Hypersurface Singularities. Nearby and Vanishing Cycles

Abstract

Some of the early applications of the theory of perverse sheaves appear in Singularity Theory, for the study of complex hypersurface singularities. In Section 10.1 we give a brief overview of the local topological structure of hypersurface singularities, as originally described by Milnor. In Section 10.2 we investigate the global topology of complex hypersurfaces by means of invariants inspired by knot theory. The nearby and vanishing cycle functors, introduced in Section 10.3, are used to glue the local topological data around the singularities. Concrete applications of nearby and vanishing cycles are presented in Section 10.4 (to the computation of Euler characteristics of complex projective hypersurfaces), in Section 10.5 (for obtaining generalized Riemann–Hurwitz-type formulae), and in Section 10.6 (for deriving homological connectivity statements for the local topology of complex singularities).