Vector fields, residues and cohomology

Abstract

1. Residue formulas. Let E be a holomorphic vector bundle on a compact complex manifold X of dimension n with structure sheaf OX , and let E be the locally free sheaf of OX -modules (or briefly, an OX -sheaf) canonically associated to E. A residue formula for E expresses the Chern numbers of E as finite sums of residues. Recall that a Chern number is associated to a symmetric OX -linear map p : EndOX (E)⊗n → OX as follows. Letting c(E) ∈ H(X,EndOX (E)⊗ΩX)) denote the Chern class of E in the sense of [At], one may apply p to c(E) to obtain a class p(E) = p(c(E)) ∈ H(X,ΩX) which may be evaluated on the fundamental cycle of X. The number (2πi)−n ∫ X p(E) is the associated Chern number of X; we will discuss computing these numbers as sums of residues. Let ΘX denote the sheaf of sections of the holomorphic tangent bundle W of X. Let L be an invertible OX -sheaf, and assume V ∈ H(X,ΘX ⊗ L) is a section that has only isolated zeros. The zero set of V can be given the structure of a possibly unreduced scheme Z, called the zero scheme of V . Namely, Z is the finite subscheme of X defined by the sheaf of ideals IZ = i(V )(ΩX ⊗ L−1) ⊂ OX , where, i(V ) : ΩX ⊗ L−1 → OX denotes the canonical contraction operator defined by viewing V as an operator V : ΩX → L (so i(V ) = V ⊗ 1). The structure sheaf of Z is denoted by OZ . Thus OZ := OX/IZ . Letting LZ := L ⊗ OZ denote the pull back to Z of L, there is a canonical C-linear map ResV : H(Z,LZ) → C called the Grothendieck residue morphism [C1], [CL2], which is based on [Be], [H], [V] (also see [L]). A residue formula for a pair (p,E) as above will consist of using V to associate a natural class pZ(E) ∈ H(Z,LZ) to p(E) (the localization to Z) such that