Categorification of Donaldson-Thomas invariants via perverse ...

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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
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