Categorification of Donaldson-Thomas invariants via perverse ...

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