Singularities at infinity and motivic integration (Q2883361)

If given a smooth complex variety \(X\) and a non constant morphism \(f : X \to \mathbb A^1\) there exists a finite set \(S \subset \mathbb A^1\), called the bifurcation set, such that the restriction of \(f\) on \(X \setminus f ^{-1} (S)\) is a locally trivial topological fibration; the critical values of \(f\) belong to \(S\). The cohomology groups with compact support \(H_c^*(f^{-1} (t),\mathbb Q)\) have a natural mixed Hodge structure ([\textit{P. Deligne}, Publ. Math., Inst. Hautes Étud. Sci. 44, 5--77 (1974; Zbl 0237.14003)]). Then could be defined the limit mixed Hodge structure ([\textit{J. Steenbrink} and \textit{S. Zucker}, Invent. Math. 80, 489--542 (1985; Zbl 0626.14007)]) whose spectrum is an invariant of \(f\), called spectrum at infinity ([\textit{A. Dimca}, NATO Sci. Ser. II, Math. Phys. Chem. 21, 257--278 (2001; Zbl 0993.32017)], [\textit{R. García López} and \textit{A. Némethi}, Compos. Math. 100, No. 2, 205--231 (1996; Zbl 0855.32016)], [\textit{R. García López} and \textit{A. Némethi}, Compos. Math. 115, No. 1, 1--20 (1999; Zbl 0947.32014)], [\textit{A. Némethi} and \textit{C. Sabbah}, Abh. Math. Semin. Univ. Hamb. 69, 25--35 (1999; Zbl 0973.32014)]). In the article under review are introduced and studied the motivic counterparts of these and other related with them notions.NEWLINENEWLINEFor any morphism \(f : X \to\mathbb A^1\) over a field \(k\) of characteristic 0, and any closed point \(x\inf^{-1}(0)\), \textit{J. Denef} and \textit{F. Loeser} [J. Algebr. Geom. 7, No. 3, 505--537 (1998; Zbl 0943.14010); Prog. Math. 201, 327--348 (2001; Zbl 1079.14003)] introduced the motivic Milnor fiber \(S_{f,x}\) in the Grothendieck ring of varieties with a \(\mathbb G_m\) action. It specializes in the Grothendieck ring of Hodge structures with a quasi unipotent endomorphism \(K_0 (SH^{mon})\) to the class of the limit mixed Hodge structure of the Milnor fiber of \(f\) at \(x\). \textit{F. Bittner} [Math. Z. 249, No. 1, 63--83 (2005; Zbl 1085.14020)] and \textit{G. Guibert, F. Loeser} and \textit{M. Merle} [Duke Math. J. 132, No. 3, 409--457 (2006; Zbl 1173.14301)] generalized the motivic Milnor fiber as an additive morphism on the localized Grothendieck ring of varieties over \(X\). A brief review of all this together with realizations of these objects (when \(k =\mathbb C\)) in some Grothendieck rings is given in Section 1.NEWLINENEWLINEIn Section 2 is defined a compactification of \(f\), so that \(X\) could be considered as an open subset of it, and is introduced the motivic Milnor fiber at infinity \(S_{f,\infty}\). Then we have the modified motivic zeta function, associated with any compactification of \(f\), and by its rationality follows that this is an invariant of \(f\) that does not depend on the compactification chosen. Applying its realization in the Grothendieck ring of mixed Hodge modules with a quasi unipotent endomorphism are calculated the class of the mixed Hodge structure at infinity and the spectrum at infinity of \(f\). Another approach to the motivic Milnor fiber at infinity using resolutions is proposed by \textit{Y. Matsui} and \textit{K. Takeuchi} [Math. Z. 268, No. 1--2, 409--439 (2011; Zbl 1264.14005)].NEWLINENEWLINEFor a Laurent polynomial \(f\) in a few variables with coefficients in a field of characteristic 0 is defined its Newton polyhedron \(\Gamma\). In Section 3 for such an \(f\) which is nondegenerated with respect to its Newton polyhedron at infinity is calculated its motivic Milnor fiber at infinity in terms of the faces of \(\Gamma\). When \(k =\mathbb C\) is obtained the spectrum at infinity of \(f\) as a decomposition of spectra of quasi homogeneous varieties.NEWLINENEWLINEIn the last section, for a dominant \(f : X\to\mathbb A^1\) and any \(a\in \mathbb A^1\) is defined the complete motivic Milnor fiber \(S_{f,a}\) of \(f\) at \(a\), which does not depend by the compactification. It generalizes the classical ``affine'' motivic Milnor fiber. This makes it possible to introduce the global motivic vanishing cycles and motivic vanishing cycles at infinity of \(f\) at \(a\). Then appear the notions of motivic atypical values and motivically tame function, the latter being the analog of a cohomologically tame polynomial ([\textit{C. Sabbah}, Port. Math. (N.S.) 63, No. 2, 173--226 (2006; Zbl 1113.14011)], [\textit{A. Parusiński}, Banach Cent. Publ. 39, 131--141 (1997; Zbl 0882.32018)]). This permits to introduce the motivic bifurcation set of \(f\) by taking all possible compactifications and their log resolutions, and is shown that any \(f\) which is non-degenerated and convenient with respect to its Newton polyhedron is motivically tame. Also, to any compactification of \(f\) is associated a global zeta function. Then is proved its rationality and are defined the global motivic vanishing cycles. In the case of a smooth \(X\) is introduced the universal discriminant for \(f\), which contains the motivic bifurcation set. In particular, for \(f\) motivically tame this inclusion is actually an equality. Finally, in the case of a homogeneous \(f\) is proved that \(S_{f,0}\) and \(S_{f,\infty}\) replace each other by some inversion.