From Chern classes to Milnor classes – A history of characteristic classes for singular varieties

Abstract

In this paper, we give a survey and recent developments about the definitions of characteristic classes for possibly singular complex analytic (or algebraic) varieties. We recall the classical construction of characteristic classes in the case of manifolds, by obstruction theory and using Schubert cycles. Then, we present various generalizations of characteristic classes to singular varieties, due to M.H. Schwartz, W.T. Wu, J. Mather, R. MacPherson, W. Fulton and K. Johnson and we discuss relations among these definitions. More recent results concern the definition and properties of so-called Milnor classes, as developped by P. Aluffi, J.P. Brasselet-D. Lehmann-J. Seade-T. Suwa, A. Parusiński-P. Pragacz and S. Yokura.