Index of a singular point of a gradient vector field
Let \(f: ({\mathbb{C}}^ n,0)\to ({\mathbb{C}},0)\) be a germ of a real (i.e. with real values on \({\mathbb{R}}^ n\subset {\mathbb{C}}^ n)\) holomorphic function with isolated critical point in the origin. The main result of the paper is a formula which expresses the index ind f of the singular point of the gradient vector field of the germ f (on the space \({\mathbb{R}}^ n)\) in terms of the action of the complex conjugation on the homology groups of Milnor fibres of the germ f. Let \(V_ z=\{x\in {\mathbb{C}}^ n:\quad f(x)=z,\quad \| x\| \leq \rho \}\) (z\(\in {\mathbb{C}}\), \(0<\| z\| <<\rho\), \(\rho\) is small enough) be the Milnor fibre of the germ f, \(\epsilon\in {\mathbb{R}}\) positive and small enough. The complex conjugation induces involutions on the Milnor fibres \(V_{\epsilon}\) and \(V_{- \epsilon}\). Let \(\sigma_+\) and \(\sigma_-\) be their actions on the homology groups \(H_{n-1}\) \((V_{\pm \epsilon};{\mathbb{R}})\). The bilinear forms \(Q_{\pm}(a,b)=<\sigma_{\pm}a,b>\) \((a,b\in H_{n-1}(V_{\pm \epsilon};{\mathbb{R}})\), \(<\), \(>\) is the intersection form on the homology group) are symmetric. - Theorem: \(ind f=(-1)^{(n+1)/2}(sgn Q_--sgn Q_+/2\) for n odd; \(sgn Q_+=-sgn Q_-\) for n even.
index of a singular point of a gradient vector field
scientific article; zbMATH DE number 3884472