On Verdier's specialization formula for Chern classes

Abstract

Let M be a complex manifold of complex dimension m, and let X C M be a proper analytic subvariety. Let D C ~ be the unit disk, and f : X ~ D a proper analytic morphism. Put X~: = f l(s), s e D. We know that, for r > 0 sufficiently small, the inclusion Xo C f I(D,) is a homotopy equivalence. Thus for small nonzero s e D we may define the specialization map on homology a. :H,(X~)~H.(Xo) as the composition ~, H,(X~) , H , ( f I(D,)) ~ n,(Xo).