Categorification of Donaldson-Thomas invariants via perverse ...

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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,
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