Functions and vector fields on $C(\mathbb{C}P^n)$-singular manifolds

Abstract

In this paper we study functions and vector fields with isolated singularities on a $C(\mathbb{C}P^n)$-singular manifold. In general, a $C(\mathbb{C}P^n)$-singular manifold is obtained from a smooth $(2n+1)$-manifold with boundary which is a disjoint union of complex projective spaces $\mathbb{C}P^n \cup\ldots \cup\mathbb{C}P^n$ and subsequent capture of the cone over each component $\mathbb{C}P^n$ of the boundary. We calculate the Euler characteristic of a compact $C(\mathbb{C}P^n)$-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 $C(\mathbb{C}P^n)$-singular manifold.