L\^e cycles and Milnor classes

Abstract

The purpose of this work is to establish a link between the theory of Chern classes for singular varieties and the geometry of the varieties in question. Namely, we show that if $Z$ is a hypersurface in a compact complex manifold, defined by the zero-scheme of a nonzero holomorphic section of a very ample line bundle, then its Milnor classes, regarded as elements in the Chow group of $Z$, determine the global L\^e cycles of $Z$; and viceversa: The L\^e cycles determine the Milnor classes. Morally this implies, among other things, that the Milnor classes determine the topology of the local Milnor fibres at each point of $Z$, and the geometry of the local Milnor fibres determines the corresponding Milnor classes.