On the Euler characteristic of complex algebraic varieties (Q1821160)

Let \(h_ 1,...,h_ s\in {\mathbb{C}}[z_ 0,...,z_ n]\) be homogeneous polynomials. Set \(X=\{z\in {\mathbb{C}}P^ n: h_ 1(z)=...=h_ s(z)=0\}.\) In this paper is presented a construction of a family of polynomials \(G_ i: ({\mathbb{R}}^{2i+1},0)\to ({\mathbb{R}},0),\) \(0\leq i\leq n\), with isolated singular points at the origin, such that the Euler characteristic \(\chi(X)=(n+1-\sum ^{n}_{i=0}\deg (dG_ i)),\) where \(\deg (dG_ i)\) is the degree of the mapping \(x\mapsto \text{grad}(G_ i(x))/\| \text{grad}(G_ i(x))\|\) from a small sphere \(S_ r^{2i}\) centered at the origin to the unit sphere of \({\mathbb{R}}^{2i+1}\).