Categorification of Donaldson-Thomas invariants via perverse ...

CATEGORIFICATION OF DONALDSON-THOMAS INVARIANTS 17Writing z k = u k + iv k , we can view D as an open subset of R 2n , where R 2nis with the coordinate variable (u 1 , · · · , u n , v 1 , · · · , v n ). By allowing u k and v k totake complex values, we embed R 2n ⊂ C 2n , thus embed D ⊂ C 2n as a (totally real)analytic subset. We call an open D C ⊂ C 2n a complexification of D if D C ∩R 2n = D.We use w to denote the complex coordinate variables of C 2n .Lemma 4.5. We can find a complexification D C ⊃ D such that the function a 0 (z)extends to a holomorphic a 0 (·) C : D C → Ω 0,1 (adE).Proof. The extension is standard. Since a 0 (z) is derived from the JS chart, it isholomorphic in z. Thus for any α = (α 1 , · · · , α n ) ∈ D, a 0 (z) equals to a convergentpower series in (z k −α k ) in a small disk centered at α with coefficients in Ω 0,1 (adE).Letting α k = a k + ib k , a k , b k ∈ R, and writing z k = u k + iv k , the power seriesbecomes a power series in (u k − a k ) and (v k − b k ).Because u k and v k are complex coordinate variables of C 2n ⊃ R 2n ⊃ D, a 0 (z)extends to a holomorphic Ω 0,1 (adE)-valued function in a small neighborhood of αin C 2n . Because the extension of a function defined on an open subset of R 2n toa germ of holomorphic function on C n is unique, the various extensions of a 0 (z)using power series expansions at various α ∈ D give a single extension of a 0 (z) toa holomorphic a 0 (w) C on some complexifications D C ⊃ D.□For D ⊂ V 0 a neighborhood of 0 = x 0 ∈ V 0 , we denoteO 0 := D ∩ X f0 = D ∩ X.For x ∈ O 0 , we write ∂ ∗ x = ∂ ∗ 0 + a 0 (x) † . The extension problem for a 0 (x) † is moredelicate because it is not defined away from O 0 .Lemma 4.6. For any y ∈ Y , there is an open neighborhood S ⊂ Y of y ∈ Y and anopen neighborhood D ⊂ V 0 of 0 ∈ V 0 so that the hermitian metric h| O0×S extendsto an L l s+2 hermitian metric on E D×S := E D | D×S , analytic in z ∈ D.Proof. Let S ⊂ Y be an open neighborhood of y so that S is biholomorphic to theunit ball in C 3 , and that E| ∼ O0×S = O ⊕rO and H| ∼ 0×S S = O S .We let k S be the hermitian norm of 1 in O S∼ = H|S of the hermitian metric ofH fixed earlier. Then k S is a smooth positive function on S. We let s 1 , · · · , s r bethe standard basis of E| ∼ O0×S = O ⊕rO . Because E ⊂ 0×S p∗ Y H⊕N is a subvector bundleover X × Y , using H| S∼ = OS , the image of s k in p ∗ Y H⊕N | O0×S has the presentations k = (s k,1 , · · · , s k,N ), where s k,j ∈ Γ(O O0×S). Then the hermitian metric form ofh on E| O0×S in the basis s 1 , · · · , s r takes the form(4.3) h(s k , s l ) = k S · ∑s k,j s l,j .To extend this expression over D×S, we will modify the semiconnection ∂ D×S :=∂ 0 + ∂ z + a 0 | D×S to an integrable semiconnection ∂ ′ D×S and extend s 1 , · · · , s r toholomorphic sections of (E D×S , ∂ ′ D×S).Let m ⊂ O D be the maximal ideal generated by z 1 , · · · , z n , and let I ⊂ O D bethe ideal sheaf of D ∩ X ⊂ D. Then F 0,2 ≡ 0 mod I. We construct ∂ ′ ∂ 0+a 0D×S bypower series expansion. We let s ′ = s + 2, and set b 0 (z) = 0. Suppose we havefound b k (z) ∈ Ω 0,1 (adE| S ) s ′ ⊗ C I such that(4.4) ∂ 0 b k (z) ∈ Ω 0,1 (adE| S ) s ′ ⊗ C I and F 0,2∂ 0+a 0(z)+b k (z) ≡ 0 mod mk ∩ I,j