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We study the formal neighborhoods at rational non-degenerate arcs of the arc scheme associated with a toric variety. The first main result of this article shows that these formal neighborhoods are generically constant on each Nash component of the variety. Furthermore, using our previous work, we attach to every such formal neighborhood, and in fact to every toric valuation, a minimal formal model (in the class of stable isomorphisms) which can be interpreted as a measure of the singularities of the base-variety. As a second main statement, for a large class of toric valuations, we compute the dimension and the embedding dimension of such minimal formal models, and we relate the latter to the Mather discrepancy. The class includes the strongly essential valuations, that is to say those the center of which is a divisor in the exceptional locus of every resolution of singularities of the variety. We also obtain a similar result for monomial curves.

In this paper, we compute the Artin part of a relative cohomological motive, introduced by Ayoub and Zucker, as a “weight zero part” in two challenging contexts. For this, we first introduce, in a very natural way, the part of punctual weight $\leqslant0$ of any complex of mixed Hodge modules and verify that the Hodge realization of the Artin part of smooth cohomological motives coincide with the part of punctual weight $\leqslant0$ of its realization. Second, we compute the Artin part of the motivic nearby sheaf, introduced by Ayoub, and relate it to the Betti cohomology of Berkovich spaces defined by tubes in non-Archimedean geometry. In particular, the former result provides a motivic interpretation of the Betti cohomology of the analytic Milnor fiber (seen as a Berkovich space).

Let k be a field of characteristic zero, R D k[[t]] the ring of formal power series and K D k((t)) its fraction field. Let X be a finite type R-scheme with smooth generic fiber. Let H be the t-adic completion of X and Hη the generic fiber of H. Let Z ⊂ Xσ bea locally closed subset of the special fiber of X. In this article, we establish a relation between the rigid motive of [Z] (the tube of Z in Hη) and the restriction to Z of the nearby motivic sheaf associated with the R-scheme X. Our main result, Theorem 7.1, can be interpreted as a motivic analog of a theorem of Berkovich. As an application, given a rational point x ∈ Xσ, we obtain an equality, in a suitable Grothendieck ring of motives, between the motivic Milnor fiber of Denef-Loeser at x and the class of the rigid motive of the analytic Milnor fiber of Nicaise-Sebag at x.

Let k be a field of characteristic 0. Let f∈k[X0,Y0] be a non-constant reduced polynomial. In this article we study the differential ideal {f}:=[f] of the differential ring k{X0,Y0}. We introduce the S-algorithm which allows, for every differential polynomial P∈k{X0,Y0}, to test if it belongs or not to the differential ideal {f}:=[f]. Another consequence of this study is to establish a precise characterization of the reducedness of the arc scheme associated with a plane curve X, defined over k, in terms of the geometry of X.

Due to exacerbated inflammation and recurring tissue injury patients with Inflammatory Bowel Disease (IBD) are at higher risk of developing colorectal cancer (CRC). Although, PMN infiltration of the intestinal mucosa is a hallmark of IBD, and is associated with tissue injury, the contribution of PMNs to CRC and more so to IBD associated CRC is not known. We recently showed that PMNs can acutely exacerbate tissue injury and promote genomic instability by promoting miR-23a and miR-155-dependent accumulation of double-strand breaks (DSBs) and inhibition of DSB-repair by homologous recombination (HR). We now demonstrate that in chronic gut inflammation and recurring epithelial injury, as seen in IBD, via similar miRNA-dependent mechanism, PMNs alter DNA damage responses (DDR) to promote CRC progression. In murine model of Colitis-associated CRC (Azoxymethane, AOM/Dextran sodium sulfate, DSS) and human CRC xenografts, intratumoral PMNs intriguingly, functional dualism, suppressing tumor onset, but promoting tumor cell survival and growth in progressive tumors. The observed PMN effects were mediated by persistent suppression of HR-mediated DSB repair. HR suppression by PMNs, initially resulted in elevated replication stress and increased tumor cell apoptosis, however, longer term, altered DDR transcriptional profile and facilitated the upregulation of DSB repair by non-homologous end-joining (NHEJ). NHEJ upregulation enhanced CRC progression and tumor cell survival. PMN depletion, CRSPER-mediated deletion of miR-155 responsive sequence in RAD51 (preserves HR activity) or inhibition of NHEJ by small molecule inhibitors increased tumor cell death and diminished tumor development. Collectively, our data define a novel link between PMN-mediated mucosal injury and colon carcinogenesis.

In this note we identify the classes of Q-homological planes in the Grothendieck group of complex varieties K0(VarC). Precisely, we prove that a connected, smooth, affine, complex, algebraic surface X is a Q-homological plane if and only if [X] = in the ring K0(VarC) and Pic(X)Q := Pic(X) ⊗z Q = 0.

In this chapter, we define the Grothendieck ring of varieties over an arbitrary base scheme. This is a ring of virtual varieties up to cut-and-paste relations; it takes a central place in the theory of motivic integration, because (after a suitable localization and/or completion) it serves as the ring where motivic integrals take their values. After the basic definitions in section 1, we define the notion of motivic measures, which are ring morphisms from the Grothendieck ring to other rings with a more explicit structure. Motivic measures are fundamental both for the understanding of Grothendieck ring itself and for extracting geometric information from its elements. Among the motivic measures, we develop in sections 3 and 5 the cohomological and motivic realizations. In sections 5 and 6, we study the main structure theorems for the Grothendieck ring over a field of characteristic zero: the theorems of Bittner and Larsen-Lunts. Bittner’s theorem gives a presentation of the Grothendieck ring in terms of smooth projective varieties and blow-up relations, which is quite useful to construct motivic measures. The theorem of Larsen and Lunts relates equalities in the Grothendieck ring to the notion of stable birational equivalence. In section 4 we discuss a process of dimensional completion for the Grothendieck ring of varieties.

Motivic integration and some of its applications take they very inspiration from results of p-adic integration, that is, integration on analytic manifolds over non-Archimedean locally compact fields.

Throughout this chapter, we denote by R a complete discrete valuation ring with maximal ideal \(\mathfrak{m}\) and residue field k. For every integer n⩾0, we set \(R_{n} = R/\mathfrak{m}^{n+1}\).

In this paper, we establish a criterion for every complete separated linearly topologized ring to be isomorphic to a ring of power series, i.e. to be a cylinder over another complete topological ring. We use this criterion to establish a general statement of cancellation in the category of local complete but non-necessarily adic topological [Formula: see text]-algebras.

In this article, we introduce the notion of A-approximations associated with a polynomial differential equation F = 0 of order n ≥ 1 and degree d ≥ 2, defined over a differential field of characteristic zero. We also give applications of this construction to the irreducible decomposition of perfect differential ideals, generated by a single element.

In this paper, we study the influence of the Cartan–Tresse linearization polynomial of a differential equation of order one in some classical problems of analysis, differential algebra and geometry, as singular solutions, the Ritt problem, or webs theory.

Resume Nous presentons une preuve geometrique du theoreme de Kolchin qui utilise l'existence de resolutions des singularites en caracteristique nulle. Nous montrons egalement les limites de cette technique en caracteristique positive en donnant un contre-exemple. Pour citer cet article : J. Nicaise, J. Sebag, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

The development of Maxim Kontsevich's initial ideas on motivic integration has unexpectedly influenced many other areas of mathematics, ranging from the Langlands program over harmonic analysis, to non-Archimedean analysis, singularity theory and birational geometry. This book assembles the different theories of motivic integration and their applications for the first time, allowing readers to compare different approaches and assess their individual strengths. All of the necessary background is provided to make the book accessible to graduate students and researchers from algebraic geometry, model theory and number theory. Applications in several areas are included so that readers can see motivic integration at work in other domains. In a rapidly-evolving area of research this book will prove invaluable. This first volume contains introductory texts on the model theory of valued fields, different approaches to non-Archimedean geometry, and motivic integration on algebraic varieties and non-Archimedean spaces.

We show how formal and rigid geometry can be used in the theory of complex singularities, and in particular in the study of the Milnor fibration and the motivic zeta function. We introduce the so-called analytic Milnor fiber associated to the germ of a morphism f from a smooth complex algebraic variety X to the affine line. This analytic Milnor fiber is a smooth rigid variety over the field of Laurent series C((t)). Its etale cohomology coincides with the singular cohomology of the classical topological Milnor fiber of f; the monodromy transformation is given by the Galois action. Moreover, the points on the analytic Milnor fiber are closely related to the motivic zeta function of f, and the arc space of X. We show how the motivic zeta function can be recovered as some kind of Weil zeta function of the formal completion of X along the special fiber of f, and we establish a corresponding Grothendieck trace formula, which relates, in particular, the rational points on the analytic Milnor fiber over finite extensions of C((t)), to the Galois action on its etale cohomology. The general observation is that the arithmetic properties of the analytic Milnor fiber reflect the structure of the singularity of the germ f., Comment: Some minor errors corrected. The original publication is available at http://www.springerlink.com

We construct a theory of motivic integration for smooth rigid varieties. As an application new invariants of degenerations are obtained., Comment: 24 pages