Lê Cycles and Milnor Classes of Compact Hypersurfaces

We determine the relation amongst the global Lê cycles and the Milnor classes of analytic hypersurfaces defined by a section of a very ample line bundle over a compact complex manifold. The key point is finding appropriate expressions for the global Lê cycles and for the Milnor classes in terms of polar varieties. Our starting points are an interpretation of the Lê cycles given by T. Gaffney and R. Gassler, a formula by A. Parusinski and P. Pragacz for the Milnor classes via McPherson’s functor, and a conjecture of J.-P. Brasselet, that we prove, stating that Milnor classes can be expressed in terms of polar varieties. We then use the work by R. Piegne for Mather classes, by J. Schürmann and M. Tibăr for MacPherson’s classes for constructible functions, and by D. Massey for an extension of the local Lê cycles for constructible sheaves.