On indices of meromorphic 1-forms

Abstract

We discuss meromorphic 1-forms on compact complex manifolds and on complete intersections with isolated singularities. We give a Poincaré-Hopf type formula for them, i.e. we express the Euler characteristic in terms of singularities of a meromorphic 1-form. For that we introduce a suitable notion of an index of a germ of a meromorphic 1-form (with an additional structure) on an isolated complete intersection singularity. For a meromorphic 1-form on a complete intersection V with isolated singularities the indices of the singular points sum up to (plus-minus) the Euler characteristic of a smoothing of V. © Foundation Compositio Mathematica 2004.