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Toric surfaces, vanishing Euler characteristic and Euler obstruction of a function. (English. French summary) Zbl 1333.14005

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 G.-M. Greuel and 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 [J. P. Brasselet et al., J. Lond. Math. Soc., II. Ser. 70, No. 1, 59–76 (2004; Zbl 1052.32026)]. By Seade, Tibar, Verjovsky [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 [N. Dutertre and 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.

MSC:

14B05 Singularities in algebraic geometry
32S05 Local complex singularities
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14M12 Determinantal varieties
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