On the Zeroes of Meromorphic Vector-Fields

Abstract

Let M be a compact complex analytic manifold and let x be a holomorphic vector-field on M. In an earlier paper by one of us (see [2]) it was shown that the behavior of x near its zeroes determined all the Chern numbers of M and the nature of this determination was explicitly given where x had only nondegenerate zeroes. The primary purpose of this note is to extend this result to meromorphic fields, or equivalently to sections s of T⊗L where T is the holomorphic tangent bundle to M and L is a holomorphic line bundle. We will also drop the non-degeneracy assumption of the zeroes of s, but we treat only the case where s vanishes at isolated points {p}.

This research was supported by grants from the National Science Foundation. The first author is partially supported by GP-6571, the second by GP-6585.