Willem Veys
Inspired by p-adic (and real) principal value integrals, we introduce motivic principal value integrals associated to multi-valued rational differential forms on smooth algebraic varieties. We investigate the natural question whether (for complete varieties) this notion is a birational invariant. The answer seems to be related to the dichotomy of the Minimal Model Program.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Motivic principal value integrals for hyperplane arrangements
本文针对超平面排列，验证了Denef-Jacobs-Veys猜想，将多值有理顶端形式的示性主值积分与秩一局部系统支撑定位的上同调联系起来。
We introduce motivic analogues of p-adic exponential integrals. We prove a basic multiplicativity property from which we deduce a motivic analogue of the Thom-Sebastiani Theorem. In particular, we obtain a new proof of the Thom-Sebastiani Theorem for the Hodge spectrum of (non isolated) singularities of functions.
We use the theory of motivic integration in order to give a geometric explanation of the behavior of some p-adic integrals.
We develop the Denef-Loeser motivic integration to the equivariant motivic integration and use it to prove the full integral identity conjecture for regular functions.
A motivic integral identity for $(-1)$-shifted symplectic stacks
本文证明了关于$(-1)$-移位辛堆栈的motivic积分恒等式，扩展了Kontsevich-Soibelman和Joyce-Song的工作，并为扩展motivic Donaldson-Thomas理论提供了新的工具。
A functional equation for the motivic integral corresponding to the Milnor number of an arc is derived using the Denef-Loeser formula for the change of variables. Its solution is a function of five auxiliary parameters, it is unique up to multiplication by a constant and there is a simple recursive algorithm to find its coefficients. The method is universal enough and gives, for example, equations for the integral corresponding to the intersection number over the space of pairs of arcs and over the space of unordered tuples of arcs.
We give an explicit formula for the motivic integrals related to the Milnor number over spaces of parametrised arcs on the plane with fixed tangency orders with the axis. These integrals are rational functions of the parameters and the class of the affine line. Using a set of natural recurrence relations between them, we prove an unexpected invariance property with respect to the simultaneous inversion of the parameters and the class of the affine line. We also discuss a generalization of this system of recurrence relations whose solutions are also symmetric and satisfy additional differential…
Motivic integration on special rigid varieties and the motivic integral identity conjecture
本文通过发展特殊刚性簇上的等变动机积分新框架，证明了Kontsevich和Soibelman的动机积分恒等式猜想，并建立了具有Fubini型性质的动机体积概念。
A motivic integral $p$-adic cohomology
本文构造了一个积分$p$-adic上同调，与刚性上同调相容，并在奇点消解条件下证明其与Ertl-Shiho-Sprang的“好”上同调一致，从而得到motivic性质和Künneth公式。
Integral cohomology of dual boundary complexes is motivic
本文给出了光滑拟射影复代数簇对偶边界复形整值上同调的动机刻画，并推论稳定仿射空间的对偶边界复形可缩，该结果已被用于证明弱几何 P=W 猜想。
Motivic measure on the space of functions was introduced by Campillo, Delgado and Gusein-Zade as an analog of the motivic measure on the space of arcs . In this paper we prove that the measure on the space of functions can be related to the motivic measure on the space of arcs by a factor, which can be defined explicitly in geometric terms. This provides a possibility to rewrite motivic integrals over the space of functions as integrals over the union of all symmetric powers of the space of arcs.
Motivic integration for singular Artin stacks
本文将光滑 Artin 叠上的母题积分变量替换公式推广到奇异情形，并借助 warping 叠典范地将任意母题积分表示为 warped 弧上的积分。
Let $\mathcal{X} \to Y$ be a birational map from a smooth Artin stack to a (possibly singular) variety. We prove a change of variables formula that relates motivic integrals over arcs of $Y$ to motivic integrals over arcs of $\mathcal{X}$. With a view toward the study of stringy Hodge numbers, this change of variables formula leads to a new notion of crepantness for the map $\mathcal{X} \to Y$ that coincides with the usual notion in the special case that $\mathcal{X}$ is a scheme.
The integral identity conjecture in motivic homotopy theory
本文基于动机Donaldson-Thomas不变量的存在性，探讨了Kontsevich和Soibelman的积分恒等猜想的泛函版本，并扩展了其应用范围。
The integral motivic dual Steenrod algebra
本文计算了在模p motivic对偶Steenrod代数符合Voevodsky公式的底方案上的整系数motivic对偶Steenrod代数。
Following the work of Gangl, Goncharov and Levin in [GGL], we will give a combinatorial framework for motivic study of iterated integrals on the affine line. We will show that under a certain genericity condition these combinatorial objects yield to elements in the motivic Hopf algebra constructed in Bloch-Kriz [BK]. It will be shown that the Hodge realization of these elements coincides with the Hodge structure induced from the fundamental torsor of path of punctured affine line.
Discriminants and motivic integration
本文利用形式形变对称幂上的母题积分研究平面曲线奇点不变量，证明判别式积分可恢复主 Hilbert 概形母题类，orbifold 积分给出母题 Igusa zeta 函数的 plethystic 指数，并建立了其与纽结 Floer 同调的联系。
We construct a new motivic integration morphism, the so-call bounded integral, that interpolates both the integration morphisms with and without volume forms of Hrushovski and Kazhdan. This is done within the framework of model theory of algebraically closed valued fields of equicharacteristic zero. As an application, we recover and extend some results of Hrushovski and Loeser about the motivic Milnor fiber.
By associating a `motivic integral' to every complex projective variety X with at worst canonical, Gorenstein singularities, Kontsevich proved that, when there exists a crepant resolution of singularities Y of X, the Hodge numbers of Y do not depend upon the choice of the resolution. In this article we provide an elementary introduction to the theory of motivic integration, leading to a proof of the result described above. We calculate the motivic integral of several quotient singularities and discuss these calculations in the context of the cohomological McKay correspondence.
We discuss duality pairings on integral étale motivic cohomology groups of regular and proper schemes over algebraically closed fields, local fields, finite fields, and arithmetic schemes.
Motivic Galois theory for one-loop Feynman integrals in momentum space
本文提出了一种动机构建一阶费曼积分在动量空间中的理论，通过关联通用动量学的动机局部系统来处理带有切口的图，并计算了这些系统的重量分量，这些分量是与最大切口商图相关的二次Artin动机的塔扭曲。
Integral Motivic Cohomology of $BSO_{4}$
本文计算了 $BSO_{4}$ 的积分动机上同调，借鉴 Guillot 和 Yagita 的方法，为 $BG_{2}$ 的类似计算奠定基础。
The goal of this article is to give an elementary proof of the double shuffle relations directly for the Goncharov and Manin motivic multiple zeta values. The shuffle relation is straightforward, but for the stuffle we use a modification of a method first introduced by P. Cartier for the purpose of proving stuffle for the real multiple zeta values via integrals and blow-up sequences.
This survey paper, to appear in he proceedings of the Miami Winter School ``Geometric Methods in Algebra and Number Theory'', is concerned with extending classical results à la Ax-Kochen-Er{š}ov to $p$-adic integrals in a motivic setting, using the framework of constructible motivic functions we introduced in math.AG/0410203. We conclude the paper by discussing briefly the relevance of our results to the study of orbital integrals and the Fundamental Lemma.
Principal value integrals are associated to multi-valued rational differential forms with normal crossings support on a non-singular algebraic variety. We prove their vanishing on rational surfaces in the context of a conjecture of Denef-Jacobs. As an application we obtain a strong vanishing result for candidate poles of p-adic and motivic Igusa zeta functions.
The Poincare series of a multi-index filtration on the ring of germs of functions can be written as a certain integral with respect to the Euler characteristic over the projectivization of the ring. Here this integral is considered with respect to the generalized Euler characteristic with values in the Grothendieck ring of varieties. For the filtration defined by orders of functions on the components of a plane curve singularity $C$ and for the so called divisorial filtration for a modification of $(\C^2,0)$ by a sequence of blowing-ups there are given formulae for this integral in terms of an…
We introduce spaces of exponential constructible functions in the motivic setting for which we construct direct image functors in the absolute and relative cases. This allows us to define a motivic Fourier transformation for which we get various inversion statements. We define also motivic Schwartz-Bruhat spaces on which motivic Fourier transformation induces an isomorphism. Our motivic integrals specialize to non archimedian integrals. We give a general transfer principle comparing identities between functions defined by integrals over local fields of characteristic zero, resp. positive, havi…
Milnor-Witt motivic decomposition of Stiefel varieties
本文计算了Stiefel流形的整MW-动机上同调群，并建立了其整MW-动机分解，从而证明了先前工作中的猜想。
We prove a formula expressing the motivic integral (\cite{ls}) of a K3 surface over $\bC((t))$ with semi-stable reduction in terms of the associated limit Hodge structure. Secondly, for every smooth variety over a non-archimedean field we define an analogue of the monodromy pairing, constructed by Grothendieck in the case of Abelian varieties, and prove that our monodromy pairing is a birational invariant of the variety. Finally, we propose a conjectural formula for the motivic integral of maximally degenerated K3 surfaces over an arbitrary non-archimedean field and prove this conjecture for K…
Recently, it is well known that the conjectural integral identity is of crucial importance in the motivic Donaldson-Thomas invariants theory for non-commutative 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 ground field algebraically closed. Technically, the result on motivic Milnor fiber by Hrushovski-Loeser using Hrushovski-Kazhdan's motivic integration and Nicaise's computations on motivic integrals on special formal schemes are main tools.
This article gives a short step-by-step introduction to the representation of parametric Feynman integrals in scalar perturbative quantum field theory as periods of motives. The application of motivic Galois theory to the algebro-geometric and categorical structures underlying Feynman graphs is reviewed up to the current state of research. The example of primitive log-divergent Feynman graphs in scalar massless $ϕ^4$ quantum field theory is analysed in detail.
With representation-theoretic applications in mind, we construct a formalism of reduced motives with integral coefficients. These are motivic sheaves from which the higher motivic cohomology of the base scheme has been removed. We show that reduced stratified Tate motives satisfy favorable properties including weight and t-structures. We also prove that reduced motives on cellular (ind-)schemes unify various approaches to mixed sheaves in representation theory, such as Soergel-Wendt's semisimplified Hodge motives, Achar-Riche's complexes of parity sheaves, as well as Ho-Li's recent category of…
We prove that if two semi-algebraic subsets of $\mathbb{Q}_p^n$ have the same $p$-adic measure, then this equality can already be deduced using only some basic integral transformation rules. On the one hand, this can be considered as a positive answer to a $p$-adic analogue of a question asked by Kontsevich-Zagier in the reals (though the question in the reals is much harder). On the other hand, our result can also be considered as stating that over $\mathbb{Q}_p$, universal motivic integration (in the sense of Hrushovski-Kazhdan) is just $p$-adic integration.
On the Integral Part of A-Motivic Cohomology
本文研究了正特征全局域上的A-动机上同调的整数部分，提出模型版本和l-adic版本的定义，并证明模型版本包含l-adic版本，同时探讨了regulated extensions的子模块。
We investigate connections between Lipschitz geometry of real algebraic varieties and properties of their arc spaces. For this purpose we develop motivic integration in the real algebraic set-up. We construct a motivic measure on the space of real analytic arcs. We use this measure to define a real motivic integral which admits a change of variables formula not only for the birational but also for generically one-to-one Nash maps. As a consequence we obtain an inverse mapping theorem which holds for continuous rational maps and, more generally, for generically arc-analytic maps. These maps app…
We prove relations between fractional linear cycles in Bloch's integral cubical higher Chow complex in codimension two of number fields, which correspond to functional equations of the dilogarithm. These relations suffice, as we shall demonstrate with a few examples, to write down enough relations in Bloch's integral higher Chow group CH^2(F,3) for certain number fields F to detect torsion cycles. Using the regulator map to Deligne cohomology, one can check the non-triviality of the torsion cycles thus obtained. Using this combination of methods, we obtain explicit higher Chow cycles generatin…
This is a sequel to our previous paper (joint with Furusho). It will give a more natural framework for constructing elements in the Hopf algebra of framed mixed Tate motives according to Bloch and Kriz. This framework allows us to extend our previous results to interpret all multiple zeta values (including the divergent ones) and the multiple polylogarithms in one variable as elements of this Hopf algebra. It implies that the pro-unipotent completion of the torsor of paths on projective line minus three points, is a mixed Tate motive in the sense of Bloch-Kriz. Also It allows us to interpret t…
Fourier transform for étale motivic cohomology
该研究利用平展态分上同调考察交换簇 Fourier 变换的整性，并证明平展 Chow 环正次数部分关于 Pontryagin 积具有 PD-结构。
This an extended version of the previous preprint dated by February 2005. We prove that the Chow motive of an anisotropic projective homogeneous variety of type F4 is isomorphic to the direct sum of twisted copies of a generalized Rost motive. In particular, we provide an explicit construction of a generalized Rost motive for a generically splitting variety for a symbol in K_3^M(k)/3. We also establish a motivic isomorphism between two anisotropic non-isomorphic projective homogeneous varieties of type F4. All our results hold for Chow motives with integral coefficients.
We discuss the structure of integral etale motivic cohomology groups of smooth and projective schemes over algebraically closed fields, finite fields, local fields, and arithmetic schemes.
We introduce spaces of exponential constructible functions in the motivic setting for which we construct direct image functors in the absolute and relative cases. This allows us to define a motivic Fourier transformation for which we get various inversion statements. We define also motivic Schwartz-Bruhat spaces on which motivic Fourier transformation induces an isomorphism. Our motivic integrals specialize to non archimedian integrals. We give a general transfer principle comparing identities between functions defined by integrals over local fields of characteristic zero, resp. positive, havi…
We associate canonical virtual motives to definable sets over a field of characteristic zero. We use this construction to show that very general p-adic integrals are canonically interpolated by motivic ones.
In the first part of this paper we study the coaction dual to the action of the cosmic Galois group on the motivic lift of the sunrise Feynman integral with generic masses and momenta, and we express its conjugates in terms of motivic lifts of Feynman integrals associated to related Feynman graphs. Only one of the conjugates of the motivic lift of the sunrise, other than itself, can be expressed in terms of motivic lifts of Feynman integrals of subquotient graphs. To relate the remaining conjugates to Feynman integrals we introduce a general tool: subdiving edges of a graph. We show that all m…
In this note, using Cluckers-Loeser's theory of motivic integration, we prove the integral identity conjecture with framework a localized Grothendieck ring of varieties over an arbitrary base field of characteristic zero.
The integral motivic Satake equivalence for ramified groups
本文构建了非阿基米德局部域上准分裂reductive群的几何萨塔克等价式，扩展了Gaussent-Littelmann的工作，并推广了Zhu的积分萨塔克同构到分裂群，定义了通用球面海格尔代数并构造了通用萨塔克和伯恩斯坦同构。
Period integrals arise from the comparison between Betti cohomologies and de Rham cohomologies. In this paper, we give a formula for the determinant of period integrals in terms of the canonical pairing between Picard group of rank 1 realizations and relative Chow group. It is compatible with the similar formula for Galois modules proved by the first author, and as a consequence, under the assumption of Gersten resolution, the determinant of the cohomology of a determinantally motivic object is motivic.
Towards Motivic Coactions at Genus One from Zeta Generators
本文通过类比亏格零单值映射与zeta生成元构造，提出亏格一全纯Eisenstein级数迭代积分的动机余作用公式，并验证其性质及推导多重模值的分解。
We study transfer principles for upper bounds of motivic exponential functions and for linear combinations of such functions, directly generalizing the transfer principles from [7] by Cluckers-Loeser and [13, Appendix B] by Shin-Templier (appendix B by Cluckers-Gordon-Halupczok). These functions come from rather general oscillatory integrals on local fields, and can be used to describe e.g. Fourier transforms of orbital integrals. One of our techniques consists in reducing to simpler functions where the oscillation only comes from the residue field.
We study the geometry and Hodge theory of the cubic hypersurfaces attached to two-loop Feynman integrals for generic physical parameters. We show that the Hodge structure attached to planar two-loop Feynman graphs decomposes into mixed Tate pieces and the Hodge structures of families of hyperelliptic, elliptic, or rational curves depending on the space-time dimension. For two-loop graphs with a small number of edges, we give more precise results. In particular, we recover a result of Bloch arXiv:2105.06132 that in the well-known double box example, there is an underlying family of elliptic cur…
Monadic resolutions for generalized spaces
本文将 Bousfield–Kan 的单子分解方法推广到 ∞-拓扑斯，证明主纤维化引理，并据此建立不稳定等变与母题同伦理论中多类单子分解的收敛性，特别证明了代数闭域上不稳定 Adams–Novikov 谱序列对幂零连通 2-有效母题空间的收敛性。
A Proof of the Integral Identity via Braden's Theorem
本文利用 Braden 的双曲局部化/限制定理，在 Morel-Voevodsky 动机稳定同伦范畴中，为 Kontsevich-Soibelman 猜想的虚拟动机积分恒等式提供了一个广义范畴化版本的简短证明。
Kontsevich-Zagier conjecture for Hensel-minimal fields with sections
本文在 Hensel-极小框架下推广了带截面的 motivic 积分，建立了 Hrushovski-Kazhdan 型双射，并证明 Cluckers-Loeser motivic 积分可由此同构导出。
We put forward in this paper a uniform narrative that weaves together several variants of Hrushovski-Kazhdan style integral, and describe how it can facilitate the understanding of the Denef-Loeser motivic Milnor fiber and closely related objects. Our study focuses on the so-called "nonarchimedean Milnor fiber" that was introduced by Hrushovski and Loeser, and our thesis is that it is a richer embodiment of the underlying philosophy of the Milnor construction. The said narrative is first developed in the more natural complex environment, and is then extended to the real one via descent. In the…
We consider parametric Feynman integrals and their dimensional regularization from the point of view of differential forms on hypersurface complements and the approach to mixed Hodge structures via oscillatory integrals. We consider restrictions to linear subspaces that slice the singular locus, to handle the presence of non-isolated singularities. In order to account for all possible choices of slicing, we encode this extra datum as an enrichment of the Hopf algebra of Feynman graphs. We introduce a new regularization method for parametric Feynman integrals, which is based on Leray coboundari…
We introduce a family of periods of mixed Tate motives called dissection polylogarithms, that are indexed by combinatorial objects called dissection diagrams. The motivic coproduct on the former is encoded by a combinatorial Hopf algebra structure on the latter. This generalizes Goncharov's formula for the motivic coproduct on (generic) iterated integrals. Our main tool is the study of the relative cohomology group corresponding to a bi-arrangement of hyperplanes.
Algebraic cycles of some Fano varieties with Hodge structure of level one
本文研究了具有Hodge结构一级的Fano簇的代数循环，通过étale动机方法证明了光滑四次双五维流形的整数Hodge猜想。
We define motivic iterated integrals on the affine line, and give a simple proof of the formula for the coproduct in the Hopf algebra of they make. We show that it encodes the group law in the automorphism group of certain non-commutative variety. We relate the coproduct with the coproduct in the Hopf algebra of decorated rooted planar trivalent trees - a planar decorated version of the Hopf algebra defined by Connes and Kreimer. As an application we derive explicit formulas for the coproduct in the motivic multiple polylogarithm Hopf algebra. We give a criteria for a motivic iterated integral…
Hodge correlators are complex numbers given by certain integrals assigned to a smooth complex curve. We show that they are correlators of a Feynman integral, and describe the real mixed Hodge structure on the pronilpotent completion of the fundamental group of the curve. We introduce motivic correlators, which are elements of the motivic Lie algebra and whose periods are the Hodge correlators. They describe the motivic fundamental group of the curve. We describe variations of real mixed Hodge structures on a variety by certain connections on the product of the variety by an afine line. We call…
Beginning with the conjecture of Artin and Tate in 1966, there has been a series of successively more general conjectures expressing the special values of the zeta function of an algebraic variety over a finite field in terms of other invariants of the variety. In this article, we present the ultimate such conjecture, and provide evidence for it. In particular, we enhance Voevodsky's Z[1/p]-category of etale motivic complexes with a p-integral structure, and show that, for this category, our conjecture follows from the Tate and Beilinson conjectures. As the conjecture is stated in terms of mot…
We study upper bounds, approximations, and limits for functions of motivic exponential class, uniformly in non-Archimedean local fields whose characteristic is $0$ or sufficiently large. Our results together form a flexible framework for doing analysis over local fields in a field-independent way. As corollaries, we obtain many new transfer principles, for example, for local constancy, continuity, and existence of various kinds of limits. Moreover, we show that the Fourier transform of an $L^2$-function of motivic exponential class is again of motivic exponential class. As an application in th…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。