Functions and vector fields on C(ℂPn)-singular manifolds

Abstract

In this paper we study functions and vector fields with isolated singularities on a C(ℂPn)-singular manifold. In general, a C(ℂPn)-singular manifold is obtained from a smooth (2n + 1)-manifold with boundary which is a disjoint union of complex projective spaces Claro ℂPn ∪ … ∪ ℂPn and subsequent capture of the cone over each component ℂPn of the boundary. We calculate the Euler characteristic of a compact C(ℂPn)-singular manifold M2n+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(ℂPn)-singular manifold.