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

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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) = ∑ = 1 ind(x i ) + ∑ = 1 ind(m i ).