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

Alice Kimie Miwa Libardi, Vladimir V. Sharko

DOI: http://dx.doi.org/10.12775/TMNA.2015.081

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 Poincare-Hopf Index Theorem for an almost smooth vector field with finite number of zeros on a~$C(\mathbb{C}P^n)$-singular manifold.

Keywords


Semi-free circle action; manifold; $S^1$-invariant Bott function; Morse number; Poincare-Hopf index

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