Functions and Vector Fields on $C(ℂP^N)$-Singular Manifolds

Abstract

Let $M^{2n+1}$ be a $C(ℂP^N)$ -singular manifold. We study functions and vector fields with isolated singularities on $M^{2n+1}$. A $C(ℂP^N)$ -singular manifold is obtained from a smooth manifold $M^{2n+1}$ with boundary in the form of a disjoint union of complex projective spaces $ℂP^n ∪ ℂP^n ∪ . . . ∪ ℂP^n$ with subsequent capture of a cone over each component of the boundary. Let $M^{2n+1}$ be a compact $C(ℂP^N)$ -singular manifold with k singular points. The Euler characteristic of $M^{2n+1}$ is equal to $X\left({M}^{2n+1}\right)=\frac{k\left(1-n\right)}{2}$. Let $M^{2n+1}$ be a $C(ℂP^n)$-singular manifold with singular points $m_1 , ... ,m_k$. Suppose that, on $M^{2n+1}$, there exists an almost smooth vector field $V(x)$ with finite number of zeros $m_1 , ... ,m_k , x_1 , ... ,x_l$. Then $X(M 2n + 1) = ∑_{i = 1}^l ind(x_i ) + ∑_{i = 1}^k ind(m_i )$.