Motivic invariants of rigid varieties, and applications to complex singularities (Q2900495)

The motivic integration was introduced by M. Kontsevich (Lecture at Orsay, 1995) in an attempt to prove some results about the topology of Calabi-Yau varieties. Originally formulated over smooth varieties, it has been developed further for some class of varieties with mild singularities in the works of [\textit{J. Denef} and \textit{F. Loeser}, Invent. Math. 135, No. 1, 201--232 (1999; Zbl 0928.14004); Prog. Math. 201, 327--348 (2001; Zbl 1079.14003); Duke Math. J. 99, No. 2, 285--309 (1999; Zbl 0966.14015); J. Algebr. Geom. 7, No. 3, 505--537 (1998; Zbl 0943.14010)]. After that more generalizations have been proposed, one among them being the motivic integration over formal schemes and rigid spaces [\textit{F. Loeser} and \textit{J. Sebag}, Duke Math. J. 119, No. 2, 315--344 (2003; Zbl 1078.14029); \textit{J. Nicaise}, Math. Ann. 343, No. 2, 285--349 (2009; Zbl 1177.14050)]. The article under review gives an introduction to this theory, with some applications to complex hypersurface singularities and rational points on varieties over a field with a non-archimedean valuation. We are working in the category of special formal schemes. For \(R\) a Noetherian adic ring, a special formal scheme over \(R\) is a separated Noetherian adic formal scheme \(\mathcal{X} \rightarrow Spf R\), such that the set \(V(\mathcal{I})\) is of finite type over \(R\) for any ideal of definition \(\mathcal{I}\). Such a scheme \(\mathcal{X}\) is separated of topologically finite type (stft) if \(\mathcal{X} \rightarrow Spf R\) is an adic morphism.NEWLINENEWLINE The Greenberg schemes [\textit{M. J. Greenberg}, Ann. Math. (2) 73, 624--648 (1961; Zbl 0115.39004); Ann. Math. (2) 78, 256--266 (1963; Zbl 0126.16704)] play the role of the arc spaces in the classic motivic integration.NEWLINENEWLINE For \((R, m)\) a DVR with residue field \(k\) and quotient field \(K\), let \(R_n = R/m^{n+1}\), and let \(X\) be a separated \(R_n\)-scheme of finite type. Using the \(R_n\)-algebra \(\mathcal{R}_n(A)\) (defined as \(R \otimes_k A\), if \(R\) has equal characteristic), there is a functor \(\mathcal{A}lg_k \rightarrow \mathcal{S}ets, A \mapsto X(\mathcal{R}_n(A))\). It could be shown that this functor is represented by a separated scheme of finite type \(Gr_n(X)\), the Greenberg scheme of \(X\). If \(\mathcal{X}\) is stft formal scheme, the Greenberg scheme of \(\mathcal{X}\) at level \(n\) is \(Gr_n(\mathcal{X} \times_R R_n)\), and there are truncation morphisms \(GR_n(X) \rightarrow GR_m(X), n \geq m\), as in the case of jet schemes. The fact these are affine morphisms permits to define \(\mathrm{Gr}(\mathcal{X}) = \lim_{\leftarrow n} GR_n(\mathcal{X})\), the Greenberg scheme of \(\mathcal{X}\), which is not Noetherian in general. It could be shown that it parametrizes the étale sections on \(\mathcal{X}\). Also, there are natural morphisms \(\theta_n \colon \mathrm{Gr}(\mathcal{X}) \rightarrow GR_n(\mathcal{X})\), and any morphism of stft formal schemes \(g \colon \mathcal{Y} \rightarrow \mathcal {X}\) induces a morphism of the corresponding Greenberg schemes \(\mathrm{Gr}(g)\). The schemes \(GR_n(\mathcal{X}), \mathrm{Gr}(\mathcal{X})\) over \(\mathcal{X}\) share the main properties of the jet schemes and arc spaces over \(X\).NEWLINENEWLINE The ring in which the motivic integral takes its value, when \(R\) has equal characteristic, is the localization of \(K_0(Var_X)\), the Grothendieck ring of varieties over \(X\), by the class of the line over \(X\) (in mixed characteristic is used the modified Grothendieck ring of varieties over \(X\)). For \(\mathcal{X}\) a smooth stft formal \(R\)-scheme, a cylinder set of level \(n\) is \(C = \theta^{-1}_n(C_m)\) for some constructible \(C_m \subset Gr_m(\mathcal{X})\). Then the motivic measure \(\mu_{\mathcal{X}}(C)\) is defined (for \(\mathcal{X}\) of pure relative dimension), which does not depend on \(n\). An integrable function \(\beta \colon \mathrm{Gr}(\mathcal{X}) \rightarrow \mathbb{Z} \cup \{\infty\}\) is one which takes finitely many values, and all its fibers are cylinders. For such \(\beta\) the motivic integral \(\int_{\mathrm{Gr}(\mathcal{X})} {\mathbb{L}}^{\beta}\) is defined, belonging to the localization of \(K_0(Var_{\mathcal{X}_0})\), where \(\mathcal{X}_0\) is the reduction of \(\mathcal{X}\). A typical example is \(\int_{\mathrm{Gr}(\mathcal{X})} {\mathbb{L}}^{-\mathrm{ord}(\omega)}\), where \(\omega\) is nowhere vanishing differential form of maximal degree, defined on each connected component of \(\mathcal{X}_{\eta}\), and called a gauge form.NEWLINENEWLINE The main tool in any theory of motivic integration is the change of variables. Let \(h \colon \mathcal{Y} \rightarrow \mathcal{X}\) be morphism of smooth stft formal \(R\)-schemes, inducing \(\mathrm{Gr}(h) \colon \mathrm{Gr}(\mathcal{Y}) \rightarrow \mathrm{Gr}(\mathcal{X})\). After defining the notion of the Jacobian of \(h\) is given a corrected form of the original change of variables [\textit{J. Sebag}, Bull. Soc. Math. Fr. 132, No. 1, 1--54 (2004; Zbl 1084.14012)].NEWLINENEWLINE The dilatation of a flat special formal \(R\)-scheme is a tool, permitting to reduce some constructions from special formal schemes to stft formal schemes [\textit{J. Nicaise}, Math. Ann. 343, No. 2, 285--349 (2009; Zbl 1177.14050)]. It is related with the notions of Néron smoothening of a special formal scheme, and that of weak Néron model for rigid variety. After their definitions given with some basic properties, for generically smooth special formal \(R\)-scheme \(\mathcal{X}\), with \(\mathcal{X}_{\eta}\) admitting a gauge form \(\omega\), is defined the motivic integral \({\int}_{\mathcal{X}} |\omega|\). For given \(X\) a separated smooth rigid \(K\)-variety, admitting a weak Néron model, and a gauge form \(\phi\), is defined the motivic integral \(\int_X |\omega|\), showing that it is independent of the choice of the weak Néron model.NEWLINENEWLINEThe applications to rational points on rigid variety are using the motivic Serre invariant [\textit{F. Loeser} and \textit{J. Sebag}, Duke Math. J. 119, No. 2, 315--344 (2003; Zbl 1078.14029)]. It gives a motivic measure of the set of rational points on a separated rigid \(K\)-variety, admitting a weak Néron model, and could be modified for algebraic varieties as well. Then is formulated a trace formula J. Nicaise [Zbl 1177.14050], which provides a cohomological interpretation of it. In the case of analytic manifolds and \(p\)-adic integration, the \(p\)-adic Serre invariant is also defined. By a result of Serre, it classifies the \(K\)-analytic manifolds, and the motivic Serre invariant specializes to it.NEWLINENEWLINEFor the applications of this theory of motivic integration to singularities, a central notion is that of the analytic Milnor fiber. If \(Z\) is a complex manifold, \(g\) is analytic function on \(Z\), and \(g(x) = 0\), take \(B\) to be an open ball centered at \(x\), and \(D^*\) to be a puncured open disc at \(0 \in \mathbb{C}\), both small enough. Then at \(x\) is defined the Milnor fibration \(g_x \colon g^{-1}(D^*) \cap B \rightarrow D^*\). There is a monodromy action on the singular cohomology \(\sum_n H^n_{\mathrm{sing}}(F_x, \mathbb{Z})\) of the universal fiber of the fibration \(F_x\). An eigenvalue of the monodromy is any \(\alpha \in \mathbb{C}\), which is eigenvalue of the monodromy on \(H^n_{\mathrm{sing}}(F_x, \mathbb{Z})\) for some \(n\). The monodromy theorem states that all eigenvalues are roots of unity.NEWLINENEWLINE Let \(X\) be smooth irreducible variety over \(k\), and \(f \colon X \rightarrow {\mathbb{A}}^1_k\) a non-constant morphism. Then is defined the motivic zeta function \(Z_f(T)\) associated to \(f\), and by a result of Denef and Loeser, when \(\mathrm{char}k = 0\), \(Z_f(T)\) is a rational function [Zbl 1079.14003]. They proposed also the motivic monodromy conjecture, relating the poles of \(Z_f(T)\) with the monodromy eigenvalues of \(f\), generalizing the Igusa's conjecture about the \(p\)-adic zeta function of \(f\) over a number field.NEWLINENEWLINE The analytic Milnor fiber is a rigid \(k((t))\)-variety, which is the non-archimedean model of the topological Milnor fibration. It contains information about the invariants of the singularity of an \(f\) as above. The points on the analytic Milnor fiber can be described in terms of some subsets of the arc space \(X_{\infty}\). In this way it connects motivic zeta functions, arc spaces and the monodromy.NEWLINENEWLINEFor the entire collection see [Zbl 1241.14001].