Looijenga motivic measures

We obtain a formula for the generating series of the push-forward under the Hilbert—Chow morphism of the Hirzebruch homology characteristic classes of the Hilbert schemes of points for a smooth quasi-projective variety of arbitrary pure dimension. This result is based on a geometric construction of a motivic exponentiation generalizing the notion of motivic power structure, as well as on a formula for the generating series of the Hirzebruch homology characteristic classes of symmetric products.

As corollaries, we obtain counterparts for the MacPherson and Aluffi Chern classes of Hilbert schemes of a smooth quasi-projective variety resp. Source Geom.

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Zentralblatt MATH identifier Keywords Hilbert scheme symmetric product generating series power structure Pontrjagin ring motivic exponentiation characteristic classes. Characteristic classes of Hilbert schemes of points via symmetric products. Abstract Article info and citation First page References Abstract We obtain a formula for the generating series of the push-forward under the Hilbert—Chow morphism of the Hirzebruch homology characteristic classes of the Hilbert schemes of points for a smooth quasi-projective variety of arbitrary pure dimension.

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You have access to this content. You have partial access to this content. You do not have access to this content. More like this.European Congress of Mathematics pp Cite as. This paper is a survey on arc spaces, a recent topic in algebraic geometry and singularity theory. The geometry of the arc space of an algebraic variety yields several new geometric invariants and brings new light to some classical invariants. Unable to display preview. Download preview PDF. Skip to main content. This service is more advanced with JavaScript available.

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Publishing, River Edge, NJ, Google Scholar. Batyrev, Non-archimedian integrals and stringy Euler numbers of log terminal pairsJournal of European Math. Batyrev, Mirror symmetry and tonic geometryDoc. DMV, Extra Vol. MathSciNet Google Scholar. Denef, Degree of local zeta functions and monodromyCompositio Math.

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Haskell ed. Denef, F. Loeser, Motivic Igusa zeta functionsJ. Algebraic Geom. Loeser, Germs of arcs on singular algebraic varieties and motivic integrationInvent. Loeser, Definable sets, motives and P-adic integralsJ. Loeser, Lefschetz numbers of iterates of the monodromy and truncated arcsTopology to appear, 10 pagesavailable at math. Gillet, C. Greenberg, Rational points in discrete valuation ringsPubl. Kontsevich, Lecture at Orsay December 7, Lejeune-Jalabert, A. Nash Jr. Pas, Uniform p-adic cell decomposition and local zeta functionsJ.

Reine Angew.To browse Academia. Skip to main content. Log In Sign Up. Download Free PDF. Positive motivic measures are counting measures Jordan Ellenberg. Download PDF. A short summary of this paper.

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Positive motivic measures are counting measures. By a K-variety, we mean a geometrically reduced, separated scheme of finite type over K. Throughout the paper, we follow the usual convention of writing L for [A1K ]. A ring homomorphism from K0 VarK to a field F is called a motivic measure. See, e. We will call all such measures counting measures. Our main result is the following: Theorem 1. Every positive motivic measure is a counting measure. Proposition 2.

If K is infinite, there are no positive motivic measures on K0 VarK. We write Fqn for the degree n extension of K. Proposition 3. We prove first that these measures coincide for varieties of the form Spec Fqd and deduce that they coincide for all affine varieties. As K0 VarFq is generated by the classes of affine varieties, this implies the theorem.

Lemma 5.Offers end pm EST. Algebraic Geom. Abstract: We introduce a new notion of -product of two integrable series with coefficients in distinct Grothendieck rings of algebraic varieties, preserving the integrability of and commuting with the limit of rational series. In the same context, we define a motivic multiple zeta function with respect to an ordered family of regular functions, which is integrable and connects closely to Denef-Loeser's motivic zeta functions.

We also show that the -product is associative in the class of motivic multiple zeta functions. Furthermore, a version of the Euler reflexion formula for motivic zeta functions is nicely formulated to deal with the -product and motivic multiple zeta functions, and it is proved for both univariate and multivariate cases by using the theory of arc spaces. As an application, taking the limit for the motivic Euler reflexion formula we recover the well-known motivic Thom-Sebastiani theorem.

References [Enhancements On Off] What's this? I Barcelona, Progr. Reine Angew. MR [16] D.Offers end pm EST.

Algebraic Geom. Abstract: Let be a smooth scheme over an algebraically closed field of characteristic zero and let be a regular function, and writeas a closed -subscheme of.

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The motivic vanishing cycle is an element of the -equivariant motivic Grothendieck ringdefined by Denef and Loeser, and Looijenga, and used in Kontsevich and Soibelman's theory of motivic Donaldson-Thomas invariants. We prove three main results: a depends only on the third-order thickenings of. References [Enhancements On Off] What's this?

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BravV. BussiD. DupontD. Joyceand B. Cisinski and F. I Barcelona, Progr. I, IIAnn. Differential Geom. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformationsarXiv Number Theory Phys.

MR [19] D. Maulik, Motivic residues and Donaldson-Thomas theoryin preparation, MR [23] Vladimir VoevodskyTriangulated categories of motives over a fieldCycles, transfers, and motivic homology theories, Ann. Press, Princeton, NJ,pp. MR Brav, V. Bussi, D. Dupont, D. Joyce, and B. MR [23] Vladimir Voevodsky, Triangulated categories of motives over a fieldCycles, transfers, and motivic homology theories, Ann.I will outline in fair detail how one sets up a theory of motivic integration for a smooth complex variety.

This is by no means the most generality in which motivic integration is defined but the smooth complex case serves as a very good model case to learn the heart of the theory that is the geometric flavor of the theory. Secondly, I will describe some applications of motivic integration to the study of invariants in birtational geometry. These results shed light on how the geometry of the arc space contains information about the birational geometry of a variety.

This connection can be used to study these invariants, most successfully it was applied to show a "inversion of adjunction" formula for the minimal log discrepancies. Computation of the motivic integral in some simple examples. Mention and motivation of the key formula for the motivic integral of a function associated to a normal crossing divisor.

The birational transformation rule. The birational transformation rule computed in an example blowup of P 2 at a point and an important case blowup at a smooth center.

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Using the weak factorization theorem hard this essentially yields a full proof of the result already. Preparations for lecture 3: relative canonical divisor, Kaehler differentials and arcs. Elementary proof of the transformation rule sketchy at times. This lecture is somewhat technical but provides a fairly detailed and fairly elementary proof of the transformation rule in the smooth case.

Applications of motivic integration to log canonical threshold. This is a typical application of motivic integration which gives a characterization of the log canonical threshold birational invariant of a pair in terms of dimensions of the jet spaces. Some useful corollaries can be drawn from this characterization.

Contact loci and valuations. Drawing from the ideas of motivic integration but not really using motivic integration itself Ein, Lazarsfeld and Mustata relate certain cylinders multi-contact loci in the arc space directly to the divisorial valuations over a smooth variety this lecture will be more expository than the previous ones.Full-text: Access denied no subscription detected We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.

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Read more about accessing full-text. Recently, it has become well known that the conjectural integral identity is of crucial importance in the motivic Donaldson—Thomas invariants theory for noncommutative Calabi—Yau threefolds. The purpose of this article is to consider different versions of the identity, for regular functions and formal functions, and to give them the positive answer for the algebraically closed ground fields.

Source Duke Math. Zentralblatt MATH identifier Proofs of the integral identity conjecture over algebraically closed fields. Duke Math. Read more about accessing full-text Buy article.

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Abstract Article info and citation First page References Abstract Recently, it has become well known that the conjectural integral identity is of crucial importance in the motivic Donaldson—Thomas invariants theory for noncommutative Calabi—Yau threefolds. Article information Source Duke Math. Export citation. Export Cancel. References [1] V. You have access to this content. You have partial access to this content.

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