Categorification of Donaldson-Thomas invariants via perverse ...

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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 .
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