Categorification of Donaldson-Thomas invariants via perverse ...

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