Characteristic classes of complex hypersurfaces

Abstract 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.