Contact loci, motivic Milnor fibers of nondegenerate singularities (Q1988506)

Let \(f \in \mathbb{C}[x_1, \dots, x_d]\) with \(f(O) = 0\) be nondegenerate with respect to its Newton polyhedron \(\Gamma\). This means that for any compact face \(\gamma \subset \Gamma\) the corresponding function \(f_{\gamma} = \Sigma_{\alpha \in \gamma}c_{\alpha} x^{\alpha}\) is smooth in \((\mathbb{C}^*)^d\). For \(n \in \mathbb{N}\) the \(n\)-iterated contact locus of \(f\) at \(O\) is defined as \(\mathfrak{X}_{n, O}(f) = \{\alpha \in (t\mathbb{C}[t]/t^{n+1})^d : f(\alpha) = t^n \pmod {t^{n+1}}\} \). Let \(\textit{S}_{f, O}\) be the motivic Milnor fiber of \(f\) at \(O\), that is, \[\textit{S}_{f, O}:= -\lim_{T \rightarrow \infty}Z_{f, O} \in K_0(Var_{\mathbb{C}, \hat{\mu}})[\mathbb{L}^{-1}]\] of the motivic zeta function of \(f\) at \(O\), where \[\hat{\mu} = \lim_{\leftarrow}\mu_n\] is the limit of the groups of roots of 1. A main result in the article under review claims that, given a sheaf \(\mathcal{F}\) on \(\mathfrak{X}_{n, O}(f)\) there is a spectral sequence of cohomology groups with compact support of \(\mathcal{F}\) which is degenerate at \(E_1\). This is used to calculate in the case of certain sheaves these cohomologies. The problem of computing \(\textit{S}_{f, O}\) in terms of \(\Gamma\) was proposed by \textit{G. Guibert} [Comment. Math. Helv. 77, No. 4, 783--820 (2002; Zbl 1046.14008)]. Another question, asked by \textit{Lê Dũng Tráng} is about the relation between the monodromy of \((f, O))\), and the monodromy of its restriction to a generic hyperplane \(H\) [J. Fac. Sci., Univ. Tokyo, Sect. I A 22, 409--427 (1975; Zbl 0355.32012)]. In the article that problem is explored in the case of nondegenerate \(f\) by using a refined decomposition of \(\textit{S}_{f, O}\), permitting to revisit \textit{G. Guibert}'s result [Comment. Math. Helv. 77, No. 4, 783--820 (2002; Zbl 1046.14008)] in a more general setting. A main result, considered as a motivic analogue of Lê's theorem in the case of nondegenerate \(f\), represents as a sum \[\textit{S}_{f, O} = \textit{S}_{\tilde{f}, \tilde{O}} + \textit{S}^{\Delta}_{f, x_d, O}\] the first term being the class of the motivic Milnor fiber at the origin of the restriction to the hyperplane \(x_d=0\) with \(\tilde{f} = f(x_1, \dots, x_{d-1}, 0)\), and the second term is the class of motivic Milnor fiber of the pair \((f, x_d)\) at \(O\). A detailed version with complete proofs of this article could be found in [\textit{Q. T. Lê} and \textit{T. T. Nguyen}, ``Contact loci and motivic nearby cycles of nondegenerate polynomials'', Preprint, \url{arXiv:1903.07262}]. There as a corollary of the latter theorem is obtained also a proof of Kontsevich-Soibelman's integral identity conjecture in the case of nondegenerate \(f\).