.Definition 2.2. Let f : V → C be a continuous function on a pseudo-manifold V .We defineA • f := φ f (Q[−1]) = RΓ {Ref≤0} Q| f −1 (0).When V is a complex manifold of dimension r and f is holomorphic, A • f[r] is aperverse sheaf on f −1 (0). (See [13, Chapter 8].) Let X f be the critical set (df = 0)of f in V . Since A • f [r] = φ f (Q[r − 1]) is zero on the smooth manifold f −1 (0) − X f ,A • f [r] is a perverse sheaf on X f , called the perverse sheaf of vanishing cycles for f.The stalk cohomology of A • f [1] at x ∈ f −1 (0) is the reduced cohomology ˜H • (M f )of the Milnor fiberM f = f −1 (δ) ∩ B ɛ (x) for 0 < δ ≪ ɛ ≪ 1.Proposition 2.3. Let f, f 0 , f 1 : V → C be continuous functions on a pseudomanifoldV .(1) Let Z be a subset of f0 −1 −1(0) ∩ f1 (0). Suppose Φ : V → V is a homeomorphismsuch that Φ| Z = id Z and f 1 ◦ Φ = f 0 . Then Φ induces an isomorphism Φ ∗ :A • ∼ =f 1| Z −→ A • f 0| Z in D b (Z) by pulling back.(2) Suppose Φ t : V → V , t ∈ [0, 1], is a continuous family of homeomorphismspreserving f, i.e. f ◦ Φ t = f for all t, such that Φ t | Z = id Z and Φ 0 = id V . Thenthe pullback isomorphism Φ ∗ 1 : A • f | ∼ =Z −→ A • f | Z is the identity morphism.Proof. By the definition of A • f , we have A• f 0= A • ∼ f 1◦Φ = Φ ∗ A • f 1. Let ı : Z ↩→ Vdenote the inclusion. Since Φ ◦ ı = ı, we have the isomorphismΦ ∗ : ı ∗ A • f 1= ı ∗ Φ ∗ A • f 1∼ =−→ ı ∗ A • f 0.Because Φ t preserves the set {Ref ≤ 0}, the isotopy {Φ t } induces a homotopyfrom the identity chain map Φ ∗ 0 = id A •f | Zto Φ ∗ 1 by choosing a flabby resolutionI • by the complex of singular cochains. Since homotopic chain maps are equal inthe derived category, we find that the induced isomorphism Φ ∗ 1 : A • f | ∼ =Z −→ A • f | Z isindeed the identity morphism.□Example 2.4. Let q = ∑ ri=1 y2 i on C r . The set {Re q > 0} ⊂ C r is a disk bundleover R r −{0} which is obviously homotopic to S r−1 . From the distinguished triangle(2.1) RΓ {Re q≤0} Q → Q → Rı ∗ ı ∗ Qwhere ı : {Re q > 0} ↩→ C r is the inclusion, we find that A • q[1] is a sheaf complexsupported at the origin satisfying A • [1] ∼ = Q[−r + 1], i.e.A • [r] ∼ = Q.

CATEGORIFICATION OF DONALDSON-THOMAS INVARIANTS 5Suppose Φ : C r → C r is a homeomorphism such that q ◦ Φ = q. Since A • [r] ∼ = Q,the isomorphism Φ ∗ : A • q[r] → A • q[r] is either 1 or −1. The sign is determined bythe change in the orientation of the sphere S r−1 in the Milnor fiber. Since q ispreserved by Φ, dΦ| 0 : T 0 C r → T 0 C r is an orthogonal linear transformation withrespect to q whose determinant is either 1 or −1. It is easy to see that these twosign changes are identical, i.e.Φ ∗ = det(dΦ| 0 ) · id .The following fact about the sheaf A • fof vanishing cycles will be useful.Proposition 2.5. (1) Let g : W → C be a holomorphic function on a connectedcomplex manifold W of dimension d and let q = ∑ ri=1 y2 i . Let V = W × Cr andf : V → C be f(z, y) = g(z) + q(y). Then the summation form of f induces anisomorphismA • f [d + r] ∼ = pr −11 A• g[d] ⊗ pr −12 A• q[r] ∼ = pr −11 A• g[d] ⊗ Q ∼ = pr −11 A• g[d]of perverse sheaves on the critical set X f of f.(2) Let Φ : V → V be a biholomorphic map such that f ◦Φ = f and Φ| W = id W ×{0} .Then Φ ∗ : A • f → A• f is det(dΦ| W ×{0}) id A •fand det(dΦ| W ×{0} ) = ±1.Proof. (1) is a result of D. Massey in [24, §2]; (2) is proved in [5, Theorem 3.1].□It is well known that perverse sheaves and isomorphisms glue.Proposition 2.6. Let X be a complex analytic space with an open covering {X α }.(1) Suppose that for each α we have P • α ∈ P erv(X α ) and for each pair α, β we haveisomorphismsσ αβ : P • α| Xα∩X β∼ =−→ P • β | Xα∩X βsatisfying the cocycle condition σ βγ ◦ σ αβ = σ αγ . Then {P α} • glue to define aperverse sheaf P • on X such that P • | Xα∼ = P•α and that σ αβ is induced by theidentity map of P • | Xα∩X β.(2) Suppose P • , Q • ∈ P erv(X) and σ α : P • ∼ =| Xα −→ Q • | Xα such that σ α | Xα∩X β=σ β | Xα∩X β. Then there exists an isomorphism σ : P • → Q • such that σ| Xα = σ αfor all α.See [5, Theorem 2.5] for precise references for proofs of Proposition 2.6. One wayto prove Proposition 2.6 is to use the elementary construction of perverse sheavesby MacPherson and Vilonen.Theorem 2.7. [22, Theorem 4.5] Let S ⊂ X be a closed stratum of complexcodimension c. The category P erv(X) is equivalent to the category of objects(B • , C) ∈ P erv(X − S) × Sh Q (S) together with a commutative triangleR −c−1 π ∗ κ ∗ κ ∗ B • R −c π ∗ γ ! γ ∗ B •msuch that ker(n) and coker(m) are local systems on S, where κ : K ↩→ L andγ : L − K ↩→ L are inclusions of the perverse link bundle K and its complementCn

6 YOUNG-HOON KIEM AND JUN LIL−K in the link bundle π : L → S. The equivalence of categories is explicitly givenby sending P • ∈ P erv(X) to B • = P • | X−S together with the natural morphismsR −c−1 π ∗ κ ∗ κ ∗ B • R −c π ∗ γ ! γ ∗ B •mR −c π ∗ ϕ ! ϕ ∗ P • nwhere ϕ : D − K ↩→ D is the inclusion into the normal slice bundle.See [22, §4] for precise definitions of K, L and D. Morally the above theoremsays that an extension of a perverse sheaf on X − S to X is obtained by adding asheaf on S. Since sheaves glue, we can glue perverse sheaves stratum by stratum.Proof of Proposition 2.6. We stratify X by complex manifolds and let X (i) denotethe union of strata of codimension ≤ i. On the smooth part X (0) , P • α are honestsheaves and hence they glue to a sheaf P • | X (0). For X (1) = X (0) ∪ S, we find thatsince P • α are isomorphic on intersections X α ∩ X β , the sheaves R −c π ∗ ϕ ! ϕ ∗ P • α glueand so do the natural triangles(2.2) R −2 π ∗ κ ∗ κ ∗ (P • | X (0)) R −1 π ∗ γ ! γ ∗ (P • | X (0))mR −1 π ∗ ϕ ! ϕ ∗ P • α.nHence we obtain a perverse sheaf P • | X (1) ∈ P erv(X (1) ). It is obvious that we cancontinue this way using Theorem 2.7 above until we obtain a perverse sheaf P • onX such that P • | Xα = P α.•The gluing of isomorphisms is similar.□Another application of Theorem 2.7 is the following rigidity property of perversesheaves.Lemma 2.8. Let P • be a perverse sheaf on an analytic variety U. Let π : T → Ube a continuous map from a topological space T with connected fibers and let T ′ bea subspace of T such that π| T ′ is surjective. Suppose an isomorphism µ : π −1 P • ∼ =π −1 P • satisfies µ| T ′ = id (π −1 P • )| T ′. Then µ = id π −1 P •.Proof. We first prove the simple case: if we let C be a locally constant sheaf overQ of finite rank on Z ⊂ U and ¯µ : π −1 C → π −1 C be a homomorphism such that¯µ| T ′ ∩π −1 (Z) = id, then ¯µ is the identity morphism. Indeed, since the issue is local,we may assume that Z is connected and that C ∼ = Q r so that ¯µ : Q r → Q r is givenby a continuous map π −1 (Z) → GL(r, Q). By connectedness, this obviously is aconstant map which is 1 along T ′ ∩π −1 (Z). We thus proved the lemma in the sheafcase.For the general case, we use Theorem 2.7. As in the proof of Proposition 2.6above, we stratify U and let U (i) be the union of strata of codimension ≤ i. Since P •is a perverse sheaf, P • | U (0)[− dim U] is isomorphic to a locally constant sheaf andhence µ| U (0) is the identity map. For U (1) = U (0) ∪S, using the notation of Theorem2.7, C = R −1 π ∗ ϕ ! ϕ ∗ P • is a locally constant sheaf and µ induces a homomorphismπ −1 C → π −1 C which is identity on T ′ ∩ π −1 (S). Therefore µ induces the identity

CATEGORIFICATION OF DONALDSON-THOMAS INVARIANTS 7morphism of the pullback of (2.2) by π to itself and hence µ is the identity on U (1) .Continuing in this fashion, we obtain Lemma 2.8.□3. Preorientation data and perverse sheavesIn this section, we collect the main results of this paper. We first introducethe notion of preorientation data which induces a family of CS charts. This willgive us a collection of local perverse sheaves and gluing isomorphisms. We identifythe cocycle condition for the gluing isomorphisms as the existence of a square rootof the determinant bundle of Ext • π(E, E) = Rπ ∗ RHom(E, E) where E denotes theuniversal bundle over X × Y −→ πX.3.1. Chern-Simons functionals on connection spaces. In this subsection webriefly recall the necessary gauge theoretic background. More details will be providedin later sections. Our presentation largely follows [12, Chapter 9].Let Y be a smooth projective Calabi-Yau 3-fold over C, with a Hodge metricimplicitly chosen. We fix a nowhere vanishing holomorphic (3, 0)-form Ω on Y . LetE be a smooth complex vector bundle on Y with a smooth hermitian metric. Inthis paper, a smooth semiconnection is a differential operator ¯∂ : Ω 0 (E) → Ω 0,1 (E)satisfying the ∂-Leibniz rule. We denote by Ω 0,k (E) the space of smooth (0, k)-forms on Y taking values in E. Following the notation in gauge theory, we denotead E = E ∨ ⊗ E; thus fixing a smooth semiconnection ∂ 0 , all other semiconnectionscan be expressed as ∂ 0 + a, with a ∈ Ω 0,1 (adE).We fix a pair of integers s ≥ 4 and l > 6, and form the completion Ω 0,k (adE) sof Ω 0,k (adE) under the Sobolev norm L l s. (L l s is the sum of L l -norms of up to s-thpartial derivatives.) We say ∂ 0 +a is L l s if a is L l s, assuming ∂ 0 is smooth. We denoteby G the gauge group of L l s+1-sections of Aut(E) modulo C ∗ , which is the L l s+1completion of C ∞ (Aut(E))/C ∗ . We denote by A the space of L l s-semiconnectionson E. We have an isomorphism of affine spaces, after a choice of smooth ∂ 0 ∈ A,via(3.1) ∂ 0 + · : Ω 0,1 (ad E) s −→ A, a ↦→ ∂ 0 + a.The gauge group G acts on A via g · (∂ 0 + a) = (g −1 ) ∗ (∂ 0 + a). Let A si be theG-invariant open subset of simple semiconnections, i.e. the automorphism groupsare all C ∗ · id E . LetB si = A si /G ⊂ A/G := B.Then B si is a complex Banach manifold.An element ¯∂ ∈ A is called integrable if the curvature F ¯∂0,2 := ( ¯∂) 2 vanishes. If¯∂ is integrable and a ∈ Ω 0,1 (ad E), then ¯∂ + a is F ¯∂+a 0,2 = ¯∂a + a ∧ a. By Sobolev inequality,F 0,2 is a continuous operator from ∂ Ω0,1 (adE) s to Ω 0,2 (adE) s−1 , analytic0+·in a. An integrable smooth semiconnection ∂ defines a holomorphic vector bundle(E, ∂) on Y .Picking a (reference) integrable ¯∂ ∈ A, the holomorphic Chern-Simons functionalis defined asCS : A → C, CS( ¯∂ + a) = 1 ( 14π∫Y2 tr2 ( ¯∂a) ∧ a + 1 )3 a ∧ a ∧ a ∧ Ω.

8 YOUNG-HOON KIEM AND JUN LIThis is a cubic polynomial in a whose quadratic part isCS 2 ( ¯∂ + a) = 14π∫Y2 tr ( 12 ( ¯∂a) ∧ a ) ∧ Ω.Since the directional derivative of CS at ∂ + a in the direction of b isδ CS(∂ + a)(b) = 14π∫Y2 tr(b ∧ F ¯∂+a 0,2 ) ∧ Ω,δ CS(∂ + a) = 0 if and only if ∂ + a is integrable. Thus the complex analyticsubspace A intsi of simple integrable smooth semiconnections in A si is the criticallocus (complex analytic subspace) of CS. Let V si = A intsi /G ⊂ B si. Since CS isG-equivariant, it descends tocs : B si → Cwhose critical locus (complex analytic subspace) in B si is V si .3.2. The universal family over X. Let X ⊂ V si ⊂ B be an open complexanalytic subspace and denote by X = X red the X set endowed with the reducedscheme structure. We assume that X is quasi-projective and that there is a universalfamily of holomorphic bundles E over X×Y that induces the morphism X → B. Ourconvention is that E is with its holomorphic structure ∂ X implicitly understood.For x ∈ X, we denote by E x = E| x×Y , the holomorphic bundle associated withx ∈ X. We denote by ∂ x the restriction of ∂ X to E x . We use adE x to denote thesmooth vector bundle E ∨ x ⊗ E x .We fix a hermitian metric h on E that is analytic in X direction (see §4.3). For x ∈X, we denote by h x the restriction of h to E x . Using h x , we form Ω 0,k (adE x ) s , thespace of L l s adE x -valued (0, k)-forms. We form the space A x of L l s semiconnectionson E x , which is isomorphic to Ω 0,1 (adE x ) s via ∂ x +· (cf. (3.1)). We form the adjoint∂ ∗ x of ∂ x using the hermitian metric h x .Since Aut(E x ) = C ∗ , a standard argument in gauge theory shows that the tangentspace of B at x is(3.2) T x B ∼ = Ω 0,1 (adE x ) s / Im(∂ x ) s∼ = ker(∂∗x) s , x ∈ X,where the first isomorphism depends on a choice of smooth E x∼ = E; the secondisomorphism is canonical using the Hodge theory of (E x , ∂ x , h x ). (We use the subscript“s” to indicate that it is the image in Ω 0,1 (adE x ) s .) For any open subsetU ⊂ X, we denote T U B = T B| U . Since X ⊂ B is a complex analytic subspace, it isa holomorphic Banach bundle over U.For x ∈ X, we form the Laplacianand its truncated eigenspace□ x = ∂ x ∂ ∗ x + ∂ ∗ x∂ x : Ω 0,1 (adE x ) s → Ω 0,1 (adE x ) s−2 ,Θ x (ɛ) = C-span {a ∈ Ω 0,1 (adE x ) s | ∂ ∗ xa = 0, □ x a = λa, λ < ɛ} ⊂ ker(∂ ∗ x) s .Note that T x X ∼ = ker(□ x ) 0,1s ⊂ Θ x (ɛ) for any ɛ > 0; since (E x , ∂ x , h x ) are smooth,Θ x (ɛ) ⊂ Ω 0,1 (adE x ) (i.e., consisting of smooth forms).

CATEGORIFICATION OF DONALDSON-THOMAS INVARIANTS 93.3. Preorientation data. We recall the quadratic form cs 2,x on T x B, x ∈ X,induced by the CS-function cs : B → C, defined explicitly bycs 2,x (a 1 , a 2 ) = 1 ∫8π 2 tr ( )a 1 ∧ ∂ x a 2 ∧ Ω, ai ∈ Ω 0,1 (adE x ) s /ker(∂ x ) s .Since T x X is the null subspace of cs 2,x , we have the induced non-degenerate quadraticform(3.3) Q x : T x B/T x X × T x B/T x X −→ C.We introduce the notion of orientation bundles and their homotopies. Let r bea positive integer. (The notion of analytic subbundles will be recalled in the nextsection.)Definition 3.1. Let U ⊂ X be open. A rank r orientation bundle on U is a rankr analytic subbundle Ξ ⊂ T U B such that(1) there is an assignment U ∋ x ↦→ ɛ x ∈ (0, 1), continuous in x ∈ U, such thatfor every x ∈ U, Θ x (ɛ x ) ⊂ Ξ x := Ξ| x ;(2) at each x ∈ U, Q x := Q x | Ξx/T xX is a non-degenerate quadratic form.We also need a notion of homotopy between orientation bundles.Definition 3.2. Let Ξ a and Ξ b be two orientation bundles over U. A homotopyfrom Ξ a to Ξ b is a family of orientation bundles Ξ t on U, such that(1) the family is analytic in t, and Ξ 0 = Ξ a and Ξ 1 = Ξ b ;(2) for all t ∈ [0, 1], Ξ t satisfy (1) of Definition 3.1 for a single ɛ·.The following is one of the key ingredients in our construction of perverse sheaveson moduli spaces.Definition 3.3. We say X is equipped with rank r preorientation data if there are(1) a locally finite open cover X = ∪ α U α ,(2) a rank r orientation bundle Ξ α on U α for each α, and(3) a homotopy Ξ αβ from Ξ α | Uαβ to Ξ β | Uαβ for each U αβ = U α ∩ U β .Since the open cover is locally finite, for each x ∈ X, we can find an open setU x which is contained in any U α with x inside. We can further choose ɛ x > 0 suchthat Ξ α ⊃ Θ x (ɛ x ) whenever x ∈ U α . To simplify the notation, from now on we willsuppress ɛ x and write Θ x for the subbundle Θ x (ɛ x ).In §4, we will prove the following.Proposition 3.4. Every quasi-projective X ⊂ V si with universal family admitspreorientation data.3.4. Families of CS charts and local trivializations. We introduce anotherkey ingredient, called CS charts.Definition 3.5. Let f be a holomorphic function on a complex manifold V suchthat 0 is the only critical value of f. The reduced critical locus (also called thecritical set) is the common zero set of the partial derivatives of f. The criticallocus of f is the complex analytic space X f defined by the ideal (df) generated bythe partial derivatives of f.

10 YOUNG-HOON KIEM AND JUN LIDefinition 3.6. An r-dimensional CS chart for X is an r-dimensional complexsubmanifold V of A si , which embeds holomorphically into B si by the projectionA si → B si , such that, letting ı : V → B be the inclusion, the critical locus X cs◦ı ⊂ Vis an open complex analytic subspace of X ⊂ V si .We say the chart (V, ι) contains x ∈ X if x ∈ ı(X cs◦ı ).Example 3.7. By [12, Theorem 5.5], for any x = ¯∂ 0 ∈ V si ,V = { ¯∂ a = ¯∂ 0 + a | ¯∂ ∗ 0a = ¯∂ ∗ 0F 0,2¯∂ 0+a = 0, ‖a‖ s< ε} ⊂ A siis a CS chart of X containing x of dimension dim T x X, for a sufficiently smallε > 0. This chart depends on x and the choice of a hermitian metric on E onwhich the adjoint ∂ ∗ 0 depends. In this paper, we call this chart the Joyce-Song chartat x. We remark that when ∂ 0 and the hermitian metric are smooth, all ∂ 0 + a ∈ Vare smooth.Definition 3.8. Let ρ : Z → X be a continuous map from a topological space Zto X. Let r be a positive integer. A family of r-dimensional CS charts (for X) is asubspace V ⊂ Z × B that fits in a diagramV Z × BπZpr Zsuch that for each x ∈ Z the fiber V x := π −1 (x) ⊂ B si and is an r-dimensional CSchart of X containing ρ(x).Given V ⊂ Z × B a family of CS charts over ρ : Z → X, we define ∆(Z) ={(x, ρ(x) x ) | x ∈ Z} ⊂ Z × V, where ρ(x) x is the unique point in V x whose imagein B is ρ(x).Definition 3.9. A local trivialization of the family Z ← V ↩→ Z × B consistsof an open U 0 ⊂ Z, an open neighborhood U of ∆(U 0 ) ⊂ V U0 := π −1 (U 0 ), and acontinuous(3.4) U 0 × V U0 ⊃ UΨ−−−−→ V U0 × U 0 ,such that Ψ commutes with the two tautological projections U ⊂ U 0 ×V U0 → U 0 ×U 0and V U0 × U 0 → U 0 × U 0 , and that(1) letting U 0 → U 0 × U 0 be the diagonal, then Ψ| U×U0 ×U 0 U 0= id;(2) for any x, y ∈ U 0 , letting U x,y = U ∩(x×V y ) and Ψ x,y := Ψ| Ux,y : U x,y → V x ,then Ψ x,y is biholomorphic onto its image; Ψ x,y restricted to U x,y ∩ X fy isan open immersion into X fx ⊂ V x , commuting with the tautological openimmersions X fx , X fy ⊂ X.We say that V ⊂ Z × B admits local trivializations if for any x 0 ∈ Z there is alocal trivialization over an open neighborhood U 0 of x 0 in Z.Definition 3.10. Let Z ⊂ R n be a (real) analytic subset defined by the vanishingof finitely many analytic functions. Let V ⊂ Z × B be a family of CS charts overρ : Z → X. A complexification of V consists of a complexification Z C ⊂ C n of Z(thus having Z C ∩ R n = Z), a holomorphic ρ C : Z C → X extending ρ : Z → X,

CATEGORIFICATION OF DONALDSON-THOMAS INVARIANTS 11and a holomorphic family of CS charts V C ⊂ Z C × B over Z C (i.e. V C is a complexanalytic subspace of Z C × B) such thatV C × Z C Z = V ⊂ Z × B.Let U ⊂ X be an open subset and V ⊂ U × B be a family of CS charts over Uwhich admits local trivializations. Let U 0 ⊂ U be an open subset and Ψ in (3.4) alocal trivialization of V over U 0 .Definition 3.11. We say that the local trivialization Ψ is complexifiable if for anyx ∈ U 0 , there is an open neighborhood x ∈ O x ⊂ U 0 such that if we let V Ox ⊂ O x ×Band Ψ Ox : U Ox → V Ox ×O x , where U Ox = U × U×V O x ×V Ox and Ψ Ox is the pullbackof Ψ, the following hold:(1) the family V Ox ⊂ O x × B admits a complexification V O C x⊂ O C x × B over acomplexification O C x of O x ;(2) there is a holomorphic local trivialization Ψ O C x: U O C x→ V O C x× O C x , i.e.U O C x⊂ O C x × V O C xis open and contains the diagonal ∆(O C x ), such that Ψ O C xis holomorphic,U O C x× O C xO x = U Ox and Ψ = Ψ O C x| U : U Ox → V Ox × O x ⊂ V O C x× O C x .In §5, we will prove the following.Proposition 3.12. Let X ⊂ B si be equipped with preorientation data (∪U α , Ξ α , Ξ αβ ).Then there are(1) a family of r-dimensional CS charts V α ⊂ U α × B with complexifiable localtrivializatons at all x ∈ U α ;(2) an open neighborhood U x and a subfamily W x of CS charts in V α | Ux foreach x ∈ U α , i.e. a subbundle W x of V α | Ux which admits compatible complexifiablelocal trivializationsU x × V α | Ux ⊃ U Ψ V α | Ux × U x U x × W x | Ux ⊃ U ′Ψ W x | Ux × U x(3) a family of CS charts V αβ parameterized by U αβ ×[0, 1] with V αβ | Uαβ ×{0} =V α | Uαβ , V αβ | Uαβ ×{1} = V β | Uαβ , which has complexifiable local trivializationsat all (x, t) ∈ U αβ × [0, 1], such that V αβ | Ux×[0,1] contains the subfamilyW x × [0, 1] of CS charts over U x × [0, 1].We call the above (V α , W x , V αβ ) CS data for X.3.5. Local perverse sheaves and gluing isomorphisms. Given CS data, wecan construct perverse sheaves P α • on each U α and gluing isomorphisms σ αβ :P α| • Uαβ → Pβ • | U αβ.We will prove the following in §6.Proposition 3.13. (1) Let π : V → U be a family of CS charts on U ⊂ X ⊂ B siwith complexifiable local trivializations at every point x ∈ U. Then the perversesheaves of vanishing cycles forcsf x : V x = π −1 (x) ⊂ B si −→ C

12 YOUNG-HOON KIEM AND JUN LIglue to a perverse sheaf P • on U, i.e. P • is isomorphic to A • f x[r] in a neighborhoodof x.(2) Let V α and V β be two families of CS charts on U with complexifiable localtrivializations. Let P α • and Pβ • be the induced perverse sheaves on U. Let V bea family of CS charts on U × [0, 1] with complexifiable local trivializations suchthat V| U×{0} = V α and V| U×{1} = V β . Suppose for each x ∈ U, there are anopen neighborhood U x ⊂ U and a subfamily W of both V α | Ux and V β | Ux such thatW × [0, 1] is a complexifiable subfamily of CS charts in V| Ux×[0,1]. Then there isan isomorphism σ αβ : P α • ∼ = Pβ • of perverse sheaves.(3) If there are three families V α , V β , V γ with homotopies among them as in (2),then the isomorphisms σ αβ , σ βγ , σ γα satisfyσ αβγ := σ γα ◦ σ βγ ◦ σ αβ = ± id .In fact, the isomorphism in (2) is obtained by gluing pullback isomorphismsvia biholomorphic maps χ αβ : V α,x → V β,x at each x. The sign ±1 in (3) is thedeterminant of the compositiondχ αβ dχ βγ dχ γαT x V α,x −→ Tx V β,x −→ Tx V γ,x −→ Tx V α,xwhere V·,x is the fiber of V· over x. By Serre duality, det T V α,x is a square rootof det Ext • π(E, E) and hence the 2-cocycle σ αβγ defines an obstruction class inH 2 (X, Z 2 ) for the existence of a square root of det Ext • π(E, E). Combining Propositions3.4, 3.12 and 3.13, we thus obtain the following.Theorem 3.14. Let X ⊂ X ⊂ V si be quasi-projective and equipped with a universalfamily E. Then there is a perverse sheaf P • on X which is locally a perverse sheaf ofvanishing cycles if and only if there is a square root of the line bundle det Ext • π(E, E)in Pic(X).It is obvious that the theorem holds for any étale cover of X.3.6. Divisibility of the determinant line bundle. In this subsection, we showthat for moduli spaces X of stable sheaves on Y , and for the universal sheaf E onX × Y , det Ext • π(E, E) has a square root possibly after taking a Galois étale coverX † → X. By Theorem 3.14, there is a globally defined perverse sheaf P • on X †which is locally the perverse sheaf of vanishing cycles.To begin with, using the exponential sequencewe see thatH 1 (X, O X ) −→ H 1 (X, O ∗ X) −→ H 2 (X, Z),(3.5) det Ext • π(E, E) = det Rπ ∗ RHom(E, E)admits a square if and only if its first Chern class in H 2 (X, Z) is even. We determinethe torsion-free part of the first Chern class of (3.5) using the Grothendieck-Riemann-Roch theorem:ch(Ext • π(E, E)) = π ∗(ch(RHom(E, E))td(Y )).Since E is flat over X, α i := c i (E) ∈ A ∗ (X × Y ) Q . Let r = rank E. Then one hasch(E) = r + α 1 + 1 2 (α2 1 − 2α 2 ) + 1 6 (α3 1 − 3α 1 α 2 + 3α 3 ) + δ 4 + · · · ,where δ 4 ∈ A 4 (X × Y ) Q , and · · · are elements in A >4 (X × Y ) Q . Thus(3.6) ch(RHom(E, E)) = ch(E)ch(E ∨ )

CATEGORIFICATION OF DONALDSON-THOMAS INVARIANTS 13= r 2 + ((r − 1)α 2 1 − 2rα 2 ) + (− α4 112 + α2 2 − α 1 α 3 + 2rδ 4 ) + · · · .Since Y is a Calabi-Yau three-fold, we havetd(Y ) = 1 + 112 c 2(T Y ).Thus by GRR, the torsion-free part of the first Chern class of (3.5) is(3.7) c 1(det Ext•π (E, E) ) = [ ch(Ext • π(E, E)) ] 1( α 4= π ∗ −112 + α2 2 − α 1 α 3 + 2rδ 4 + ((r − 1)α1 2 − 2rα 2 ) 112 c 2(T Y ) )where [·] 1 denotes the degree one part in A 1 X Q .We now suppose that X is the moduli of one dimensional sheaves. Then r = 0and α 1 = 0. Hence (3.7) reduces toc 1(det Ext•π (E, E) ) = π ∗ (α 2 2).We let [α 2 ] be the torsion-free part of the image of α 2 in H 4 (X × Y, Z). We applythe Künneth formula [34]. By the canonical isomorphism (modulo torsions)H 4 (X × Y, Z)/tor ∼ = [H ∗ (X, Z) ⊗ H ∗ (Y, Z)] 4 /tor.We write [α 2 ] = ∑ 4 ∑i=0 j a i,j ⊗ b 4−i,j , where b 4−i,j ∈ H 4−i (Y, Z), etc. Then[α 2 ] 2 ≡4∑ ∑a i,j ⊗ b i−4,j ∧ a i,j ⊗ b i−4,j ≡ ∑ a 2 2i,j ⊗ b 2 4−2i,j mod 2,ji,ji=0whose part in · ⊗ [Y ] ∨ is trivial. This proves that the torsion-free part of (3.7) iseven. More generally, by a similar calculation, it is proved in §9 that the torsion-freepart of (3.7) is even for any perfect complex E on X × Y . See Theorem 9.1.Lemma 3.15. Let X be a fine moduli scheme of simple sheaves on Y . There is atorsion line bundle L on X = X red with L ⊗k ∼ = O X for some k > 0 such that thepullback of det Ext • π(E, E) by the cyclic étale Galois cover X † = {s ∈ L | s k = 1} →X admits a square root L on X † .Proof. Since the torsion-free part of c 1 (det Ext • π(E, E)) is even, there is a torsionline bundle L on X such that L ⊗ det Ext • π(E, E) is torsion-free and admits a squareroot. Note that the pullback of L to X † is trivial.□For a moduli space X of stable sheaves with universal bundle, we can applythe Seidel-Thomas twists ([12]) so that we can identify X as a complex analyticsubspace of B si . By Theorem 3.14, we have the following.Theorem 3.16. If X is a fine moduli scheme of stable sheaves on Y , there exist acyclic étale Galois cover X † → X and a perverse sheaf P • on X † which is locallythe perverse sheaf of vanishing cycles.4. Existence of preorientation dataIn this section we prove Proposition 3.4. We construct orientation bundles, theirhomotopies, and their complexifications.

14 YOUNG-HOON KIEM AND JUN LI4.1. The semiconnection space. We continue to use the convention introducedat the beginning of Subsection 3.2. Thus, E is the universal family of locally freesheaves on X × Y and (E x , ∂ x ) is the restriction E| x×Y , and we use adE x to denotethe smooth vector bundle Ex ∨ ⊗ E x .Since X is quasi-projective, X has a stratification according to the singularitytypes of points in X. We fix such a stratification. We say a continuous function(resp. a family) on an open U ⊂ X is smooth if its restriction to each stratum issmooth.We now choose a smooth hermitian metric h on E. Since X is quasi-projective,by replacing E by its twist via a sufficiently negative line bundle from X, for asufficiently ample H on Y , we can make E ∨ ⊗ p ∗ Y H generated by global sections,where p Y : X×Y → Y is the projection. Thus for an integer N, we have a surjectivehomomorphism of vector bundles O ⊕NX×Y→ E ∨ ⊗ p ∗ Y H; dualizing and untwistingH ∨ , we obtain a subvector bundle homomorphism E ⊂ p ∗ Y H⊕N . We then endow Ha smooth hermitian metric; endow H ⊕N the direct sum metric, and endow p ∗ Y H⊕Nthe pullback metric. We define h to be the induced hermitian metric on E via theholomorphic subbundle embedding E ⊂ p ∗ Y H⊕N . For x ∈ X, we denote by h x therestriction of h to E x .For the integers s and l chosen before, we denote the L l s-completion of Ω 0,k (adE x )by Ω 0,k (adE x ) s . We form the formal adjoint ∂ ∗ x of ∂ x using the hermitian metrich x . Then ∂ x and ∂ ∗ x extend to differential operators∂ x (resp. ∂ ∗ x) : Ω 0,k (adE x ) s+1−k → Ω 0,k+1 (adE x ) s−k (resp. Ω 0,k−1 (adE x ) s−k ).We use ker(∂ x ) 0,ks+1−k to denote the kernel of ∂ x in Ω 0,k (adE x ) s+1−k ; likewise,ker(∂ ∗ x) 0,ks+1−k. We form the Laplacian□ x = ∂ x ∂ ∗ x + ∂ ∗ x∂ x : Ω 0,k (adE x ) s+1−k → Ω 0,k (adE x ) s−1−k .We denote by □ −1x (0) 0,k the set of harmonic forms (the kernel of □ x ) in Ω 0,k (adE x ). 1It will be convenient to fix a local trivialization of E. Given x 0 ∈ X, we realizean open neighborhood U 0 ⊂ X of x 0 as U 0 = X f0 , where (V 0 , f 0 ) := (V x0 , f x0 )is the JS chart. We fix a smooth isomorphism E ≃ E x0 . Since V 0 is a complexmanifold, by shrinking x 0 ∈ V 0 if necessary, we assume that V 0 is biholomorphic toan open subset of C n for some n. We let z = (z 1 , · · · , z n ) be the induced coordinatevariables on V 0 . By abuse of notation, we also use z to denote a general element inV 0 .Let ∂ 0 + a 0 (z) be the family of semiconnections on E of the chart (V 0 , f 0 ).Because a 0 (z) satisfies the system in Example 3.7, ∂ z a 0 (z) = 0. Let E V0 = V 0 × E,as a vector bundle over V 0 × Y . Let ∂ z be the ∂-operator along the z direction ofthe product bundle E V0 = V 0 × E. Then ∂ V0 := ∂ 0 + ∂ z + a 0 (z) is a semiconnectionon E V0 . It is known that (cf. [12, Chapter 9], [26]) X f0 = (F 0,2 = 0), which is∂ 0+a 0(z)the same as ((∂ V0 ) 2 = 0) since ∂ z a 0 (z) = 0.Using U 0 = X f0 , the restriction (E V0 , ∂ V0 )| U0 is a holomorphic bundle over U 0 ×Y . By construction, it is biholomorphic to E U0 := E| U0 . Let(4.1) ζ : (E U0 , ∂ V0 | U0 ) −→ E U01 If we don’t put any subscript at Ω 0,·(·), it means the set of smooth sections.

CATEGORIFICATION OF DONALDSON-THOMAS INVARIANTS 15be a biholomorphism. Since E x0 is simple, we can assume that ζ extends the smoothisomorphism E ≃ E x0 we begun with. In the following, we call ζ a framing of Eover U 0 .Via this framing, we identify Ω 0,k (adE x ) s with Ω 0,k (adE) s ; the connection forma 0 (z) locally has convergent power series expansion in z everywhere over V 0 , withcoefficients smooth adE-valued (0, 1)-forms.Another application of this local framing is that it gives local trivializations ofT X B. Using the holomorphic family of semiconnections ∂ 0 + a 0 (z), we embeds V 0into B as a complex submanifold. (Since (V 0 , f 0 ) is a CS chart, V 0 ⊂ B si .) UsingB = A/G, we have an induced surjective homomorphism of holomorphic Banachbundles(4.2) V 0 × Ω 0,1 (adE) s −→ T V0 B.By shrinking 0 ∈ V 0 if necessary, the induced V 0 × ker(∂ ∗ 0) 0,1s → T V0 B is an isomorphismof holomorphic Banach bundles. Since for x ∈ X, ∂ ∗ x : Ω 0,1 (adE x ) s →Ω 0 (adE x ) s−1 /C is surjective, ker(∂ ∗ ) 0,1X := ∐ x∈X ker(∂∗ x) 0,1sbundle. The same holds for ker(∂) 0,2X ⊂ Ω0,2 X (adE) s−1.is a smooth Banach4.2. Truncated eigenspaces. We form the (partially) truncated eigenspaceΘ x (ɛ):= C-span {a ∈ Ω 0,1 (adE x ) s | ∂ ∗ xa = 0, □ x a = λa, λ < ɛ} ⊂ ker(∂ ∗ x) 0,1and its (0, 2)-analogueΘ x (ɛ) ′ := C-span {a ∈ Ω 0,2 (adE x ) s | ∂ x a = 0, □ x a = λa, λ < ɛ} ⊂ ker(∂ x ) 0,2 .Let U ⊂ X be an open subset. We formΘ U (ɛ) = ∐ Θ x (ɛ) ⊂ ker(∂ ∗ ) 0,1X | U and Θ ′ U (ɛ) = ∐ Θ ′ x(ɛ) ⊂ ker(∂) 0,2X | U .x∈Ux∈URecall that each □ x has discrete non-negative spectrum. We say an ɛ > 0 separateseigenvalues of □ x for x ∈ U if for any x ∈ U, ɛ is not an eigenvalue of □ x .Using that all E x in X are simple, we have the following vanishing result.Lemma 4.1. There is a continuous function X ∋ x ↦→ ɛ x ∈ (0, 1) such that forany x ∈ X, □ x has no eigenforms in ker(∂ x ) s 0,1 and ker(∂ ∗ x) 0,2s−1 of eigenvaluesλ ∈ (0, ɛ x ).Proof. For any x 0 ∈ X, we pick a connected open x 0 ∈ U 0 ⊂ X so that the closureŪ 0 ⊂ X is compact. We show that we can find ɛ 0 > 0 so that the first statementholds for x ∈ U 0 and ɛ x replaced by ɛ 0 .Suppose not. Then we can find a sequence x n ∈ U 0 , α n ∈ ker(∂ xn ) 0,1 and λ n > 0such that □ xn α n = λ n α n and λ n → 0. Using the spectral theory of self-adjointoperators and Hodge theory, and that ∂ xn α n = 0, we conclude that α n = ∂ xn β nfor a β n ∈ ker(∂ ∗ (x) 0 ∗)s. Since □ xn α n = λ n α n , ∂ xn ∂ x n∂ xn β n − λ n β n = 0. Further,by subtracting a constant multiple of id E , we can assume ∫ Y tr β n ∗1 = 0 for all n.Since Ū0 is compact, by passing through a subsequence we can assume x n →x ∈ Ū0. Then after normalizing the L l norm of β n to be one, using elliptic estimatewe conclude that (after passing through a subsequence) β n converges to β ≠ 0 ∈ker(∂ ∗ x) 0 s and satisfying ∂ x ∂ ∗ x∂ x β = 0 and ∫ Y tr β ∗1 = 0. From ∂ x∂ ∗ x∂ x β = 0, weconclude ∂ x β = 0. Because E x is simple (by assumption on X), ∫ tr β ∗1 = 0 forcesβ = 0, contradicting to ‖β ‖ L l= 1.

CATEGORIFICATION OF DONALDSON-THOMAS INVARIANTS 17Writing z k = u k + iv k , we can view D as an open subset of R 2n , where R 2nis with the coordinate variable (u 1 , · · · , u n , v 1 , · · · , v n ). By allowing u k and v k totake complex values, we embed R 2n ⊂ C 2n , thus embed D ⊂ C 2n as a (totally real)analytic subset. We call an open D C ⊂ C 2n a complexification of D if D C ∩R 2n = D.We use w to denote the complex coordinate variables of C 2n .Lemma 4.5. We can find a complexification D C ⊃ D such that the function a 0 (z)extends to a holomorphic a 0 (·) C : D C → Ω 0,1 (adE).Proof. The extension is standard. Since a 0 (z) is derived from the JS chart, it isholomorphic in z. Thus for any α = (α 1 , · · · , α n ) ∈ D, a 0 (z) equals to a convergentpower series in (z k −α k ) in a small disk centered at α with coefficients in Ω 0,1 (adE).Letting α k = a k + ib k , a k , b k ∈ R, and writing z k = u k + iv k , the power seriesbecomes a power series in (u k − a k ) and (v k − b k ).Because u k and v k are complex coordinate variables of C 2n ⊃ R 2n ⊃ D, a 0 (z)extends to a holomorphic Ω 0,1 (adE)-valued function in a small neighborhood of αin C 2n . Because the extension of a function defined on an open subset of R 2n toa germ of holomorphic function on C n is unique, the various extensions of a 0 (z)using power series expansions at various α ∈ D give a single extension of a 0 (z) toa holomorphic a 0 (w) C on some complexifications D C ⊃ D.□For D ⊂ V 0 a neighborhood of 0 = x 0 ∈ V 0 , we denoteO 0 := D ∩ X f0 = D ∩ X.For x ∈ O 0 , we write ∂ ∗ x = ∂ ∗ 0 + a 0 (x) † . The extension problem for a 0 (x) † is moredelicate because it is not defined away from O 0 .Lemma 4.6. For any y ∈ Y , there is an open neighborhood S ⊂ Y of y ∈ Y and anopen neighborhood D ⊂ V 0 of 0 ∈ V 0 so that the hermitian metric h| O0×S extendsto an L l s+2 hermitian metric on E D×S := E D | D×S , analytic in z ∈ D.Proof. Let S ⊂ Y be an open neighborhood of y so that S is biholomorphic to theunit ball in C 3 , and that E| ∼ O0×S = O ⊕rO and H| ∼ 0×S S = O S .We let k S be the hermitian norm of 1 in O S∼ = H|S of the hermitian metric ofH fixed earlier. Then k S is a smooth positive function on S. We let s 1 , · · · , s r bethe standard basis of E| ∼ O0×S = O ⊕rO . Because E ⊂ 0×S p∗ Y H⊕N is a subvector bundleover X × Y , using H| S∼ = OS , the image of s k in p ∗ Y H⊕N | O0×S has the presentations k = (s k,1 , · · · , s k,N ), where s k,j ∈ Γ(O O0×S). Then the hermitian metric form ofh on E| O0×S in the basis s 1 , · · · , s r takes the form(4.3) h(s k , s l ) = k S · ∑s k,j s l,j .To extend this expression over D×S, we will modify the semiconnection ∂ D×S :=∂ 0 + ∂ z + a 0 | D×S to an integrable semiconnection ∂ ′ D×S and extend s 1 , · · · , s r toholomorphic sections of (E D×S , ∂ ′ D×S).Let m ⊂ O D be the maximal ideal generated by z 1 , · · · , z n , and let I ⊂ O D bethe ideal sheaf of D ∩ X ⊂ D. Then F 0,2 ≡ 0 mod I. We construct ∂ ′ ∂ 0+a 0D×S bypower series expansion. We let s ′ = s + 2, and set b 0 (z) = 0. Suppose we havefound b k (z) ∈ Ω 0,1 (adE| S ) s ′ ⊗ C I such that(4.4) ∂ 0 b k (z) ∈ Ω 0,1 (adE| S ) s ′ ⊗ C I and F 0,2∂ 0+a 0(z)+b k (z) ≡ 0 mod mk ∩ I,j

18 YOUNG-HOON KIEM AND JUN LIwhere F 0,2∂ 0+a 0(z)+b k (z) := (∂ 0 + a 0 (z) + b k (z)) 2 | D×S . Then by the Bianchi identity,using a 0 (0) = b k (0) = 0, we have(4.5) ∂ 0 F 0,2∂ 0+a 0(z)+b k (z) ≡ (∂ 0 + a 0 (z) + b k (z))F 0,2∂ 0+a 0(z)+b k (z) ≡ 0 mod mk+1 ∩ I.We let φ α ∈ m k ∩ I, α ∈ Λ k , be a C-basis of (m k ∩ I)/(m k+1 ∩ I); we writeF 0,2 ≡ ∑A∂ 0+a 0(z)+b k (z) α φ α mod m k+1 ∩ I.α∈Λ kBecause a 0 (z) is from the family on the JS chart, a 0 (z) ∈ Ω 0,1 (adE) ⊗ C I (i.e. issmooth). Thus using (4.4) and (4.5), we conclude that A α ∈ Ω 0,2 (adE| S ) s ′, and∂ 0 A α = 0. Since S is biholomorphic to the unit ball in C 3 , it is strictly pseudoconvexwith smooth boundary. Applying a result of solutions to the ∂-equationwith L l s estimate (cf. [16, Theorem 6.11]), there is a constant C depending only onS such that for each α ∈ Λ k , we can find B α ∈ Ω 0,1 (adE| S ) s ′ such that(4.6) ∂ 0 B α = A α and ‖B α ‖ L 2 (S)≤ C ‖A α ‖s L ′ 2s ′ (S) .We define δ k (z) = ∑ α∈Λ kB α φ α , and let b k+1 (z) = b k (z) + δ k (z). Then (4.4) holdswith k replaced by k + 1.We consider the infinite sum ∑ ∞k=1 δ k(z). Using the estimate (4.6), and the Morrey’sinequality ‖ u ‖ C (S)≤ C ′ ‖ u ‖ 0,1−6/l L l1 (S) for the domain S, a standard powerseries convergence argument (cf. [17, Section 5.3(c)]) shows that possibly aftershrinking 0 ∈ D and y ∈ S, ∑ ∞k=1 δ k| D×S converges to a b(z) ∈ Ω 0,1 (adE| S ) s ′ ⊗ C I.Then the semiconnection∂ ′ D×S := ∂ 0 + ∂ z + a 0 (z) + b(z)on E D×S is integrable and is the desired modification.We now extend the metric. Possibly after shrinking 0 ∈ D and y ∈ S, we canassume that the subbundle homomorphism E| O0×S → p ∗ Y H⊕N | O0×S extends to asubbundle homomorphismg : (E D×S , ∂ ′ D×S) −→ p ∗ Y H ⊕N | D×S ;the sections s 1 , · · · , s r extend to holomorphic sections ˜s 1 , · · · , ˜s r of (E D×S , ∂ ′ D×S)that span the bundle E D×S . Using H| S∼ = OS , and writing g(˜s k ) = (˜s k,1 , · · · , ˜s k,N ),we defineN∑(4.7) h S (˜s k , ˜s l ):= k S · ˜s k,j ˜s l,j ,which defines a hermitian metric h S of E D×S , extending the metric (4.3).It remains to express the metric h S in a basis constant along D. We let e k =s k | 0×S ; e 1 , · · · , e r form a smooth basis of E| S . We let ẽ k be the pullback of e k viathe tautological projection E D×S → E| S . Under this basis, a 0 (z) + b(z) becomesan r × r-matrix with entries Ω 0,1 (adE| S ) s ′-valued holomorphic functions over D.Let c kj be functions so that ˜s k = ∑ j c kjẽ j . Because ˜s k are ∂ ′ D×S holomorphic,using ∂ z ẽ k = ∂ 0 ẽ k = 0, we havej=1∂ z c kj + (a 0 (z) + b(z)) ki c ij + ∂ 0 c kj = 0.Since the only the term ∂ z c kj takes value in (0, 1)-forms of D, (others take valuein (0, 1)-forms of S,) we have ∂ z c kj = 0. Therefore, c kj are holomorphic in z.

CATEGORIFICATION OF DONALDSON-THOMAS INVARIANTS 19This proves that the hermitian metric form of h S under the basis ẽ 1 , · · · , ẽ r is realanalytic in (Re z, Im z). Finally, we add that ˜s i and ẽ j are L l s ′, thus c kj lie inL l s ′ = Ll s+2. This proves that the metric h S is L l s+2. □Corollary 4.7. There is an open neighborhood 0 ∈ D ⊂ V 0 and an L l s+2 hermitianmetric ˜h D on E D such that ˜h D is analytic in z ∈ D and extends h| O0×Y .Proof. By the pervious Lemma, for any y ∈ Y , we can find an open neighborhoodD ×S ⊂ V 0 ×Y of (0, y) ∈ V 0 ×Y and a hermitian metric ˜h S on E D×S that extendsh| O0×S, and is analytic in z ∈ D. Because Y is compact, we can cover Y by finitelymany such opens S a , a = 1, · · · , l, paired with 0 ∈ D a ⊂ V 0 . Let D 0 = ∩D a . Then0 ∈ D 0 ⊂ V 0 is open and ˜h Sa are defined over D 0 × S a .We then pick a smooth partition of unity ∑ la=1 χ a = 1 with χ a : Y → [0, 1] suchthat the closure (χ a > 0) lies in S a . Then ˜h D = ∑ la=1 χ a · ˜h Sa is an L l s+2 hermitianmetric on E D0 that is analytic in z ∈ D 0 , and extends h| Ox×Y . □Lemma 4.8. Let the hermitian metric ˜h D on E D be given by Corollary 4.7, andlet ∂ ∗ z = ∂ ∗ 0 + a 0 (z) † , z ∈ D, be the formal adjoint of ∂ z using the hermitian metric˜h z := ˜h D | z×Y . Then we can find a complexification D C ⊃ D such that the functiona 0 (z) † extends to a holomorphic a 0 (·) † C : DC → Ω 0 (adE ⊗ C T 0,1Y ) s.Proof. Using the explicit dependence of ∂ ∗ z := (∂ 0 +a 0 (z)) ∗ on the metric ˜h z , we seeimmediately that a 0 (z) † := ∂ ∗ z − ∂ ∗ 0 ∈ Ω 0 (adE ⊗ C T 0,1Y) s is analytic in (Re z, Im z).Following the proof of Lemma 4.5, there is a complexification D C ⊃ D such thata 0 (z) † extends to a 0 (w) † C , defined over DC and holomorphic in w ∈ D C . Here wehave used that ˜h D is L l s+2 to ensure that a 0 (w) † C are Ll s.□In the remainder of this section, we fix a complexification D C ⊃ D so that botha 0 (z) and a 0 (z) † extend holomorphically to a 0 (w) C and a 0 (w) † C on DC . We define(4.8) O C 0 = ( F 0,2∂ 0+a 0(w) C= 0 ) red ⊂ DC .For w ∈ D C , we define ∂ ∗ w = ∂ 0 + a 0 (w) † C .Corollary 4.9. We have (∂ ∗ w) 2 | O C0= 0.Proof. Via the holomorphic map η : D C → V 0 , w = (u 1 , · · · , u n , v 1 · · · , v n ) ↦→ z =(u 1 +iv 1 , · · · , u n +iv n ), we see that the pullback a 0 (η(w)) is a holomorphic extensionof a 0 (z). Thus by the uniqueness of holomorphic extension, we have a 0 (w) C =a 0 (η(w)). Thus O C 0 = η −1 (O 0 ). In particular, every irreducible component A ⊂ O 0has its complexification A C = η(A), and vice versa (cf. [27, Proposition 5.3]).Since (∂ ∗ w) 2 is holomorphic and vanishes along O 0 , by studying its vanishing neara general point of any irreducible component A of O 0 , and noticing that O C 0 is withthe reduced analytic subspace structure, we conclude that (∂ ∗ w) 2 | O C0= 0. □We now complexify Θ D (ɛ) using the span of generalized eigenvectors of the“Laplacian” of ∂ w . We define□ w = ∂ w ∂ ∗ w + ∂ ∗ w∂ w : Ω 0,j (adE) s −→ Ω 0,j (adE) s−2 , w ∈ D C .This is a family of second order elliptic operators, holomorphic in w ∈ D C , whosesymbols are identical to that of □ x0 .

20 YOUNG-HOON KIEM AND JUN LILemma 4.10. Let the notation be as before. Suppose ɛ 0 > 0 separates eigenvaluesof ∂ z and ɛ 0 < ɛ(z) for all z ∈ O 0 . Then we can choose D C ⊃ D such thatΘ O0 (ɛ 0 ) ⊂ O 0 × Ω 0,1 (adE) s extends to a holomorphic subbundle Θ O C0(ɛ 0 ) ⊂ O C 0 ×Ω 0,1 (adE) s .Proof. We extend Θ D (ɛ 0 ) to D C using the generalized eigenforms of □ w . Since □ wis holomorphic in w, and since 1 + □ z , z ∈ D, are invertible, by [14, page 365] aftershrinking D C ⊃ D if necessary, the family(1 + □ w ) −1 : Ω 0,1 (adE) s −→ Ω 0,1 (adE) s , w ∈ D C ,is a holomorphic family of bounded operators.We now extend Θ D (ɛ 0 ). First, note that λ is a spectrum of □ z if and only if(1 + λ) −1 is a spectrum of (1 + □ z ) −1 , and they have identical associated spacesof generalized eigenforms. Since □ z , z ∈ D, has discrete spectrum (eigenvalues)and ɛ is not its eigenvalue, for any x ∈ D, we can pick a small open neighborhoodD x ⊂ D of x ∈ D and a sufficiently small δ, ɛ 0 ≫ δ > 0, so that no eigenvalues of(1 + □ z ) −1 for z ∈ D x lie in ∣ ∣|λ| − (1 + ɛ 0 ) −1∣ ∣ < δ. Then by the continuity of thespectrum, we can find an open D C x ⊂ D C , D C x ∩ D = D x , such that no (1 + □ w ) −1 ,w ∈ D C x , contains spectrum in the region ∣ ∣|λ| − (1 + ɛ 0 ) −1∣ ∣ < δ/2. Applying [14,Theorem VII-1.7], over D C x we have decompositions Ω 0,1 (adE) s = E 1,w ⊕ E 2,wsuch that E 1,w and E 2,w are holomorphic in w, invariant under (1 + □ w ) −1 , andT i,w := (1 + □ w ) −1 | Ei,w : E i,w → E i,w has its spectrum in |λ| < (1 + ɛ 0 ) −1 for i = 1and in |λ| > (1 + ɛ 0 ) −1 for i = 2.For us, the key property is that E 1,w = Θ w (ɛ 0 ) when w ∈ D x . For w ∈ D C x ,we define Θ w (ɛ 0 ) = E 1,w . Then Θ D C x(ɛ 0 ) := ∐ w∈D C x Θ w(ɛ 0 ) extends Θ D (ɛ 0 ) holomorphicallyto D C x . By covering D by open subsets like D x , and using that theholomorphic extensions of Θ D (ɛ 0 ) are unique, when they exist, we conclude thatfor a complexification D C ⊃ D,Θ D C(ɛ 0 ):= ∐w∈D C Θ w (ɛ 0 ) ⊂ D C × Ω 0,1 (adE) sextends Θ D (ɛ 0 ) and is a holomorphic bundle over D C .Corollary 4.11. Θ U0 (ɛ 0 ) is an analytic subbundle of T U0 B.Proof. Applying the surjective homomorphism (4.2), and using the complexificationconstructed, the conclusion follows.□4.4. The existence of orientation bundles. We prove Proposition 3.4. Webegin with a rephrasing of the non-degeneracy condition under cs 2,x . We define apairing(4.9) (·, ·) x : Ω 0,1 (adE x ) s × Ω 0,2 (adE x ) s−1 −→ C, x ∈ X,via (a 1 , a 2 ) x = 18π 2 ∫tr(a1 ∧ a 2 ) ∧ Ω. It relates to the quadratic form cs 2,x via(4.10) cs 2,x (a, b) = (a, ∂ x b) x , a, b ∈ T x B ∼ = ker(∂ ∗ x) 0,1s .Given a subspace W ⊂ T x B ∼ = ker(∂ ∗ x) 0,1sits companion spaces bythat contains T x X ∼ = □ −1x (0) 0,1 , we define(4.11) W ′ = □ −1x (0) 0,2 ⊕ ∂ x (W ) and W ′′ := □ −1x (0) 0,1 ⊕ □ x (W ).Recall that Q x is the descent of cs 2,x to T x B/T x X.□

CATEGORIFICATION OF DONALDSON-THOMAS INVARIANTS 21Lemma 4.12. Let W ⊂ T x B be a subspace containing T x X, and let W ′ be itscompanion space. Then Q x | W/TxX is non-degenerate if and only if the restrictedpairing (·, ·) x : W × W ′ → C is a perfect pairing.Proof. Since Y is a Calabi-Yau threefold, by Serre duality, the pairing (·, ·) x restrictedto □ −1x (0) 0,1 ×□ −1x (0) 0,2 is perfect. Let e 1 , · · · , e l ∈ W be so that ∂ x e 1 , · · · ,∂ x e l form a basis of ∂ x (W ). By Hodge theory, e 1 , · · · , e l and □ −1x (0) 0,1 span W .By (4.10), and that □ −1s (0) 0,1 is orthogonal to Im(∂ x ) under (·, ·) x , Q x is nondegenerateon W/T x X = W/□ −1x (0) 0,1 if and only if (e i , ∂ x e j ) x form an invertiblel × l matrix, which is equivalent to that (·, ·) x on W × W ′ is perfect. This provesthe lemma.□Lemma 4.13. Let x ∈ X and r ≥ d x = dim T x X be an integer. Then we can findan open neighborhood U ⊂ X of x such that there exists a rank r orientation bundleover U.Proof. We first recall an easy fact. Let q be a non-degenerate quadratic form onC n . Let 0 < l ≤ n be an integer, and let Gr(l, C n ) be the Grassmannian of ldimensional subspaces of C n . We introduce(4.12) Gr(l, C n ) ◦ = {[S] ∈ Gr(l, C n ) | q| S is non-degenerate}.Since q is non-degenerate, it is direct to check that Gr(l, C n ) ◦ is the complementof a divisor in Gr(l, C n ), and thus is connected and smooth.We now prove the lemma. We first treat the case r = d x . Since □ x has nonnegativediscrete eigenvalues, there is an ɛ > 0 so that it has no eigenvalues in(0, 2ɛ). By Corollary 4.11, over an open neighborhood U ⊂ X of x, Θ U (ɛ) is ananalytic subbundle of T U B. To show that it is an orientation bundle, we only needto verify that for any y ∈ U, Q y restricting to Θ y (ɛ)/□ −1y (0) 0,1 is non-degenerate.By Lemma 4.12, this is equivalent to that(·, ·) y : Θ y (ɛ) × Θ y (ɛ) ′ −→ Cis perfect. Because it is perfect at y = x, and because being perfect is an opencondition, possibly after shrinking x ∈ U if necessary, it is perfect for every y ∈ U.This proves the case r = d x .In case l = r − d x > 0, applying the discussion at the beginning of this proof,we find an l-dimensional subspace W ⊂ T x B/T x X so that Q x | W is non-degenerate.We let Ξ x be the preimage of W under the quotient map T x B → T x B/T x X. Tocomplete the proof, we extend Ξ x to a neighborhood of x ∈ X and show that it isan orientation bundle.We pick an open neighborhood U ⊂ X of x ∈ X such that the isomorphism ζ in(4.1) has been chosen; we pick a basis of W , say u 1 , · · · , u l ∈ ker(∂ ∗ x) 0,1s /□ −1x (0) 0,1 =Im(∂ ∗ x) 0,1s . We then extend u k to be the constant section of U × Ω 0,1 (adE) s and letũ k be its image sections in T U B. It is a holomorphic extension of u k . By shrinkingx ∈ U if necessary, Θ U (ɛ) and the sections ũ 1 , · · · , ũ l span a subbundle Ξ of T U B.Because Θ U (ɛ) is an analytic subbundle of U × Ω 0,1 (adE) s , and because ˜s k areholomorphic, we see that Ξ is an analytic subbundle of T U B and contains Θ U (ɛ).It remains to check that for any y ∈ U, Q y restricted to Ξ y /T y X is nondegenerate.By the previous lemma, this is equivalent to the fact that the pairing(·, ·) y : Ξ y × Ξ ′ y → C is perfect, where Ξ ′ y is the companion space of Ξ y (cf. (4.11)).Because this pairing is perfect when y = x, by shrinking x ∈ U if necessary, we canmake it perfect for all y ∈ U. Therefore, Ξ is an orientation bundle over U. □

22 YOUNG-HOON KIEM AND JUN LILemma 4.14. Suppose Ξ α and Ξ β are two rank r orientation bundles over an openU. Then for any x ∈ U, there is an open neighborhood U 0 ⊂ U of x such that thereis a homotopy from Ξ α | U0 to Ξ β | U0 .Proof. We begin with an easy fact. For any finite dimensional subspace W 0 ⊂ W :=T x B/T x X, there is a finite dimensional subspace W ⊂ W, containing W 0 , such thatQ x | W is non-degenerate. Indeed, let N ⊂ W 0 be the null-subspace of Q x | W0 . SinceQ x is non-degenerate, we can find a subspace M ⊂ W such that W 0 ∩ M = 0 andQ x : N × M → C is perfect. Then the space W = W 0 ⊕ M ⊂ W is the desiredsubspace.We now construct the desired homotopy. We first find a finite dimensionalsubspace W ⊂ W so that it contains both Ξ α | x /T x X and Ξ β | x /T x X, and Q x isnon-degenerate on W . Let l = r − d x . We form the Grassmannian Gr(l, W )and its Zariski open subset Gr(l, W ) ◦ as in (4.12) (with Q x in place of q). Thenboth [Ξ α | x /T x X] and [Ξ β | x /T x X] are in Gr(l, W ) ◦ . Because Gr(l, W ) ◦ is a smoothconnected quasi-projective variety, we can find an analytic arc [S t ] ∈ Gr(l, W ) ◦ ,t ∈ [0, 1], such that S 0 = Ξ α | x /T x X and S 1 = Ξ β | x /T x X. We let Ξ t,x be thepreimage of S t under the quotient homomorphismπ x : T x B → T x B/T x X.Then [S t ] form an analytic family of subspaces in T x B, interpolating between Ξ α | xand Ξ β | x .We extend this to an analytic family of orientation bundles in a neighborhood ofx ∈ U. As before, we realize an open neighborhood U 0 ⊂ U of x ∈ U as U 0 = X f0 ,x = 0 ∈ V 0 , for the JS chart (V 0 , f 0 ). Then we have the isomorphism of holomorphicBanach bundles T U0 B ∼ = U 0 × ker(∂ ∗ x) 0,1s . By choosing U 0 sufficiently small, we canfind an ɛ > 0 so that Θ U0 (ɛ) ⊂ Ξ α | U0 , Θ U0 (ɛ) ⊂ Ξ β | U0 , and Θ x (ɛ) = □ −1x (0) 0,1 .Next, for i = α and β, we find l analytic sections s i 1, · · · s i l of T U 0B such thatΘ U0 (ɛ) and s i 1, · · · s i l span Ξ i| U0 . Because s α k (x) and sβ k(x) all lie in π−1 x (W ), wecan find arcs ξ k (t) ∈ πx−1 (W ), analytic in t ∈ [0, 1], such that ξ k (0) = s α k(x) andξ k (1) = s β k (x), and Ξ t,x = Θ x (ɛ) ⊕ C-span ( ξ 1 (t), · · · ξ l (t) ) .Using the isomorphism T U0 B ∼ = U 0 × ker(∂ ∗ x) 0,1s , we can view ξ k (t) as analytic arcsin ker(∂ ∗ x) 0,1s ⊂ Ω 0,1 (adE) s . We then defines t k(y) = ξ k (t) + (1 − t)(s α k (y) − s α k (x)) + t(s β k (y) − sβ k (x)).Clearly, they are analytic in t, and s 0 k (y) = sα k (y) and s1 k (y) = sβ k(y) for all y ∈U 0 . Therefore, by shrinking x ∈ U 0 if necessary, for every t ∈ [0, 1], the sectionss t 1(y), · · · s t l (y) and Θ U 0(ɛ) span a rank r analytic subbundle Ξ pret ⊂ U 0 ×Ω 0,1 (adE) s .Because the arcs ξ k (t) are analytic in t, the family Ξ pret is analytic in t. Finally,because [S t ] all lie in Gr(l, W ) ◦ , by shrinking x ∈ U 0 if necessary,Ξ t := image of Ξ pret under U 0 × Ω 0,1 (adE) s → T U0 Bform an analytic family of orientation bundles providing the desired homotopybetween Ξ α | U0 and Ξ β | U0 . □Proof of Proposition 3.4. We first pick a locally finite cover U α so that each U α hasan orientation bundle Ξ α . For each x ∈ X, we pick an open neighborhood U x ofx ∈ X so that (1) U x ⊂ U α whenever x ∈ U α , and that (2) for every pair α, β with

CATEGORIFICATION OF DONALDSON-THOMAS INVARIANTS 23x ∈ U αβ = U α ∩ U β , we have a homotopy from Ξ α | Ux to Ξ β | Ux . Then we can picka locally finite refinement of the covering as follows: For each x ∈ X, we fix anyα(x) such that U x ⊂ U α(x) . Since X is quasi-projective, we have a metric d(·, ·)on X induced from projective space. By shrinking U x if necessary, we may assumeU x is the ball B(x, 2ɛ x ) of radius 2ɛ x > 0 centered at x. Let O x = B(x, ɛ x ) andΞ x = Ξ α(x) | Ox . Then {O x } is an open cover of X and Ξ x is an orientation bundle onO x . Suppose that O x ∩ O y ≠ ∅. Without loss of generality, we may assume ɛ x ≤ ɛ y .Then O x ⊂ B(y, 2ɛ y ) = U y ⊂ U α(y) . Also we have x ∈ O x ⊂ U x ⊂ U α(x) . HenceO x ⊂ U α(x) ∩ U α(y) and thus we have a homotopy from Ξ x | Ox∩O yto Ξ y | Ox∩O yasdesired.□5. CS data from preorientation dataIn this section we prove Proposition 3.12. We construct CS charts from orientationbundles, their local trivializations, and complexifications.5.1. Constructing families of CS charts. Let Ξ be an orientated bundle on U.We generalize Joyce-Song’s construction in [12] to form a Ξ-aligned family of CScharts.Given Ξ, for any x ∈ U, we view Ξ x ⊂ ker(∂ ∗ x) 0,1s and denote its companion spaceΞ ′′x ⊂ Ω 0,1 (adE x ) be as defined in (4.11) with W replaced by Ξ x . Using condition(1) of Definition 3.1, one sees that Ξ ′′ := ∐ x∈U Ξ′′ x is an analytic subbundle ofΩ 0,1X (adE) s−2| U .We define the quotient homomorphism of Banach bundles(5.1) P : Ω 0,1X (adE) s−2| U −→ Ω 0,1X (adE) s−2| U/Ξ ′′ ,whose restriction to x ∈ U is denoted by P x : Ω 0,1 (adE x ) s−2 → Ω 0,1 (adE x ) s−2 /Ξ ′′x.For x ∈ U, we form the elliptic operator(5.2) L x : Ω 0,1 (adE x ) s −→ Ω 0,1 (adE x ) s−2 /Ξ ′′x, L x (a) = P x(□x a + ∂ ∗ x(a ∧ a) ) .For a continuous ε(·) : U → (0, 1) to be specified shortly, we define(5.3) V x = {a ∈ Ω 0,1 (adE x ) s | L x (a) = 0, ‖a‖ s < ε(x)}.(Here ‖ · ‖ s is defined using h x .) Letting Π x : Ω 0,1 (adE x ) s → B be the compositeof the tautological isomorphism ∂ x + · : Ω 0,1 (adE x ) s∼ = Ax (cf. (3.1)) with thetautological projection A x → B, we define(5.4) V x = Π x (V x ).We comment that V x only depends on (Ξ x , h x , ε(x)).Let f x : V x → C (or f x : V x → C) be the composite of V x ↩→ B and cs : B → C.Proposition 5.1. Let U ⊂ X be open and Ξ a rank r orientation bundle onU. Then there is a continuous ε(·) : U → (0, 1) such that the family V x , x ∈U, constructed via (5.3) using ε(·) is a smooth family of complex manifolds ofdimension r, and such that all (V x , f x ) are CS charts of X.Proof. We relate V x to the JS charts by first showing that L x (a) = 0 if and only if(5.5) ∂ ∗ xa = 0 and P x ◦ ∂ ∗ xF 0,2∂ x+a = 0.Indeed, it is immediate that (5.5) implies L x (a) = 0. For the other direction,suppose L x (a) = 0. Since Ξ ′′x ⊂ ker(∂ ∗ x) 0,1s−2 , applying ∂∗ x to L x (a) = 0, we obtain

24 YOUNG-HOON KIEM AND JUN LI∂ x ∂ ∗ xa = 0, which forces ∂ ∗ xa = 0. Having this, we obtain P x ◦ ∂ ∗ xF 0,2proves the equivalence.By direct calculation, the linearization of L x at a = 0 isδL x = P x ◦ □ x : Ω 0,1 (adE x ) s −→ Ω 0,1 (adE x ) s−2 /Ξ ′′x,∂ x+a= 0. Thiswhich is surjective with kernel Ξ x . Since the operators L x depend smoothly onx ∈ U, applying the implicit function theorem, for a continuous ε(·) : U 0 → (0, 1)(taking sufficiently small values), the solution spaces V x , x ∈ U, form a family ofmanifolds of real dimensions 2r. Since the operators L x are holomorphic in a, eachsolution space V x is a complex submanifold of Ω 0,1 (adE x ) s and their images in Blie in B si . Finally, because the family □ x is smooth in x ∈ U, the family V x is asmooth family of complex manifolds. Using local isomorphism ζ in (4.1), we seethat the family V x , x ∈ U, form a smooth family of complex submanifolds of B si .This proves the first part of the Proposition.For the second part, we first show that by choosing ε(x) small enough, V x ∩ Xcontains an open complex analytic subspace of X containing x. At individual x ∈ U,this follows from that each V x contains (an open neighborhood of x in) the JS chart(VxJS , fx js ). However, to prove that we can choose ε(x) continuously in x, we arguedirectly.As (E x , ∂ x ) are simple, the tautological Π x : ker(∂ ∗ x) 0,1s → B is biholomorphicnear Π x (0) = x. We letF x : Ω 0,1 (adE x ) s → Ω 0,2 (adE x ) s−1 , F x (a) = F 0,2∂ x+a ,be the curvature section. Then ker(∂ ∗ x) 0,1s ∩ (F x = 0) contains an open complexanalytic subspace of X containing x (cf. [12, Chapter 9], [26]). Because ker(∂ ∗ x) 0,1s ∩(F x = 0) is contained in (L x = 0), V x ∩ X contains an open neighborhood of x ∈ X.We now prove that for ε(x) small, X fx = Π −1x (V x ∩ X). We follow the proof of[12, Prop. 9.12]. Let x ∈ U. We define a subbundleR x := {(∂ x + a, b) | P x ◦ ∂ ∗ xb = ∂ ∗ x(∂ x b − b ∧ a − a ∧ b) = 0} ⊂ V x × Ω 0,2 (adE x ) s .Applying the implicit function theorem, because the two equations in the bracketare holomorphic in a, for ε(x) small enough, R x is a holomorphic subbundle ofV x × Ω 0,2 (adE x ) s over V x . Then the Bianchi identity coupled with the equivalencerelations (5.5) ensures that the restriction of the curvature section to V x , namelyF x | Vx , is a section of R x . Since X locally near ∂ x is defined by the vanishing of F x ,we conclude(5.6) Π −1x (V x ∩ X) = V x ∩ (F x | Vx = 0).It remains to show that (F x | Vx = 0) = (df x = 0). We define a bundle mapϕ x : R x → T ∨ V x , (∂ x + a, b) ↦→ (∂ x + a, α b ),where α b ∈ T ∨ V ∫∂x is α b (·) = 1x+a 4π tr(· ∧ b) ∧ Ω. Clearly, 2 ϕx is holomorphicand ϕ x ◦ F x | Vx = df x . We show that by choosing ε(x) small, we can make ϕ x anisomorphism of vector bundles over V x . We first claim that restricting to x we have(5.7) R x | x = {b ∈ Ω 0,2 (adE x ) s−1 | P x ◦ ∂ ∗ xb = ∂ ∗ x∂ x b = 0} = ∂ x (Ξ x ) ⊕ □ −1x (0) 0,2 .Indeed the first identity follows from the definition of R x . We prove the secondidentity. For any b ∈ R x | x , since ∂ ∗ x∂ x b = 0, we have ∂ x b = 0. Thus we can write

CATEGORIFICATION OF DONALDSON-THOMAS INVARIANTS 25b = b 0 + ∂ x c with □ x b 0 = 0. Then P x ◦ ∂ ∗ xb = 0 implies that we may take c ∈ Ξ x .This proves (5.7).Applying Lemma 4.12, we know that the pairing (·, ·) x (cf. (4.9)) restricted toT x V x × R x | x is perfect and thus ϕ x | x is an isomorphism. Therefore, by choosingε(x) sufficiently small, ϕ x is an isomorphism over V x and thus (V x , f x ) is a CS chartof X. Note that the choice of ε(x) is to make sure that ϕ x are isomorphisms overV x . Since this relies on the openness argument, we can choose ε(·) : U → (0, 1)continuously to ensure this. This proves the proposition.□The family of CS charts are canonical.Lemma 5.2. Let Ξ a and Ξ b be two orientation bundles over U. Suppose Ξ a ⊂ Ξ bis a vector subbundle. Let ε(·) : U → (0, 1) be the size function that produces afamily of CS charts V(Ξ b ) ⊂ U × B using Ξ b . Then the same ε(·) produces a familyof CS charts V(Ξ a ) ⊂ U × B, and V(Ξ a ) ⊂ V(Ξ b ) ⊂ U × B.Proof. This follows from the construction because the only data needed to constructV x are Ξ x and ε(x).□5.2. Local trivializations. In this subsection, we proveProposition 5.3. The family V ⊂ U × B constructed in Proposition 5.1 is locallytrivial everywhere.The study is local. For any x 0 ∈ U, we pick an open neighborhood U 0 ⊂ U ofx 0 ∈ U so that U 0 = X f0 for the JS chart (V 0 , f 0 ), coupled with the isomorphism ζin (4.1). Using the induced E x∼ = E for x ∈ U0 , the CS charts V x become complexsubmanifolds of Ω 0,1 (adE) s . We define(5.8) V U0 = ∐x × V x ⊂ U 0 × Ω 0,1 (adE) s .x∈U 0By shrinking x 0 ∈ U 0 if necessary, we assume that U 0 lies in X fx ⊂ X for all x ∈ U 0 .To keep the notation transparent, for x, y ∈ U 0 , we denote by y x ∈ X fx ⊂ V xthe point whose image in X is y, i.e. the point in X fx associated to y ∈ U 0 . Wedenote by ∂ x + a x (y x ) the connection form of y x ∈ V x . Because U 0 ⊂ X fy , for anyx, y, z ∈ U 0 , as (E, ∂ x + a x (z x )) ∼ = (E, ∂ y + a y (z y )), there is a unique g ∈ G so that(5.9) ∂ y + a y (z y ) = g ( ∂ x + a x (z x ) ) = ∂ x − ∂ x g · g −1 + ga x (z x )g −1 .As before, for open U ⊂ U 0 × V U0 and x, y ∈ U 0 , we denote U x,y := U ∩ (x × V y ),viewed as a subset of V y . We denote ∆(U 0 ) = {(x, x x ) | x ∈ U 0 } ⊂ U 0 × V U0 .Lemma 5.4. We can find an open U ⊂ U 0 × V U0 containing ∆(U 0 ) ⊂ U 0 × V U0and a smooth g : U → G such that the following hold:(1) for any x, y ∈ U 0 , g x,y := g| Ux,y : U x,y → G is holomorphic and g x,x (·) = 1;(2) for x, y ∈ U 0 , (x, z) ∈ U x,y , and letting a(x, y, z) = g x,y (z)(∂ y +a y (z))−∂ x ,we have ∂ ∗ xa(x, y, z) = 0, and a(x, x, x x ) = 0,Proof. For α ∈ G, and for x, y ∈ U 0 and z ∈ V y , we formc x,y,z (α):= α(∂ y + a y (z)) − ∂ x ,and defineR x,y,z (·) = ∂ ∗ xc x,y,z (·) : G s+1 −→ Ω 0 (adE) s−1 /C.Note that R x,x,z (1) = 0.

26 YOUNG-HOON KIEM AND JUN LIWe calculate the linearization of the operator R x,y,z (·) at (x, x, x x ) and α = 1:δR x,x,xx | α=1 = −∂ ∗ x∂ x : Ω 0 (adE) s+1 /C −→ Ω 0 (adE) s−1 /C.Because (E, ∂ x ) are simple, δR x,x,xx | α=1 are isomorphisms. Thus by the implicitfunction theorem, for an open U ⊂ U 0 × V U0 containing the diagonal ∆(U 0 ) ⊂U 0 × V U0 , we can find a unique smoothg·,·(·) : U −→ Gsuch that R x,y,z (g x,y (z)) = 0, and g x,x (z) = 1. Because the equation R x,y,z (α) isholomorphic in z and α, g x,y (z) is holomorphic in z ∈ U x,y ⊂ V y . Thus both (1)and (2) of the lemma are satisfied.□We remark that by the uniqueness of the solution g x,y (z), by an extension of(5.9) (to the non-reduced case), we have a(x, y, ·)| Ux,y∩X fy= a y (·)| Ux,y∩X fy.Proof of Proposition 5.3. We take the family g x,y (z) defined on U ⊂ U 0 × V U0constructed in the previous Lemma. To find a local trivialization Ψ : U → V U0 ×U 0 ,we solve b(x, y, z) satisfying the system(5.10) L x(a(x, y, z) + b(x, y, z))= 0, (b(x, y, z), ν)x = 0, ∀ν ∈ Ξ ′ x,where L x is defined in (5.2). By the remark at the end of the previous proof, we seethat restricting to U x,y ∩ X fy , b(x, y, ·) = 0 are solutions. Also, since a(x, x, z) =a x (z) and L x (a x (z)) = 0, b(x, x, z) = 0 are solutions.We denote (Ξ ′ x) ⊥ = {b | (b, ν) x = 0, ∀ν ∈ Ξ ′ x} ⊂ Ω 0,1 (adE) s . We defineM x,y,z (·) = L x (a(x, y, z) + ·) : (Ξ ′ x) ⊥ −→ Ω 0,1 (adE) s−2 /Ξ ′′x.By our construction, the linearization δM x,x,xx at 0 isδM x,x,xx | 0 = P x ◦ □ x : (Ξ ′ x) ⊥ −→ Ω 0,1 (adE) s−2 /Ξ ′′x,which is an isomorphism. Thus applying the implicit function theorem, for an openU ′ ⊂ U containing the diagonal ∆(U 0 ), we can solve the system M x,y,z (b(x, y, z)) =0 uniquely and smoothly in (x, y, z). Further, since g x,y (z) is holomorphic in z, andthe operator M x,y,z is a linear operator holomorphic in z, b(x, y, z) is holomorphicin z.We set Ψ : U ′ → U 0 × V U0 to beΨ x,y (∂ y + a y (z)) = g x,y (z) ( ∂ y + a y (z) ) + b(x, y, z) − ∂ x .Because M x,y,·(0)| Ux,y∩X fy= 0, we conclude that Ψ x,y is an open embedding ofU x,y ′ ∩ X fy into X fx ⊂ V x . Therefore, Ψ is the desired local trivialization. □The local trivializations are canonical.Lemma 5.5. Let Ξ a ⊂ Ξ b be two orientation bundles over U, as in Lemma 5.2.Suppose U 0 ⊂ U is an open subset and Ψ i : U i → V(Ξ i ) U0 × U 0 , i = a, b, belocal trivializations constructed. Let V(Ξ a ) U0 ⊂ V(Ξ b ) U0 be the inclusion given byLemma 5.2. ThenΨ a | Ua∩U b= Ψ b | Ua∩U b: U a ∩ U b −→ V(Ξ a ) U0 × U 0 ⊂ V(Ξ b ) U0 × U 0 .Proof. This follows from that (Ξ ′ b,x )⊥ ⊂ (Ξ ′ a,x) ⊥ , and thus Ψ a is also the solution tothe equations for Ψ b when restricted to V a ⊂ V b . By the uniqueness of the solution,we have the identity.□

CATEGORIFICATION OF DONALDSON-THOMAS INVARIANTS 27Corollary 5.6. Let Ξ (resp. Ξ t ) be an orientation bundle (resp. a homotopy oforientation bundles) on U. Then Ξ (resp. Ξ t ) can be complexified locally everywhere.Proof. The proof is straightforward, knowing that both Ξ and Ξ t are analytic.5.3. Complexifications. We construct complexifications of a family of CS charts.Proposition 5.7. The family of CS charts constructed in Proposition 5.1 andthe local trivialization constructed in Proposition 5.3 can be complexified locallyeverywhere.Proof. Given an orientation Ξ on an open U ⊂ X, for any x 0 ∈ U, we pick an openneighborhood U 0 ⊂ U of x 0 ∈ U so that U 0 = X f0 for the JS chart (V 0 , f 0 ), wherex 0 = 0 ∈ V 0 . By Lemma 4.8, we pick an open 0 ∈ D ⊂ V 0 so that ∂ z and ∂ ∗ z entendto holomorphic ∂ w = ∂ 0 + a 0 (w) C and ∂ ∗ w = ∂ ∗ 0 + a 0 (w) † Cover a complexificationD C of D. By Corollary 5.6, we extend Θ D (ɛ) and Ξ| D to holomorphic subbundlesΘ D C(ɛ) and Ξ D C of T D CB. (Here we follow the convention that for any Z → B, wedenote T Z B = T B × B Z.)We form the projection P w as in (5.1), with E (resp. Ξ ′′x) replaced by E (resp.Ξ ′′ w). We form V w using (5.3), with E x replaced by E and subscript “x” replacedby the subscript “w”. Since the proof that the resulting family V ⊂ U × B is asmooth family of complex manifolds only uses the isomorphism property of thelinearization of L x and the implicit function theorem, the same study extends tosmall perturbations of L x . Thus possibly after shrinking D C ⊃ D if necessary, thefamilyV D C = ∐w × V w ⊂ D C × Ω 0,1 (adE) sw∈D Cis a smooth family of complex manifolds. Because all ∂ w , ∂ ∗ w and Ξ ′′ Dare holomorphicin w ∈ D C , the system L w is holomorphic in w. Therefore, V D C is a complexCmanifold and is a complex submanifold of D C × Ω 0,1 (adE) s .Next we study the issue of being CS charts. Let Π w : V w → B be defined by∂ w + ·, and define f w = Π w ◦ cs : V w → C. We show that (V w , f w ) are CS chartsfor w ∈ O0 C , where O0 C (cf. (4.8)) is the complexification of O 0 = D ∩ X.Going through the proof of Proposition 5.1, we first need to check that forw ∈ O0 C , a ∈ V w satisfies the system (cf. (5.5))(5.11) ∂ ∗ wa = 0 and P w ◦ ∂ ∗ w(∂ w a + a ∧ a) = 0.First because Ξ O C0is the complexification of Ξ O0 , and Ξ x ⊂ ker(∂ ∗ x) 0,1s for x ∈ O 0 ,we have that Ξ w ⊂ ker(∂ ∗ w) 0,1s for w ∈ O0 C . On the other hand, a ∈ V w meansthat ∂ w ∂ ∗ wa + ∂ ∗ w(∂ w a + a ∧ a) ≡ 0 mod Ξ ′′ w. Thus applying ∂ ∗ w to this relation,we obtain ∂ ∗ w∂ w ∂ ∗ wa = 0. Finally, since ∂ ∗ x∂ x : Ω 0 (adE) s /C → Ω 0 (adE) s−2 /C areisomorphisms for x ∈ D, by shrinking D C ⊃ D if necessary, ∂ ∗ w∂ w are isomorphismsfor w ∈ O0 C , thus ∂ ∗ wa = 0. This proves that for all w ∈ O0 C , all a ∈ V w satisfy(5.11).After this, mimicking the proof of Proposition 5.1, we see that (V w , f w ) is a CSchart if for the similarly defined subbundle R w ⊂ V w × Ω 0,2 (adE) s , the similarlydefined vector bundle homomorphism ϕ w : R w → T ∨ V w is an isomorphism. Butthis follows from that ϕ x are isomorphism for x ∈ O 0 , and that being isomorphism is□

28 YOUNG-HOON KIEM AND JUN LIan open condition. Thus by shrinking D C ⊃ D if necessary, all ϕ w are isomorphismsfor w ∈ O0 C . This proves that the V O C0is a complexification of V O0 .The proof that the local trivialization Ψ can be complexified is similar, usingthat all the operators and families used to construct Ψ are holomorphic on D C , andthat the only tool used to construct Ψ is the implicit function theorem. We skipthe repetition here.□5.4. Homotopy of family of CS charts. Let U ⊂ X be open and let Ξ t , t ∈ [0, 1],be a homotopy between the orientation bundles Ξ 0 and Ξ 1 on U. For any x 0 ∈ U,we pick an open neighborhood x 0 ∈ U 0 ⊂ U such that U 0 = X f0 for the JS chart(V 0 , f 0 ), that U 0 has compact closure in U, and that there is an ɛ > 0 such thatΘ U0 (ɛ) is an orientation bundle contained in Ξ t for all t ∈ [0, 1].Because of the compactness of the closure of U 0 in U, we can find a sufficientlysmall ε > 0 (in place of ε(·) : U 0 → (0, 1)) such that for each t ∈ [0, 1], we have thefamily of CS charts V t ⊂ U 0 × B by applying Proposition 5.1 using Ξ t | U0 and thesize function ε(·) = ε. We denote V [0,1] = ∐ t∈[0,1] V t, and call it the homotopy ofthe families V 0 and V 1 .Applying Proposition 5.1 to the orientation bundle Θ U0 (ɛ), using the same ε(·) =ε, we obtain the family of CS charts W ⊂ U 0 × B.Proposition 5.8. We have tautological inclusion [0, 1] × W ⊂ V U0 as subspace of[0, 1] × U 0 × B. Further, this pair can be complexified locally everywhere.Proof. The proof is similar to the complexification of the family of CS charts andtheir local trivializations studied in the previous subsection. We omit the repetitions.□6. Perverse sheaves from CS dataIn this section we prove Proposition 3.13.6.1. A perverse sheaf from a family of CS charts. In this subsection weprove (1) of Proposition 3.13. Let V ⊂ U × B si be a family of CS charts over U andΨ : U × V ⊃ U → V × U be a local trivialization. Let(6.1) f : V −→ ⊂pr csU × B si −→ B si −→ Cand let π V be the projection from U × V or V × U to V. We need a technical resultthat there is a homeomorphism interpolating (f ◦ π V ) ◦ Ψ and f ◦ π V ,Proposition 6.1. There is an open subset U ′ ⊂ U containing the diagonal ∆(U) ⊂U and an injective local homeomorphism Φ : U ′ → U preserving the projections toU × U, such that(f ◦ π V ) ◦ Ψ ◦ Φ = f ◦ π V : U ′ −→ C.Proof of Proposition 3.13 (1). For x ∈ U, we can choose a sufficiently small openneighborhood O x of x in U such that O x is contained in the critical set X fz ∩U ′ ⊂ V z ∩ U ′ for any z ∈ O x . Let ψ x : O x × V x → V be the restriction of Ψ toO x × V x ⊂ U × V (cf. (6.2)). By shrinking V x if necessary and restricting Φ toO x × V x ⊂ O x × V| Ox over O x × {x} ∼ = O x , we have a homeomorphismΦψ xλ x : O x × V x −→ Ox × V x −→ Vthat pulls back f to f x ◦ pr 2 where pr 2 denotes the projection onto the secondcsfactor and f x : V x ⊂ B si −→ C. If we further restrict it to {z} × V x and vary z

CATEGORIFICATION OF DONALDSON-THOMAS INVARIANTS 29in O x , we obtain a continuous family of homeomorphisms λ x,z : V x∼ = Vz , definedby λ x,z = λ x | z , that pulls back f z to f x and thus we have a continuous family ofisomorphisms λ ∗ x,z : A • ∼f z = A•fxby Proposition 2.3.We have to show that the perverse sheaves P x• := A • f x[r] on O x ⊂ V x glue togive us a perverse sheaf P • on U. Let y ∈ U such that O x ∩ O y ≠ ∅. For anyz ∈ O x ∩ O y , we have isomorphismsP • xλ ∗ x,z←− P z• λ ∗ y,z−→ P y•over a neighborhood of z. For another z ′ in the connected component of O x ∩ O ycontaining z, we choose a path z t for t ∈ [0, 1] with z 0 = z, z 1 = z ′ . Thenλ −1y,z t◦ λ x,zt : V x → V y is a continuous family of homeomorphisms that pulls backf y to f x . By Proposition 2.3, we have the equalityλ ∗ y,z ◦ (λ ∗ x,z) −1 = λ ∗ y,z ′ ◦ (λ∗ x,z ′)−1 : P • x∼ =−→ P • y .By Proposition 2.6, λ ∗ y,z ◦ (λ ∗ x,z) −1 glue to give us an isomorphism τ xy : P x • ∼ = P y•over O x ∩ O y .Now the cocycle condition for gluing {P x • } is obviously satisfied because τ vx ◦τ yv ◦τ xy at any z ∈ O x ∩O y ∩O v is λ ∗ x,z ◦(λ ∗ v,z) −1 ◦λ ∗ v,z ◦(λ ∗ y,z) −1 ◦λ ∗ y,z ◦(λ ∗ x,z) −1 = 1.Therefore the perverse sheaves P x • glue to give us a perverse sheaf P • on U. □The proof of Proposition 6.1 requires techniques for complex analytic subspaces.For this, we use the complexification of the family of CS charts and its local trivializations.Let x ∈ U. By assumption, we have an open neighborhood O x of xin U, a complex manifold D C x containg O x as a closed real analytic subset and aholomorphic family V D C x→ D C x of CS charts such that V D C x| Ox∼ = V ×U O x and thatthere is a holomorphic local trivializationD C x × V D C x⊃ U D C xΨ C−→ V D C x× D C x .By shrinking O x and V around x if necessary, we may assume that U D C x= Dx C × V xand O x ⊂ V z for all z ∈ O x . If we restrict Ψ to Dx C × {x}, we get a holomorphicmap ψ x fitting into the commutative diagram(6.2) D C x × V xψ x≃V D C xD C x .which is a fiberwise open embedding. (We use ≃ to mean injective holomorphiccsmaps.) Let f : V D C x→ C be similary defined as in (6.1), and let f x : V x ⊂ B si −→ C.We letf 0 = (f ◦ π V ) ◦ (id D C x×ψ x ) and f 1 = (f ◦ π V ) ◦ Ψ C ◦ (id D C x×ψ x );both are holomorphic functions D C x × D C x × V x → C.Proposition 6.2. After shrinking D C x and V x if necessary, there is an open subsetU ′ C ⊂ DC x × D C x × V x containing the diagonal ∆ = {(y, y, y) | y ∈ D C x } and aninjective local homeomorphism Φ C : U ′ C → DC x × D C x × V x commuting with theprojection pr 12 : D C x × D C x × V x → D C x × D C x such that f 1 ◦ Φ C = f 0 .

30 YOUNG-HOON KIEM AND JUN LIBy cancelling the isomorphism id D C x×ψ x and restricting to the real part O x ⊂D C x , Proposition 6.1 is an immediate consequence of Proposition 6.2.We now prove Proposition 6.2. Letf t = (1 − t)f 0 + tf 1be holomorphic functions on D C x × D C x × V x . Let (d V f t ) be the ideal generated bythe partial derivatives of f t in the direction of V x . We will find a homeomorphism Φin a neighborhood of (x, x, x) such that f 1 ◦ Φ = f 0 by introducing a vertical vectorfield along the fibers of pr 12 : D C x × D C x × V x → D C x × D C x . We need a few lemmas.Lemma 6.3. Let V be a smooth complex manifold and S a complex analytic subspace.Suppose W 1 ⊂ W 2 are two closed complex analytic subspaces of V × S suchthat the induced projection W 1 ⊂ V × S → S is flat, and that there is a closedcomplex analytic subspace S 0 ⊂ S such that W 1 × S S 0 = W 2 × S S 0 . Then there isan open subset U 0 ⊂ V × S that contains V × S 0 such that W 1 ∩ U 0 = W 2 ∩ U 0 ascomplex analytic subspaces in U 0 .Proof. Let K be the kernel of the surjection O W2 → O W1 . Since O W1 is flat overO S , we have an exact sequence0 −→ K ⊗ OS O S0 −→ O W2 ⊗ OS O S0φ−→ O W1 ⊗ OS O S0 −→ 0.Since W 1 × S S 0 = W 2 × S S 0 , φ is an isomorphism, thus K ⊗ OS O S0 = 0. As K is acoherent sheaf of O V ×S -modules, there is an open U 0 ⊂ V × S, containing V × S 0 ,such that K| U0 = 0. This proves that W 1 ∩ U 0 = W 2 ∩ U 0 , as complex analyticsubspaces of U 0 .□Lemma 6.4. Let Z t be the complex analytic subspace of D C x × D C x × V x defined bythe ideal (d V f t ). Then Z t is independent of t in an open neighborhood of (x, x, x) ∈D C x × D C x × V x .Proof. Let Z = Dx C ×Dx C ×X fx where X fx is the critical locus of f x : V x ⊂ B si −→ Cin V x defined by the ideal (df x ). We first show Z 0 = Z 1 = Z. Since the criticallocus of f i | {z}×{z′ }×V xis X fx by the definition of local trivialization for i = 0, 1,Z ⊂ Z i for i = 0, 1. By Lemma 6.3, Z = Z i for i = 0, 1.Let F, G : A 1 × Dx C × Dx C × V x → C be functions defined byF (t, z, z ′ , p) = (1 − t)f 0 (z, z ′ , p) + tf 1 (z, z ′ , p) and G(t, z, z ′ , p) = f 0 (z, z ′ , p).Since f t is a linear combination of f 0 and f 1 , we have the inclusionX F := (d V F = 0) ⊃ X G := (d V G = 0) = A 1 × Zof analytic subspaces. Also the restrictions of F and G to Γ = A 1 ×{(z, z) | z ∈ Dx C }coincide and hence X F | Γ = X G | Γ . By Lemma 6.3, X F = X G = A 1 × Z in an openneighborhood of Γ. Restricting to t ∈ [0, 1], we obtain the lemma.□We need another lemma. Let V and S be as before. We assume V ⊂ C r is anopen subset. We endow V the standard inner and hermitian product.Lemma 6.5. Let the notation be as stated. Let Z ⊂ V be a closed complex analyticsubspace. Suppose f : V × S → C is a holomorphic function with only one fiberwisecritical value 0 such that the vanishing locus (complex analytic subspace) of d V fcs

CATEGORIFICATION OF DONALDSON-THOMAS INVARIANTS 31is Z × S. Then for any convergent p n → p 0 ∈ V × S such that d V f(p n ) ≠ 0 andd V f(p n ) → 0, we havef(p n )limn→∞ |d V f(p n )| = 0.Proof. We prove the case S = pt. The general case is exactly the same. Letν : Ṽ → V be the resolution of the ideal sheaf of Z so that ν−1 (Z) is a normalcrossing divisor in Ṽ . Let ˜p n be the unique lifting of p n in Ṽ − ν−1 (Z). Suppose n kis a subsequence so that ˜p nk → q ∈ ν −1 (Z). We investigate the limiting behaviorof |f(p nk )|/|df(p nk )|.By our choice of resolution ν, locally near q ∈ Ṽ , the pullback ν∗ (df) of theideal (df) is a principal ideal sheaf generated by some monomial ϕ = z k11 · · · zkm mwith k i > 0 for holomorphic functions z 1 , · · · , z r on V such that restricting to thefiber Ṽ they give a local coordinates of Ṽ centered at q. Let w 1, · · · , w r be localcoordinates of V centered at p 0 . Then ϕ divides ν ∗ ∂fi∂w ifor all i.We claim that ν ∗ f is divisible by z k1+11 · · · zm km+1 . Indeed if we expand nearz 1 = 0, ν ∗ f = c m1 z m11 + c m1+1z m1+11 + · · · , where c j are holomorphic in z 2 , · · · , z r ,and c m1 ≠ 0. Then ϕ divides∑ ∂f ∂w i=∂ ν ∗ f = m 1 c m1 z m1−11 + (m 1 + 1)c m1+1z m11 + · · · .∂w i ∂z 1 ∂z 1Hence m 1 ≥ k 1 + 1 and z k1+11 divides ν ∗ f. Likewise z ki+1i divides ν ∗ f for eachi. Therefore ν ∗ (f) ⊂ ν ∗ (df) √ ν ∗ (df) where √ ν ∗ (df) is the radical of ν ∗ (df). Wewrite ν ∗ f = ∑ g i · ν ∗ ∂f∂w iwith g i | ν −1 (Z) = 0. Thereforeν ∗ |f(p n k)||df(p nk )| = |ν ∗ f(˜p nk )|(∑( ∑ |ν ∗ ∂f∂w i(˜p nk )| 2 ) ≤ |gi (˜p 1/2 nk )| 2) 1/2−→ 0.Because Ṽ → V is proper, and because this convergence holds for all convergentf(psubsequence ˜p nk , we find that lim n)n→∞ ||df(p = 0.□n)||Proof of Proposition 6.2. We use the inner product and the hermitian metric on V xvia embedding V x∼ = Vx ⊂ Ω 0,1 (adE x ) s . We let ∇ V f t be the relative gradient vectorfield of f t on D C x ×D C x ×V x as the metric dual of d V f t , the differential of f t in the V xdirection. Note that ∇ V f t is a vertical vector field with respect to the projectionpr 12 : D C x × D C x × V x → D C x × D C x which is differentiable on each fiber. We define atime dependent vector field on the complement of Z = Z t = (d V f t = 0) by(6.3) ξ t = f 0 − f 1|∇ V f t | 2 ∇ V f t .We claim that it extends to a well defined vector field on some neighborhood of(x, x, x) in D C x × D C x × V x . It suffices to show that|ξ t | = |f 0 − f 1 ||∇ V f t |= |f 0 − f 1 ||d V f t |approaches zero as a point approaches Z. Since Z t = Z 0 = Z 1 by Lemma 6.4, wehave an inclusion (d V f t ) ⊃ (d V f i ) of ideals for i = 0, 1. Hence we can express thevertical partial derivatives of f 0 and f 1 as linear combinations of the vertical partialderivatives of f t . Thus in a neighborhood of (x, x, x), we have|d V f 0 | ≤ C |d V f t | and |d V f 1 | ≤ C |d V f t |

32 YOUNG-HOON KIEM AND JUN LIfor some C > 0. By Lemma 6.5, we have|ξ t | = |f 0 − f 1 ||d V f t |(≤ C −1 |f0 ||d V f 0 | + |f )1||d V f 1 |→ 0, as d V f 0 , d V f 1 → 0.This proves that the vector field ξ t is well defined in a neighborhood of (x, x, x).Let x t for t ∈ [0, 1] be an integral curve of the vector field ξ t , so thatdx tdt = ẋ t = ξ t (x t ).Since ξ t is a vertical vector field for pr 13 , x t lies in a fiber of pr 13 . Then f t (x t ) isconstant in t becauseddt f t(x t ) = d V f t (ẋ t ) + f 1 − f 0 = ∇ V f t · ẋ t + f 1 − f 0f 0 − f 1= ∇ V f t ·|∇ V f t | 2 ∇ V f t + f 1 − f 0 = 0.Therefore the flow of the vector field ξ t from t = 0 to t = 1 gives a homeomorphismΦ of a neighborhood of (x, x, x) into Dx C × Dx C × V x such that f 1 ◦ Φ = f 0 . Becauseξ t = 0 for all t over the diagonal {(z, z)} ⊂ Dx C × Dx C , Φ is the identity map overthe diagonal.□6.2. Gluing isomorphisms. In this subsection we prove (2) of Proposition 3.13.We have a family V of CS charts on U ×[0, 1] with V| U×{0} = V α and V| U×{1} =V β . For x ∈ U, there exist an open neighborhood O x of x in U and a subfamilyW × [0, 1] of CS charts in V| Ox×[0,1]. Moreover, we have perverse sheaves P α•and Pβ• on U given by V α and V β respectively, whose restrictions to O x are theperverse sheaves of vanishing cycles A • f [r] and x α A• [r] respectively where f α fxβ x :csV α | x ⊂ B si −→ C and fx β cs: V β | x ⊂ B si −→ C.Recall that P α • was obtained by gluing A • f[r] by the isomorphismsx αA • f [r] (λα x,z←− )∗A • x α f [r] (λα y,z−→ )∗A • z α f [r]y αfor x, y ∈ U and z ∈ O x ∩ O y .Lemma 6.6. Suppose for each x ∈ U we have an isomorphism χ ∗ x : A • ∼ =−→ A • fxβ fxαsuch that for z ∈ O x we have a commutative diagram of isomorphisms(6.4) A • f α x(λ α x,z )∗A • f α zχ ∗ xχ ∗ zA • f β x(λ β x,z )∗A • f β zThen the isomorphisms χ ∗ x glue to give an isomorphismσ αβ : P • α → P • β .Proof. If z ∈ O x ∩ O y , we have a commutative diagram.A • f α x(λ α x,z )∗A • f α z(λ α y,z )∗ A • fyαχ ∗ xχ ∗ zχ ∗ yA • f β x(λ β x,z )∗A • f β z(λ β y,z )∗ A • .fyβ

CATEGORIFICATION OF DONALDSON-THOMAS INVARIANTS 33By Proposition 2.6 (2), we have an isomorphism σ αβ that restricts to χ ∗ x.Lemma 6.7. Let W ⊂ V be a complex submanifold in a complex manifold. Letf : V → C be a holomorphic function and g = f| W . Suppose the critical loci X fand X g are equal. Let x be a closed point of X f = X g . Then there is a coordinatesystem {z 1 , · · · , z r } of V centered at x such that W is defined by the vanishing ofz 1 , · · · , z m andm∑f = q(z 1 , · · · , z m ) + g(z m+1 , · · · , z r ) where q(z·) = zi 2 , and g = f| W .Proof. We choose coordinates {y 1 , · · · , y r } of V centered at x such that W is definedby the vanishing of y 1 , · · · , y m . Let I be the ideal generated by y 1 , · · · , y m . SinceX f = X g , i.e. (df) = (dg) + I, we have∂f∂y i∣ ∣∣W=r∑j=m+1a ij∂g∂y j,i=1i = 1, · · · , mfor some functions a ij regular at x. By calculus, we havem∑ ∂f∣ ∣∣Wf = g(y m+1 , · · · , y r ) + · y i + I 2∂y i= g(y m+1 , · · · , y r ) +r∑j=m+1i=1= g(z m+1 , · · · , z r ) +( m)∂g ∑a ij y i + I 2∂y ji=1m∑b ik y i y ki,k=1where z j = y j + ∑ mi=1 a ijy i for j ≥ m + 1 and b ik are some functions holomorphicnear x. Since the kernel of the Hessian of f at x is the tangent space of X f =X g ⊂ W at x, the quadratic form q = ∑ mi,k=1 b iky i y k is nondegenerate near x.Hence we can diagonalize q = ∑ mi=1 z2 i by changing the coordinates y 1 , · · · , y m tonew coordinates z 1 , · · · , z m . It follows that z 1 , · · · , z r is the desired coordinatesystem.□Let V t = V| U×{t} so that V 0 = V α and V 1 = V β . Let δ t : U → V t be thetautologial section which sends x ∈ U to x in the fiber Vx t ⊂ B si of V t over x. Letf t : V t cs⊂ U × B si −→ C be the family CS functional.Lemma 6.8. For x ∈ U, there exist an open neighborhood U 0 of δ 0 (x) in V 0 anda homeomorphism χ of U 0 × [0, 1] into V such that(1) the restriction χ t (of χ) to U 0 × {t} is a holomorphic map to an open setin V t ;(2) χ t | U0∩W maps U 0 ∩ W into W ⊂ V t and is the identity map;(3) χ 0 : U 0 → V 0 is the identity map;(4) f t ◦ χ t = f 0 .Proof. Since [0, 1] is compact, it suffices to find such a χ over an interval [t 0 , t 0 + ɛ]at each t 0 ∈ [0, 1] with 0 < ɛ ≪ 1. For t 0 ∈ [0, 1], by choosing a complexification ofthe pair W × [0, 1] ⊂ V near (x, t 0 ), we can find coordinate functions y 1 , · · · , y r ofthe fibers of V → U × [0, 1] at (x, t 0 ) such that y 1 , · · · , y m are holomorphic alongfibers of V → U × [0, 1], and W × [0, 1] is defined by the vanishing of y 1 , · · · , y m□

34 YOUNG-HOON KIEM AND JUN LI(cf. Proposition 5.8). Let t be the coordinate for [0, 1]. We can repeat the proof ofLemma 6.7 withf : V ↩→ U × [0, 1] × B sipr 3−→ Bsics−→ Cand g = f| W×[0,1] . By Lemma 6.3, (d V f) = (d V g) + I where I = (y 1 , · · · , y m )and (d V f) (resp. (d V g)) denotes the ideal generated by the partial derivativesin the fiber direction of V → U × [0, 1] (resp. W × [0, 1] → U × [0, 1]). Thenwe obtain a new coordinate system {z i } of V over U × [0, 1] at (x, t 0 ) such thatf = ∑ mi=1 z2 i + g(z m+1, · · · , z r ) with z j | W×[0,1] = y j for j ≥ m + 1. Then thecoordinate change from {z j | t0 } to {z j | t } defines the desired map χ.□csLet fx t : Vx t ⊂ B si −→ C be the CS functional on the CS chart Vx.tLemma 6.9. The isomorphism χ ∗ 1 : A • f→ A • x1 fx0the choice of χ.induced from χ 1 is independent ofProof. If χ ′ is another such homeomorphism, then χ −1t ◦χ ′ t : Vx 0 → Vx 0 is a homotopyof homeomorphisms which is id at t = 0. By Proposition 2.3, (χ −11 ◦ χ ′ 1) ∗ : A • f→x 0is the identity. This proves the lemma.□A • f 0 xLemma 6.10. For each x ∈ U, we have an isomorphism χ ∗ x : A • ∼ =−→ A • fxβ fxαthat (6.4) holds.suchProof. By Proposition 6.2, we have a homeomorphism ξ α : O x × V α | Ox → V α | Ox ×O x that gives the gluing isomorphism (λ α x,z) ∗ over (x, z). By construction, therestriction of ξ α to (O x × V α | Ox ) × Ox×O xO x to the diagonal O x ⊂ O x × O x is theidentity map. The composition of homeomorphisms(6.5) O x × V α | Oxid ×χ O x × V β | Oxξ −1αV α | Ox × O xχ −1 ×idξ βV β | Ox × O xis the identity over the diagonal O x ⊂ O x × O x . Upon fixing a local trivializationV α | Ox∼ = Ox ×V α,x , we have A • ∼f α = pr−12 A• f because the analytic space O x α x is locallycontractible. By Lemma 2.8 for O x × V x ⊂ O x × O x × V x → V x , we obtain thecommutativity of the diagram of isomorphisms(6.6) pr −11 A• f αpr1 −1 A• f βpr −12 A• f αpr −12 A• f βRestricting to the fiber over (x, z), we obtain (6.4) possibly after shrinking O x .□By Lemma 6.6 and Lemma 6.10, we have the desired isomorphism σ αβ : P • α → P • βover U α ∩ U β and thus we proved (2) of Proposition 3.13.

CATEGORIFICATION OF DONALDSON-THOMAS INVARIANTS 356.3. Obstruction class. In this subsection we prove (3) of Proposition 3.13 andcomplete our proof of Theorem 3.14.Let x ∈ U α ∩ U β ∩ U γ . By our construction in §6.2, in a neighborhood of x,σ γα ◦ σ βγ ◦ σ αβ is given by a biholomorphic map ϕ : V α,x → V α,x preserving fxαwhose restriction to W is the identity. By Proposition 2.5, σ γα ◦ σ βγ ◦ σ αβ isdet(dϕ| x ) · id and det(dϕ| x ) = ±1. Thus we obtain a Z 2 -valued Cech 2-cocycle{σ αβγ } of the covering {U α }. One checks directly that the cocycle is closed andits cohomology class σ ∈ H 2 (X, Z 2 ) is the obstruction class for gluing the perversesheaves {P α}.•Proposition 6.11. Given preorientation data {Ξ α } on X, the perverse sheaves{P • α} in §6.1 glue to give a globally defined perverse sheaf P • on X if and only ifthe obstruction class σ ∈ H 2 (X, Z 2 ) vanishes.The cocycle σ αβγ is by definition the cocycle for gluing the determinant linebundles det(T V α,x ) by the isomorphisms induced from the biholomorphic mapsV α,x∼ = Vβ,x . In particular, det(T V α,x ) glue to a globally defined line bundle on Xif and only if σ ∈ H 2 (X, Z 2 ) is zero.Since a neighborhood O x of x in X is the critical locus of the CS functional onV α,x , we have a symmetric obstruction theoryF := [T Vα,x → Ω Vα,x ] −→ L • X| Ox .Hence the determinant line bundle det F is the inverse square of the determinantline bundle of the tangent bundle T Vα,x . On the other hand, by [35], F ∼ =Ext • π(E, E)[2] on O x where π : X × Y → X is the projection and E is the universalbundle. Hence the determinant bundle of T Vα,x is a square root of the determinantbundle of Ext • π(E, E)[1]. Therefore if the obstruction class σ ∈ H 2 (X, Z 2 ) is zero,then det Ext • π(E, E) has a square root.Suppose det Ext • π(E, E) has a square root L. Then the local isomorphismsdet T Vα,x∼ = L|Ox induce gluing isomorphisms for {det T Vα,x }. Therefore the obstructionclass σ to gluing them is zero. This implies the gluing of {P α} • by Proposition6.11. This completes the proof of Theorem 3.14.We add that if a square root L of det Ext • π(E, E) exists, then any two such squareroots differ by tensoring a 2-torsion line bundle (i.e. a Z 2 -local system) on X, andvice versa.Remark 6.12. In [19], Kontsevich and Soibelman defined orientation data aschoices of square roots of det Ext • π(E, E) satisfying a compatibility condition. SeeDefinition 15 in [19]. For the existence of a perverse sheaf which is the goal of thispaper, it suffices to have a square root. However for wall crossings in the derivedcategory, the compatibility condition seems necessary. See §5 of [19] for furtherdiscussions.7. Mixed Hodge modulesIn this section, we prove that the perverse sheaf P • in Theorem 1.1 lifts to amixed Hodge module (MHM for short) of Morihiko Saito ([31]).A mixed Hodge module M • consists of a Q-perverse sheaf P • , a regular holonomicD-module A • with DR(A • ) ∼ = P • ⊗ Q C, a D-module filtration F and a weightfiltration W , with polarizability and inductive construction. Like perverse sheaves,

36 YOUNG-HOON KIEM AND JUN LImixed Hodge modules form an abelian category MHM(X). There is a forgetfulfunctorrat : MHM(X) −→ P erv(X)which is faithful and exact. Moreover, if f : V → C is a holomorphic function on acomplex manifold V of dimension r, there is a MHM Mf • = φm f(Q[r − 1]) such thatrat(M • f ) = φ f (Q[r − 1]) = A • f [r]is the perverse sheaf of vanishing cycles of f. Further, if Φ : V → V is a biholomorphicmap and g = f ◦ Φ, then Φ induces an isomorphism Φ ∗ : M • f → M • g . Wealso note that the category MHM(X) is a sheaf, i.e. gluing works ([33]).The goal of this section is to prove the following.Theorem 7.1. The perverse sheaf P • in Theorem 1.1 lifts to a MHM M • . Namely,there exists a MHM M • such that rat(M • ) = P • .Like Theorem 3.14, Theorem 7.1 is a direct consequence of the following analogueof Proposition 3.13 together with Propositions 3.4 and 3.12.Proposition 7.2. (1) Let π : V → U be a family of CS charts on U ⊂ X ⊂ B siwith complexifiable local trivializations at every point x ∈ U. Then the MHM ofvanishing cycles forf x : V x = π −1 cs(x) ⊂ B si −→ Cglue to a MHM M • on U, i.e. M • is isomorphic to Mf • xin a neighborhood of x.(2) Let V α and V β be two families of CS charts on U with complexifiable localtrivializations. Let M α • and Mβ • be the induced MHMs on U. Let V be a family of CScharts on U × [0, 1] with complexifiable local trivializations such that V| U×{0} = V αand V| U×{1} = V β . Suppose for each x ∈ U, there are an open U x ⊂ U anda subfamily W of both V α | Ux and V β | Ux such that W × [0, 1] is a complexifiablesubfamily of CS charts in V| Ux×[0,1]. Then there is an isomorphism σαβ m : M α • ∼ = Mβ•of MHMs.(3) If there are three families V α , V β , V γ with homotopies among them as in (2),then the isomorphisms σαβ m , σm βγ , σm γα satisfyσ m αβγ := σ m γα ◦ σ m βγ ◦ σ m αβ = ± id .The proof of Proposition 7.2 is a line by line repetition of the proof of Proposition3.13 in §6, with perverse sheaves replaced by MHMs if homeomorphismsare replaced by biholomorphic maps. Notice that the use of Propositions 2.3 and2.5 are justified by the fact that rat is a faithful functor. For example, Proposition3.13 (3) immediately implies Proposition 7.2 (3) because rat(σ m αβ ) = σ αβ andrat(± id) = ± id.Now the only non-holomorphic map used in §6 is the homeomorphism Φ in theproof of Proposition 6.2 which was defined by the integral flow of the vector field(6.3). Therefore we have a proof of Theorem 7.1 as soon as we can replace (6.3) bya holomorphic vector field ξ t which vanishes on X and satisfies(7.1) d V f t (ξ t ) = f 0 − f 1 .in the notation of §6.1.To see this, we first recall that by Lemma 6.4, the ideal I = (d V f t ) is indepedentof t and defines an analytic space Z.Lemma 7.3. f 1 − f 0 ∈ I 2 .

CATEGORIFICATION OF DONALDSON-THOMAS INVARIANTS 37Proof. We let ι : Dx C × V D C x→ A s and ι ′ : V D C x× Dx C → A s be the compositionsof the projections to V D C xwith the tautological map V D C x→ A s constructed in §6.Then f 0 = cs ◦ ι ◦ (id D C x×ψ x ) and f 1 = cs ◦ ι ′ ◦ Ψ C ◦ (id D C x×ψ x ). Since the problemis local, we can reduce the proof to the following case.Let ξ ∈ Dx C × Dx C × V x be any point in the subspace defined by the ideal I.We pick an open neighborhood ξ ∈ W ⊂ Dx C × Dx C × V x , so that W is endowedwith holomorphic coordinates z = (z 1 , · · · , z m ) with ξ = (0, · · · , 0) ∈ W . LetI = I ⊗ OD OC x ×Dx C W denote the ideal of W ∩ Z. Then it suffices to show that×Vx(7.2) f 0 | W − f 1 | W ∈ I 2 .We now describe the difference f 0 | W − f 1 | W . For simplicity, we abbreviate(id D C x×ψ x )| W to ˜ψ x . By the construction of Ψ C , we know that there are holomorphicg : W → G and ɛ : W → Ω 0,1 (adE) s satisfying ɛ| W ∩Z ≡ 0 such thatι ′ ◦ Ψ C ◦ (id D C x×ψ x )| W = ι ′ ◦ Ψ C ◦ ˜ψ x = g · (ι ◦ ˜ψ x ) + ɛ = g · (ι ◦ ˜ψ x + ɛ ′ ),where g · (−) denotes the gauge group action; · + ɛ is via the affine structureA s × Ω 0,1 (adE) s → A s , and ɛ ′ : W → Ω 0,1 (adE) s is the holomorphic map makingthe third identity hold, which satisfies ɛ ′ | W ∩Z ≡ 0. Since cs is invariant undergauge transformations, (7.2) is equivalent to(7.3) cs ◦ ι ◦ ˜ψ x − cs ◦ (ι ◦ ˜ψ x + ɛ ′ ) ∈ I 2 .We use finite dimensional approximation to reduce this to a familiar problem inseveral complex variables. First, since ɛ ′ takes values in C ∞ -forms, we can lift itto ˜ɛ : W → Ω 0,1 (adE) L 2tfor a large t so that Ω 0,1 (adE) L 2t⊂ Ω 0,1 (adE) s . SinceΩ 0,1 (adE) L 2tis a separable Hilbert space, we can approximate it by an increasing sequenceof finite dimensional subspaces R k ⊂ Ω 0,1 (adE) L 2t. Let q k : Ω 0,1 (adE) L 2t→W k ⊂ Ω 0,1 (adE) s be the orthogonal projection. Then we have a convergence ofholomorphic functionslim cs ◦ (ι ◦ ˜ψ x + q k ◦ ˜ɛ) = cs ◦ (ι ◦ ˜ψ x + ɛ ′ )k→∞uniformly on every compact subset of W . We claim that(7.4) cs ◦ ι ◦ ˜ψ x − cs ◦ (ι ◦ ˜ψ x + q k ◦ ˜ɛ) ∈ I 2 .Note that the claim and the uniform convergence imply (7.3).We prove (7.4). For a fixed k, we pick a basis e 1 , · · · , e n of R k ; we introducecomplex coordinates w = (w 1 , · · · , w n ), and form a holomorphic functionF k : W × C n −→ C; F k (z, w) = cs ◦ (ι ◦ ˜ψn∑x + w j e j ).If we write q k ◦˜ɛ = δ 1 e 1 +· · ·+δ n e n : W → R k , then all δ j are holomorphic functionslying in I. Therefore,(cs ◦ (ι ◦ ˜ψx + q k ◦ ˜ɛ) ) (z) = F (z, δ 1 (z), · · · , δ n (z)).Since δ j ∈ I, applying Taylor expansion along (z, 0), we conclude thatn∑ ∂F kF k (z, δ 1 (z), · · · , δ n (z)) ≡ F k (z, 0) + (z, 0) · δ j (z) mod I 2 .∂w jj=1j=1

38 YOUNG-HOON KIEM AND JUN LISince ∂F k∂w j(z, 0) involve the partial derivatives of the cs via the chain rule, we concludethat ∂F k∂w j(z, 0) ∈ I. This proves that F k (z, δ 1 (z), · · · , δ n (z)) − F k (z, 0) ∈ I 2 ,which is (7.4). This proves the Lemma.□Since f 0 −f 1 ∈ I 2 and I = (d V f t ), we can always find a time dependent holomorphicvertical vector field ξ t which satisfies (7.1) and vanishes along X (cf. [5]). Thisgives us the desired biholomorphic map Φ. This completes our proof of Theorem7.1. Note that we don’t need Lemma 6.5 for this choice of ξ t .Remark 7.4. (Gluing of polarization) To use the hard Lefschetz property of [30],we also need the gluing of polarization and monodromy for M • and the graduationgr W M • . Note that our gluing isomorphisms arise from continuous families of CScharts V . Since a continuous variation of rational numbers is constant, the obviouspolarizations for the constant sheaves Q V on V is constant in a continuous familyof CS charts. The induced polarizations on the perverse sheaves A • f [r] = φ f Q[r−1],defined in [30, §5.2], are functorial. Therefore the polarizations on A • f[r] glue togive us a polarization on P • . By the same argument, we have a polarization ongr W M • and the monodromy operators also glue .8. Gopakumar-Vafa invariantsIn this section, we provide a mathematical theory of the Gopakumar-Vafa invariantas an application of Theorem 3.16.8.1. Intersection cohomology sheaf. As before, let Y be a smooth projectiveCalabi-Yau 3-fold over C. From string theory ([9, 15]), it is expected that(1) there are integers n h (β), called the Gopakumar-Vafa invariants (GV forshort) which contain all the information about the Gromov-Witten invariantsN g (β) of Y in the sense that∑(8.1)N g (β)q β λ 2g−2 = ∑n h (β) 1 (2 sin( kλ ) 2h−2k 2 ) q kβg,βk,h,βwhere β ∈ H 2 (Y, Z), q β = exp(−2π ∫ β c 1(O Y (1)));(2) n h (β) come from an sl 2 × sl 2 action on some cohomology theory of themoduli space X of one dimensional stable sheaves on Y ;(3) n 0 (β) should be the Donaldson-Thomas invariant of the moduli space X.By using the global perverse sheaf P • constructed above and the method of [10],we can give a geometric theory for GV invariants.We recall the following facts from [30, 31].Theorem 8.1. (1) (Hard Lefschetz theorem) If f : X → Y is a projectivemorphism and P • is a perverse sheaf on X which underlies a pure (polarizable)MHM M • , then the cup product induces an isomorphismω k : p H −k Rf ∗ P • −→ p H k Rf ∗ P •where ω is the first Chern class of a relative ample line bundle.

CATEGORIFICATION OF DONALDSON-THOMAS INVARIANTS 39(2) (Decomposition theorem) If f : X → Y is a proper morphism and P • asabove, thenRf ∗ P • ∼ = ⊕k p H k Rf ∗ P • [−k]and each summand p H k Rf ∗ P • [−k] is a perverse sheaf underlying a MHMwhich is again polarizable semisimple and pure.Let X be the moduli space of stable one-dimensional sheaves E on Y with χ(E) =1 and [E] = β ∈ H 2 (Y, Z). In particular, the rank of E is zero and c 1 (E) = 0. By[23, Theorem 6.11], there is a universal family E on X × Y . Let X = X red be thereduced complex analytic subspace of X.Let ˜X be the semi-normalization of X and let S be the image of the morphism˜X → Chow(Y ) to the Chow scheme of curves in Y . By [18], the morphism ˜X →X is one-to-one and hence a homeomorphism because ˜X is projective and X isseparated. The natural morphism f : ˜X −→ S is projective and the intersectioncohomology sheaf IC • = IC ˜X(C) • underlies a pure simple MHM. In [10], S. Hosono,M.-H. Saito and A. Takahashi show that the hard Lefschetz theorem applied to fand c : S → pt gives us an action of sl 2 × sl 2 on the intersection cohomologyIH ∗ ( ˜X) = H ∗ ( ˜X, IC • ) as follows: The relative Lefschetz isomorphismp H −k Rf ∗ (IC • ) −→ p H k Rf ∗ (IC • )for f gives an action of sl 2 , called the left action, via the isomorphismsH ∗ ( ˜X, IC • ) ∼ = H ∗ (S, Rf ∗ IC • ) ∼ = ⊕ k H ∗ (S, p H k Rf ∗ (IC • )[−k])from the decomposition theorem. On the other hand, since p H k Rf ∗ (IC • )[−k]underlies a MHM which is again semisimple and pure, H ∗ (S, p H k Rf ∗ (IC • )[−k])is equipped with another action of sl 2 , called the right action, by hard Lefschetzagain. Therefore we obtain an action of sl 2 × sl 2 on the intersection cohomologyIH ∗ ( ˜X) of ˜X.If C ∈ S is a smooth curve of genus h, the fiber of f over C is expected to bethe Jacobian of line bundles on C whose cohomology is an sl 2 -representation space(( 1 ) ⊗h2 ) ⊕ 2(0) ,where ( 1 2 ) denotes the 2-dimensional representation of sl 2 while (0) is the trivial1-dimensional representation. In [10], the authors propose a theory of theGopakumar-Vafa invariants by using the sl 2 × sl 2 action on IH ∗ ( ˜X, C) as follows:By the Clebsch-Gordan rule, it is easy to see that one can uniquely write thesl 2 × sl 2 -representation space IH ∗ ( ˜X, C) in the formIH ∗ ( ˜X, C) = ⊕ (( 1 ) ⊗h2 ) L ⊕ 2(0) L ⊗ R h ,hwhere ( k 2 ) L denotes the k + 1 dimensional irreducible representation of the left sl 2action while R h is a representation space of the right sl 2 action. Now the authorsof [10] define the GV invariant as the Euler number T r Rh (−1) H Rof R h where H Ris the diagonal matrix in sl 2 with entries 1, −1.However it seems unlikely that the invariant n h (β) defined using the intersectioncohomology as in [10] will relate to the GW invariants of Y as proposed byGopakumar-Vafa because intersection cohomology is unstable under deformation.We propose to use the perverse sheaf P • on X constructed above instead of IC • .

40 YOUNG-HOON KIEM AND JUN LI8.2. GV invariants from perverse sheaves. In this subsection, we assume thatdet Ext • π(E, E) admits a square root so that we have a perverse sheaf P • and aMHM M • which are locally the perverse sheaf and MHM of vanishing cycles for alocal CS functional.Remark 8.2. In [11], it is proved that if the Calabi-Yau 3-fold Y is simply connectedand H ∗ (Y, Z) is torsion-free, then det Ext • π(E, E) admits a square root. Forinstance, when Y is a quintic threefold, we have the desired perverse sheaf andMHM.Since the semi-normalization γ : ˜X → X is bijective, the pullback ˜P • of P ••is a perverse sheaf and γ ∗ ˜P ∼ = P • . By Theorem 7.1, P • lifts to a MHM M •and its pullback ˜M • satisfies rat( ˜M • ) = ˜P • since rat preserves Grothendieck’s sixfunctors ([31]). Let ˆM • = gr W ˜M • be the graded object of ˜M • with respect to theweight filtration W . Then ˆM • is a direct sum of polarizable Hodge modules ([32]).Let ˆP • = rat( ˆM • ) which is the graduation gr W ˜P • by the weight filtration of ˜P •because rat is an exact functor ([31]).By [30, §5], the hard Lefschetz theorem and the decomposition theorem holdfor the semisimple polarizable MHM ˆM • . Hence by applying the functor rat, weobtain the hard Lefschetz theorem and the decomposition theorem for ˆP • . Therefore,we can apply the argument in §8.1 to obtain an action of sl 2 × sl 2 on thehypercohomology H ∗ ( ˜X, ˆP • ) to writeH ∗ ( ˜X, ˆP • ) ∼ = ⊕ (( 1 ) ⊗h2 ) L ⊕ 2(0) L ⊗ R h .hDefinition 8.3. We define the Gopakumar-Vafa invariant asn h (β) := T r Rh (−1) H R.The GV invariant n h (β) is integer valued and defined by an sl 2 × sl 2 representationspace H ∗ ( ˜X, ˆP • ) as expected from [9].Proposition 8.4. The number n 0 (β) is the Donaldson-Thomas invariant of X.Proof. Recall that the DT invariant is the Euler number of X weighted by theBehrend function ν X on X and that ν X (x) for x ∈ X is the Euler number of thestalk cohomology of P • at x. Therefore the DT invariant of X is the Euler numberof H ∗ (X, P • ).Since the semi-normalization γ : ˜X → X is a homeomorphism, γ∗ ˜P• ∼ = P • andH ∗ (X, P • ) ∼ = H ∗ (X, γ ∗ ˜P • ) ∼ = H ∗ ( ˜X, ˜P • ) so thatDT (X) = ∑ k(−1) k dim H k (X, P • ) = ∑ k(−1) k dim H k ( ˜X, ˜P • ).Since ˜P • has a filtration W with ˆP • = gr W ˜P • , we have the equality of alternatingsums ∑(−1) k dim H k ( ˜X, ˜P • ) = ∑ (−1) k dim H k ( ˜X, ˆP • ).kkSince the Euler number of the torus part ( ( 1 2 ) ) ⊗hL ⊕ 2(0) L is zero for h ≠ 0,∑(−1) k dim H k ( ˜X, ˆP • ) = T r R0 (−1) H R= n 0 (β).This proves the proposition.k□

CATEGORIFICATION OF DONALDSON-THOMAS INVARIANTS 41Furthermore, we propose the following conjecture.Conjecture 8.5. (1) The GV invariants n h (β) are invariant under deformationof the complex structure of Y .(2) The GV invariants n h (β) depend only on β and are independent of the constantterm χ(E) of the Hilbert polynomial.(3) The GV invariants n h (β) are independent of the choice of a polarization of Y .(4) The identity (8.1) holds.Note that for h = 0, (1) follows from Proposition 8.4 and [35]. Also by [12] and[35], (3) is known for h = 0. Of course, (1)-(3) are consequences of (4). Furthermore,establishing the identity (8.1) will equate Definition 8.3 with that introduced byPandharipande-Thomas [29] for a large class of CY 3-folds (cf. [28]).8.3. K3-fibered CY 3-folds. In this last subsection, we show that Conjecture8.5 holds for a primitive fiber class of K3 fibered CY 3-folds.We first consider the local case. We let ∆ ⊂ C be the unit disk, t ∈ Γ(O ∆ ) thestandard coordinate function, and let p : Y → ∆ be a smooth family of polarized K3surfaces. We suppose the central fiber Y 0 contains a curve class β 0 ∈ H 1,1 (Y 0 , R) ∩H 2 (Y 0 , Z), not proportional to the polarization, such that β 0 ceases to be (1, 1) inthe first order deformation of Y 0 in Y , which means that if we let ˜β ∈ Γ(∆, R 2 p ∗ Z Y )be the continuous extension of β 0 and let ˜ω ∈ Γ(∆, p ∗ Ω 2 Y/∆) be a nowhere vanishingsection of relative (2, 0)-form, then p ∗ (˜ω ∧ ˜β) ∉ t 2 O ∆ .For c ∈ ∆, we let ι c : Y c → Y be the closed embedding. We let β ∈ H 4 (Y, Z)be such that β 0 = ι ! 0β 0 . Since Y → ∆ is a family of polarized K3 surfaces, thefamily of relative ample line bundle is ample on Y . We form the moduli M Y (−β, 1)(resp. M Y0 (β 0 , 1)) of one dimensional stable sheaves E of O Y -modules (resp. O Y0 -modules) with c 2 (E) = −β (resp. c 1 (E) = β 0 ) and χ(E) = 1. Since β is a fiberclass, the moduli M Y (−β, 1) is well defined and is a complex scheme.Because the polarization of Y restricts to the polarization of Y 0 , we have a closedembedding(8.2) M Y0 (β 0 , 1) −→ ⊂M Y (−β, 1).Lemma 8.6. Suppose β 0 ceases to be (1, 1) in the first order deformation of Y 0 inY , and there are no c ≠ 0 ∈ ∆ such that ι ∗ c ˜β ∈ H 1,1 (Y c , R). Then the embedding(8.2) is an isomorphism of schemes.Proof. We first claim that for any sheaf [E] ∈ M Y (−β, 1), E = ι 0∗ E ′ for a sheaf[E ′ ] ∈ M Y0 (β 0 , 1). Indeed, let spt(E) be the scheme-theoretic support of E. SinceE is stable, it is connected and proper, thus its underlying set is contained in aclosed fiber Y c ⊂ Y for some closed c ∈ ∆. Denoting by the same t ∈ Γ(O Y )the pullback of t ∈ O ∆ , since E is coherent, there is a positive integer k so thatspt(E) ⊂ ((t − c) k = 0). In particular, E is annihilated by (t − c) k .Since t − c ∈ Γ(O Y ), multiplying by t − c defines a sheaf homomorphism ·(t − c) :E → E, which has non-trivial kernel since (t − c) k annihilates E. Since E is stable,this is possible only if E is annihilated by t − c. Therefore, letting E ′ = E/(t − c) · E,which is a sheaf of O Yc -modules, we have E = ι c∗ E ′ . It remains to show that c = 0.If not, then c 1 (E ′ ) = ι ! cβ will be in H 1,1 (Y c , R), a contradiction. This prove theclaim.We now prove that (8.2) is an isomorphism. Indeed, by the previous argument,we know that (8.2) is a homeomorphism. To prove that it is an isomorphism, we

42 YOUNG-HOON KIEM AND JUN LIneed to show that for any local Artin ring A with quotient field C and morphismϕ A : Spec A → M Y (−β, 1), ϕ A factors through M Y0 (β 0 , 1). By an induction onthe length of A, we only need to consider the case where there is an ideal I ⊂ Asuch that dim C I = 1 and the restriction ϕ A/I : Spec A/I → M Y (−β, 1) alreadyfactors through M Y0 (β 0 , 1).Let E be the sheaf of A × O Y -modules that is the pullback of the universalfamily of M Y (−β, 1) via ϕ A . As ϕ A/I factors through M Y0 (β 0 , 1), t · E ⊂ I · E. Ift · E = 0, then ϕ A factors, and we are done. Suppose not. Then since E 0 = E ⊗ A Cis stable, there is a c ∈ I so that t · E = c · E ⊂ E. Thus (t − c) · E = 0. Wenow let ψ : Spec A → ∆ be the morphism defined by ψ ∗ (t) = c, and let Y A =Y × ∆,ψ Spec A. Then Y A → Spec A is a family of K3 surfaces with a tautologicalembedding ι A : Y A → Y ×Spec A. Then (t−c)·E = 0 means that there is an A-flatfamily of sheaves E ′ of O YA -modules so that ι A∗ E ′ = E.Let q be the projection and ι be the tautological morphism fitting into theCartesian squareY A⏐↓ qSpec Aι−−−−→ Y⏐ ⏐↓pψ−−−−→ ∆Then c 1 (E ′ ) = ι ∗ ˜β. Thus for the relative (2, 0)-form ˜ω, we have0 = q ∗(c1 (E ′ ) ∧ ι ∗ ˜ω ) = q ∗ ι ∗ ( ˜β ∧ ˜ω) = ψ ∗ p ∗ ( ˜β ∧ ˜ω).By the assumption that p ∗ ( ˜β ∧ ˜ω) is not divisible by t 2 , the above vanishing impliesthat ψ factors through 0 ∈ ∆. This proves that c = 0 and ϕ A factors throughM Y0 (β 0 , 1). This proves the proposition.□By the above lemma, we find that the moduli scheme X = X = M Y (−β, 1) =M Y0 (β 0 , 1) is a smooth projective variety of dimensiondim X = β 2 0 + 2 = 2k,because the obstruction space Ext 2 (E, E) 0 is trivial for any stable sheaf E on a K3surface Y 0 . Hence we can set P • = Q X and thus H ∗ (X, P • ) = H ∗ (X, Q). LetπY 0 = S −→ P 1be an elliptic K3 surface and β 0 = C + kF where C is a section and F is a fiber.We can calculate the GV invariants in this case by the same calculation as in [10,Theorem 4.7]. Since the details are obvious modifications of those in [10, §4], webriefly outline the calculation. Indeed by Fourier-Mukai transform, X is isomorphicto the Hilbert scheme S [k] of points on S and the Chow scheme in this case is acomplete linear system P k . The Hilbert-Chow morphism isS [k]hcπ−→ S (k) −→ (P 1 ) (k) ∼ = P k .It is easy to see that the cohomology H ∗ (S, Q) of S as an sl 2 × sl 2 representationspace by (relative) hard Lefschetz applied to S → P 1 is( 1 2 ) L ⊗ ( 1 2 ) R + 20 · (0) L ⊗ (0) R .If we denote by t L (resp. t R ) the weight of the action of the maximal torus for theleft (resp. right) sl 2 action, we can write H ∗ (S, Q) as (t L + t −1L )(t R + t −1R ) + 20.

CATEGORIFICATION OF DONALDSON-THOMAS INVARIANTS 43Hence H ∗ (S (k) , Q) is the invariant part of(( 1 2 ) L ⊗ ( 1 2 ) R + 20 · (0) L ⊗ (0) R) kby the symmetric group action. In terms of Poincaré series, we can write∑P tL ,t R(S (k) )q k 1=(1 − t L t R q)(1 − t −1L t Rq)(1 − t L t −1.q)(1 − t−1 q)(1 − q)20kRL t−1 RApplying the decomposition theorem ([2]) for the semismall map S [k] → S (k) , wefind that ∑ k P t L ,t R(S [k] )q k is∏1(1 − t L t R q m )(1 − t −1L t Rq m )(1 − t L t −1R qm )(1 − t −1L t−1 R qm )(1 − q m ) 20m≥1which gives∑(8.3) P tL ,t R(S [k] )| tR =−1q k = ∏ 1(1 + tm≥1 L q m ) 2 (1 + t −1L qm ) 2 (1 − q m ) . 20kBy definition, the GV invariants are defined by writing (8.3) as∑(8.4) q k (t L + t −1L+ 2)h ⊗ R h (S [k] )| tR =−1 = ∑ q k n h (k)(t L + t −1L + 2)hh,kh,kBy equating (8.3) and (8.4) with t L = −y, we obtain∑(−1) h n h (k)(y 1 2 − y− 1 2 ) 2h q k−1 1=q ∏ m≥1 (1 − yqm ) 2 (1 − y −1 q m ) 2 (1 − q m ) 20h,kwith β0 2 = 2k − 2. On the other hand, by [25, Theorem 1], we have∑(−1) h r h (k)(y 1 2 − y− 1 2 ) 2h q k−1 1=q ∏ m≥1 (1 − yqm ) 2 (1 − y −1 q m ) 2 (1 − q m ) 20h,kwhere r h (k) are the BPS invariants from the Gromov-Witten theory for Y → ∆.Combining these two identities, we find thatn h (k) = r h (k),which verifies Conjecture 8.5 for the local Calabi-Yau 3-fold Y → ∆.obtainWe thusProposition 8.7. Let Y → P 1 be a K3 fibered projective CY threefold and let ι 0 :Y 0 ⊂ Y be a smooth fiber. Let β 0 ∈ H 2 (Y 0 , Z) be a curve class so that its Poincaredual β0∨ ∈ H 2 (Y 0 , Z) ceases to be (1, 1) type in the first order deformation of Y 0 inthe family Y c , c ∈ P 1 . Then X 0 = X 0 := M Y0 (β0 ∨ , 1) ⊂ X := M Y (−(ι 0∗ β 0 ) ∨ , 1) isa (smooth) open and closed complex analytic subspace, and (8.1) holds for the GVinvariants of the perverse sheaf P • (of X) restricted to X 0 where N g (β 0 ) in (8.1)are the GW invariants contributed from the connected components of stable mapsto Y that lie in Y 0 .It will be interesting to extend the constructions in this paper to the setting ofstable pairs. Then it may be possible to extend the theory of Gopakumar-Vafa invariantsto the moduli scheme of stable pairs. Let M be the moduli scheme of stablepairs (F, s) with fixed topological type, where F is a pure sheaf of one dimensional

44 YOUNG-HOON KIEM AND JUN LIsupport and s ∈ H 0 (F ) which is α-stable for some α > 0. We consider the morphismM → S to the Chow scheme possibly after taking the semi-normalization,sending (F, s) to the support of F . The general fiber over a curve C of genus g isexpected to be the symmetric product C (d) for a suitable d defined by the topologicaltype. Let J g = [{( 1 2 ) L + 2g (0) L } d ] S dbe the cohomology of C (d) as an sl 2representation space. We can write the perverse hypercohomology of M in the form⊕Jg ⊗ R g and define the GV invariant as the Euler number of R g .9. Appendix: Square root of determinant line bundleThe purpose of this appendix is to give a direct proof of the following theoremof Hua [11], where it is stated only for sheaves. The argument presented below isa simplification of the proof in [11]. A byproduct of this simplification is that theproof now works for any perfect complexes, not just sheaves as in [11].Theorem 9.1. [11, Theorem 3.1] Let E → X × Y be a perfect complex of vectorbundles; let π, ρ be the projections from X ×Y to X, Y respectively; let Ext • π(E, E) =Rπ ∗ RHom(E, E). Then the torsion-free part of c 1 (Ext • π(E, E)) ∈ H 2 (X, Z) is divisibleby 2.For a proof, we need two lemmas.Lemma 9.2. Theoroem 9.1 holds when E is a line bundle L.Proof. Since the Chern character of RHom(L, L) ∼ = O is 1, by Grothendieck-Riemann-Roch together with the Todd classT d Y = 1 + c 2(Y )12 ,we find that c 1 (Ext • π(L, L)) = 0.Lemma 9.3. We may assume c 1 (E) = 0.Proof. Let L = (det E) −1 and F = E ⊕ L. ThenMoreover we havec 1 (F) = c 1 (E) + c 1 (L) = c 1 (E) − c 1 (E) = 0.c 1 (Ext • π(F, F)) = c 1 (Ext • π(E, E))+c 1 (Ext • π(L, L))+c 1 (Ext • π(E, L))+c 1 (Ext • π(L, E)).The last two terms cancel by Serre duality and the second term is zero by Lemma9.2. Hence c 1 (Ext • π(F, F)) is divisible by 2 if and only if c 1 (Ext • π(E, E)) is divisibleby 2.□Proof of Theorem 9.1. Let α i = c i (E). By Lemma 9.3, we may assume α 1 = 0.Then we havech(E) = r − α 2 + α 32 + α2 2 − 2α 412where r is the rank of E. Since RHom(E, E) = E ∨ ⊗ E, we havech(RHom(E, E)) = r 2 − 2rα 2 + α 2 2 + r 6 (α2 2 − 2α 4 ).By the Grothendieck-Riemann-Roch formulas ch(π ! E) = ∫ Y ch(E) · T d Y and∫ch(Ext • π(E, E)) = ch(Rπ ∗ RHom(E, E)) = ch(RHom(E, E)) · T d Y ,Y□

CATEGORIFICATION OF DONALDSON-THOMAS INVARIANTS 45we have∫c 1 (Ext • π(E, E)) =∫c 1 (π ! E) =Y( α22 − 2α 4− α 2 · c2(Y ))12Y 12(α2 2 + r 6 (α2 2 − 2α 4 ) − 2rα 2 · c2(Y )12and) ∫=Yα 2 2 + 2r c 1 (π ! E).So it suffices to show that ∫ Y α2 2 is divisible by 2. By the Künneth formula, we canwriteα 2 = π ∗ A 2 + π ∗ A 1 · ρ ∗ B 1 + ρ ∗ B 2modulo torsion, where A i ∈ H 2i (Y, Z) and B i ∈ H 2i (X, Z). Then we have∫ ∫α2 2 = 2( A 2 A 1 ) · B 1which is obviously divisible by 2.YYReferences1. K. Behrend. Donaldson-Thomas type invariants via microlocal geometry. Ann. of Math. (2)170 (2009), no. 3, 1307-1338.2. A. Beilinson, J. Bernstein and P. Deligne. Faisceaux pervers. Astrisque, 100, Soc. Math.France, Paris, 1982.3. J. Bernstein and V. Lunts. Equivariant sheaves and functors. Lecture Notes in Mathematics,1578. Springer-Verlag, Berlin, 1994.4. A. Borel et al. Intersection cohomology. Notes on the seminar held at the University of Bern,Bern, 1983.5. C. Brav, V. Bussi, D. Dupont, D. Joyce and B. Szendroi. Symmetries and stabilization forsheaves of vanishing cycles. Preprint, arXiv:1211.3259.6. C. Brav, V. Bussi, D. Dupont and D. Joyce. Shifted symplectic structures on derived schemesand critical loci. Preprint.7. A. Dimca. Sheaves in topology. Universitext. Springer-Verlag, Berlin, 2004.8. D. Gieseker. A construction of stable bundles on an algebraic surface. J. Differential Geom.27 (1988), no. 1, 137-154.9. R. Gopakumar and C. Vafa. M-Theory and Topological Strings II. ArXiv: 9812127.10. S. Hosono, M.-H. Saito and A. Takahashi. Relative Lefschetz action and BPS state counting.Internat. Math. Res. Notices (2001), no. 15, 783-816.11. Z. Hua. Spin structure on moduli space of sheaves on Calabi-Yau threefold. Preprint,arXiv:1212.3790.12. D. Joyce and Y. Song. A theory of generalized Donaldson-Thomas invariants. Mem. Amer.Math. Soc. 217 (2012), no. 1020.13. M. Kashiwara and P. Schapira. Sheaves on manifolds. Grundlehren der Mathematischen Wissenschaften292, Springer, 1994.14. T. Kato. Perturbation theory for linear operators. Springer-Verlag 1980.15. S. Katz. Genus zero Gopakumar-Vafa invariants of contractible curves. J. Differential Geom.79 (2008), no. 2, 185-195.16. M. Khenkin. The method of integral representations in complex analysis, Several ComplexVariables 1, 19–116. Encyclopaedia of Mathematical Sciences, Vol. 7, Springer-Verlag.17. K. Kodaira. Complex manifolds and deformation of complex structures. Classics in Mathematics.Springer-Verlag, Berlin, 2005.18. J. Kollar. Rational curves on algebraic varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete.3. Folge. A Series of Modern Surveys in Mathematics, 32. Springer-Verlag, Berlin,1996.19. M. Kontsevich and Y. Soibelman. Stability structures, motivic Donaldson-Thomas invariantsand cluster transformations. Preprint, arXiv:0811.2435.20. M. Kontsevich and Y. Soibelman. Motivic Donaldson-Thomas invariants: summary of results.Mirror symmetry and tropical geometry, 55-89, Contemp. Math., 527, Amer. Math. Soc.,Providence, RI, 2010.□

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