Functions and Vector Fields on C(CP^n)-singular manifolds

Abstract

In this paper we study functions and vector fields with isolated singularities on a C(CP n)-singular manifold. In general, a C(CP n)-singular manifold is obtained from a smooth (2n+1)-manifold with boundary which is a disjoint union of complex projective spaces CP n ∪. .. ∪ CP n and subsequent capture of the cone over each component CP n of the boundary. We calculate the Euler characteristic of a compact C(CP n)-singular manifold M 2n+1 with finite isolated singular points. We also prove a version of the Poincaré-Hopf Index Theorem for an almost smooth vector field with finite number of zeros on a C(CP n)-singular manifold.