Toric surfaces, vanishing Euler characteristic and Euler obstruction of a function
For a normal toric surface \(X_{\sigma}\), the notion of the vanishing Euler characteristic is introduced here as \(V(X_{\sigma} ):= \chi ( X(\Delta )) -1\), where \(\Delta\) denotes a fan refining \(\sigma\) in an appropriate way related to a Hirzebruch-Jung continued fraction. This continued fraction gives rise to a simple explicit formula for \(V(X_{\sigma} )\). Furthermore, this number is related to the 2nd polar multiplicity of \(X_{\sigma}\). The notion of the vanishing Euler characteristic can be interpreted in the following way: Let \(Y\) be the generic fibre of the smoothing associated to the minimal resolution of the toric surface \(X_{\sigma}\) with isolated singularities. Then \(\beta_1(Y)=0\) by \textit{G.-M. Greuel} and \textit{J. Steenbrink} [Proc. Symp. Pure Math. 40, 535--545 (1983; Zbl 0535.32004)], and \(\dim H_2(Y) = \chi (Y)-1\). Thus, in the case that \(X_{\sigma}\) has a unique smoothing the vanishing Euler characteristic of \(X_{\sigma}\) turns out to be the Milnor number. Let \(f:(X,0) \to ({\mathbb C},0)\) be a germ of an analytic function with isolated singularity at the origin, where \((X,0)\) is a germ of an analytic singular space embedded in \({\mathbb C}^n\). Then the Euler obstruction of \(f\) is defined by Brasselet, Massey, Parameswaran and Seade [\textit{J. P. Brasselet} et al., J. Lond. Math. Soc., II. Ser. 70, No. 1, 59--76 (2004; Zbl 1052.32026)]. By Seade, Tibar, Verjovsky [\textit{J. Seade} et al., Bull. Braz. Math. Soc. (N.S.) 36, No. 2, 275--283 (2005; Zbl 1082.32018)] this invariant is essentially the number of Morse points of a Morsification of \(f\) on the regular part of \(X\). Thus it may be considered a generalization of the Milnor number of \(f\). The article under review gives some formula for the Euler obstruction of a function \(f: X_{\sigma} \to {\mathbb C}\) with isolated singularitiy at \(0\) and a formula for the difference between the Euler obstruction of \(X_{\sigma}\) and the Euler obstruction of \(f\). The authors point out that the latter may be interesting also in the case \(f\) has a non-isolated singularity (as noticed in [\textit{N. Dutertre} and \textit{N. G. Grulha}, Adv. Math. 251, 127--146 (2014; Zbl 1291.14009)], [Zbl 0535.32004]). As a special case, the Euler obstruction of \(f\) is computed for certain polynomials on a family of determinantal surfaces.
scientific article; zbMATH DE number 6418000
scientific article