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

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) = ∑ ^l _i = 1 ind(x _i ) + ∑ ^k _i = 1 ind(m _i ).