Euler obstruction and indices of vector fields
Let \((X,0)\subset (\mathbb{C}^N,0)\) be the germ of an equidimensional complex analytic singularity and \(V_i\) \(1\leq i\leq x\), the connected strata of a Whitney stratification of a small representative of \((X,0)\). Here the authors prove the existence of a Zariski open dense subset \(\Omega\) in the space of complex linear forms on \(\mathbb{C}^N\) such that for every \(f\in\Omega\) there is \(\varepsilon_0>0\) such that for any \(\varepsilon\) with \(0<\varepsilon < \varepsilon_0\) and \(t\neq 0\) sufficiently small \[ Eu(X,0)= \sum_{1\leq i\leq x}\chi \bigl(V_i\cap {\mathbf B}_\varepsilon\cap f^{-1}(t) \bigr)Eu_{V_i}(X) \] where \(Eu (X,0)\) is the Euler obstruction of \(X\) at 0, \({\mathbf B}_\varepsilon\) is the \(\varepsilon\)-ball around \(0,\chi\) denotes the Euler-Poincaré characteristic and \(Eu_{V_i}(X)\) is the value of the Euler obstruction of \(X\) at any point of \(V_i\).
scientific article; zbMATH DE number 1526236
scientific article