Characteristic classes of complex hypersurfaces

The Milnor-Hirzebruch class of a locally complete intersection X in an algebraic manifold M measures the difference between the (Poincare dual of the) Hirzebruch class of the virtual tangent bundle of X and, respectively, the Brasselet-Schurmann-Yokura (homology) Hirzebruch class of X. In this note, we calculate the Milnor-Hirzebruch class of a globally defined algebraic hypersurface X in terms of the corresponding Hirzebruch invariants of vanishing cycles and singular strata in a Whitney stratification of X. Our approach is based on Schurmann's specialization property for the motivic Hirzebruch class transformation of Brasselet-Schurmann-Yokura. The present results also yield calculations of Todd. Chern and L-type characteristic classes of hypersurfaces. (C) 2010 Elsevier Inc. All rights reserved.