GENERALIZATION OF VARIATIONS AND BAUM–BOTT RESIDUES FOR HOLOMORPHIC FOLIATIONS ON SINGULAR VARIETIES

Abstract

There are already several kinds of residues for holomorphic singular foliations on singular varieties. In this paper, combining the two techniques of using coherent sheaves and the extension of the normal bundle to a local complete intersection, we would like to define: — A generalization to the case of arbitrary dimension, of the variation, defined in [15] for foliations by complex curves on complex surfaces, — and, closely related to the previous one, a generalization to the case of foliations on singular varieties, of the usual Baum-Bott residues for singular holomorphic foliations on complex manifolds. Here, by a (singular) foliation on a singular subvariety V of a complex manifold M, we mean a foliation Ty on the regular part of V, that we assumed to be induced by a (singular) foliation f o n M leaving V invariant; in fact, the data is not only !Fy, but also the germ of T along V.