F. Loeser, J. Sebag
We construct a theory of motivic integration for smooth rigid varieties. As an application new invariants of degenerations are obtained.
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
We prove that stable rationality specializes in regular families whose fibers are integral and have at most ordinary double points as singularities. Our proof is based on motivic specialization techniques and the criterion of Larsen and Lunts for stable rationality in the Grothendieck ring of varieties.
This is a survey on motivic zeta functions associated to abelian varieties and Calabi-Yau varieties over a discretely valued field. We explain how they are related to Denef and Loeser's motivic zeta function associated to a complex hypersurface singularity and we investigate the relation between the poles of the zeta function and the eigenvalues of the monodromy action on the tame $\ell$-adic cohomology of the variety. The motivic zeta function allows to generalize many interesting arithmetic invariants from abelian varieties to Calabi-Yau varieties and to compute them explicitly on a model wi…
We use a formula of Bultot to compute the motivic zeta function for the toric degeneration of the quartic K3 and its Gross-Siebert mirror dual degeneration. We check for this explicit example that the identification of the logarithm of the monodromy and the mirror dual Lefschetz operator works at an integral level.
We prove a 1999 conjecture of Veys, which says that the opposite of the log canonical threshold is the only possible pole of maximal order of Denef and Loeser's motivic zeta function associated with a germ of a regular function on a smooth variety over a field of characteristic zero. We apply similar methods to study the weight function on the Berkovich skeleton associated with a degeneration of Calabi-Yau varieties. Our results suggest that the weight function induces a flow on the non-archimedean analytification of the degeneration towards the Kontsevich-Soibelman skeleton.
We study function germs on toric varieties which are nondegenerate for their Newton diagram. We express their motivic Milnor fibre in terms of their Newton diagram. We extend a formula for the motivic nearby fibre to the case of a toroidal degeneration. We illustrate this by some examples.
We construct motivic invariants of a subvariety of an algebraic torus from its tropicalization and initial degenerations. More specifically, we introduce an invariant of a compactification of such a variety called the "tropical motivic nearby fiber." This invariant specializes in the schon case to the Hodge-Deligne polynomial of the limit mixed Hodge structure of a corresponding degeneration. We give purely combinatorial expressions for this Hodge-Deligne polynomial in the cases of schon hypersurfaces and smooth tropical varieties. We also deduce a formula for the Euler characteristic of a gen…
Euler characteristic and signature of real semi-stable degenerations
本文通过动机性近纤维方法，证明了非奇异实热带完全交集中实部欧拉示性数等于复部符号差，并推广至具有Q-非奇异热带极限的实解析族。
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。
Let D be a central division algebra over a field F. We study in this note the rigidity of the motivic decompositions of the Severi-Brauer varieties of D, with respect to the ring of coefficients and to the base field. We first show that if the ring of coefficient is a field, these decompositions only depend on its characteristic. In a second part we show that if D remains division over a field extension E/F, the motivic decompositions of several Severi-Brauer varieties of D remain the same when extending the scalars to E.
We use the motivic obstruction to stable rationality introduced by Shinder and the first-named author to establish several new classes of stably irrational hypersurfaces and complete intersections. In particular, we show that very general quartic fivefolds and complete intersections of a quadric and a cubic in $\mathbb P^6$ are stably irrational. An important new ingredient is the use of tropical degeneration techniques.
Motivic realization of rigid G-local systems on curves and tamely ramified geometric Langlands
本文证明了具有拟单值单值性和有限阶阿贝尔化的复不可约刚性 G-局部系统是动形的，推广了 Katz 的定理，并在 de Rham 几何 Langlands 设定下证明了 Hecke 特征层的存在性、谱分解及刚性与自守刚性的关系。
We establish a trace formula for rigid varieties $X$ over a complete discretely valued field, which relates the set of unramified points on $X$ to the Galois action on its étale cohomology. We develop a theory of motivic integration for formal schemes of pseudo-finite type over a complete discrete valuation ring $R$, and we introduce the Weil generating series of a regular formal $R$-scheme $\mathfrak{X}$ of pseudo-finite type, via the construction of a Gelfand-Leray form on its generic fiber. Our trace formula yields a cohomological interpretation of this Weil generating series. When $\mathfr…
Let $k$ be a non-Archimedean rational valued field. We construct the moduli space of linearly rigidified polarized analytic tori over $k$ that admit rigid-analytic uniformization by an algebraic torus and observe that it is in definable rigid subanalytic bijection with a $PGL_N$-bundle over a polyhedral domain in an algebraic torus. We use this observation to prove that the Hrushovski-Kazhdan motivic volume of a non-Archimedean semi-algebraic family of Abelian varieties admitting such a uniformization fibrewise vanishes. This question is motivated by the conjectural geometric interpretation of…
We extend results of Colliot-Thélène and Raskind on the $\mathcal{K}_2$-cohomology of smooth projective varieties over a separably closed field $k$ to the étale motivic cohomology of smooth, not necessarily projective, varieties over $k$. Some consequences are drawn, such as the degeneration of the Bloch-Lichtenbaum spectral sequence for any field containing $k$.
Motivic p-adic periods of 1-motives
本文为1-动机显式构造了p-adic de Rham比较同构，并利用刚性解析动机构造了新的动机p-adic周期环，通过实例计算了周期配对。
We introduce a "limiting Frobenius structure" attached to any degeneration of projective varieties over a finite field of characteristic p which satisfies a p-adic lifting assumption. Our limiting Frobenius structure is shown to be effectively computable in an appropriate sense for a degeneration of projective hypersurfaces. We conjecture that the limiting Frobenius structure relates to the rigid cohomology of a semistable limit of the degeneration through an analogue of the Clemens-Schmidt exact sequence. Our construction is illustrated, and conjecture supported, by a selection of explicit ex…
We present a formal scheme based cycle model for the motivic cohomology of the fat points defined by the truncated polynomial rings $k[t]/(t^m)$ with $m \geq 2$, in one variable over a field $k$. We compute their Milnor range cycle class groups when the field has sufficiently many elements. With some aids from rigid analytic geometry and the Gersten conjecture for the Milnor $K$-theory resolved by M. Kerz, we prove that the resulting cycle class groups are isomorphic to the Milnor $K$-groups of the truncated polynomial rings, generalizing a theorem of Nesterenko-Suslin and Totaro.
We introduce a quotient of the Grothendieck ring of varieties by identifying classes of universally homeomorphic varieties. We show that the standard realization morphisms factor through this quotient, and we argue that it is the correct value ring for the theory of motivic integration on formal schemes and rigid varieties in mixed characteristic. The present note is an excerpt of a detailed survey paper which will be published in the proceedings of the conference "Motivic integration and its interactions with model theory and non-archimedean geometry" (ICMS, 2008).
Log motivic nearby cycles
本文定义了对数动机邻近圈函子，证明其将DVR分式域上光滑真概形的动机映为对数光滑模型边界动机，并在特征0下建立了与刚性解析动机范畴的等价及与Ayoub函子的联系。
Stable rationality of hypersurfaces of mock toric variety II
本文用模拟环面簇替代环面簇构造退化族，借助母题方法证明了射影空间中非常一般超曲面的稳定非有理性可推出格拉斯曼簇中相应超曲面的稳定非有理性。
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, th…
Stable rationality of hypersurfaces in schön affine varieties
本文通过动机方法，将[Nicaise--Ottem'22]的结果从代数环面扩展到schön仿射簇中的超曲面，还应用该方法证明了复Grassmann簇Gr(2,n)中某些超曲面的无理性，与先前研究结果一致。
Hodge Atoms at Conifold Degenerations: F-Bundles, Limiting Mixed Hodge Modules, and the Rigid-Flexible Decomposition
研究锥点退化中Calabi-Yau三fold的Hodge原子分解，通过刚柔分解构造修正的混合Hodge模，并利用Stokes-扩展识别确定非分裂结构。
We construct a motivic homotopy theory for rigid analytic varieties with the rigid analytic affine line $\mathbb{A} ^1_\mathrm{rig}$ as an interval object. This motivic homotopy theory is inspired from, but not equal to, Ayoub's motivic homotopy theory for rigid analytic varieties. Working in the so constructed homotopy theory, we prove that a homotopy classification of vector bundles of rank $n$ over rigid analytic quasi-Stein spaces follows from $\mathbb{A}^1_\mathrm{rig}$-homotopy invariance of vector bundles. This $\mathbb{A}^1_\mathrm{rig}$-homotopy invariance is equivalent to a rigid ana…
The present paper is devoted to the description of rigid solvable Leibniz algebras. In particular, we prove that solvable Leibniz algebras under some conditions on the nilradical are rigid and we describe four-dimensional solvable Leibniz algebras with three-dimensional rigid nilradical. We show that the Grunewald-O'Halloran's conjecture "any $n$-dimensional nilpotent Lie algebra is a degeneration of some algebra of the same dimension" holds for Lie algebras of dimensions less than six and for Leibniz algebras of dimensions less than four. The algebra of level one, which is omitted in the 1991…
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。
Complexes of stable birational invariants
本文引入了一个新的稳定双有理不变量，以链复形同伦类型的形式分类化动机体积，并由此构造了回缩有理性的动机障碍及一类普遍的不变量。
Representability of continuous K-theory in rigid analytic motivic $\mathbb{A}^1$-homotopy theory
该研究证明了刚性解析空间的连续K-理论与解析K-理论在$\boldsymbol{\rm A}^1$-同伦范畴中可表示，还得到轻凝聚谱系数的可表示性结论，并证明了Weibel消失定理及连续K-理论在局部Tate对上的$\boldsymbol{\rm A}^1$-不变性。
We prove an effective Hilbert Irreducibility result for residual realizations of a family of motives with motivic Galois group G2.
In this article, we introduce the notion of periodic de Rham bundles over smooth complex curves. We prove that motivic de Rham bundles over smooth complex curves are periodic. We conjecture that irreducible periodic de Rham bundles over smooth complex curves are motivic. We show that the conjecture holds for rank one objects and rigid objects.
Privileged $\mathrm{SL}_3$-local systems are motivic
针对复数域上光滑射影簇$X$，研究证明Esnault与Groechenig定义的不可约特权$\boldsymbol{\rm SL}_3$局部系统$\boldsymbol{\rm V}$具有几何起源，并给出特权但非刚性的$\boldsymbol{\rm SL}_3$局部系统实例
For a simple, rigid vector bundle $F$ on a Calabi-Yau $3$-fold $Y$, we construct a symmetric obstruction theory on the Quot scheme $\textrm{Quot}_Y(F,n)$, and we solve the associated enumerative theory. We discuss the case of other $3$-folds. Exploiting the critical structure on $\textrm{Quot}_{\mathbb A^3}(\mathscr O^r,n)$, we construct a virtual motive (in the sense of Behrend-Bryan-Szendrői) for $\textrm{Quot}_Y(F,n)$ for an arbitrary vector bundle $F$ on a smooth $3$-fold $Y$. We compute the associated motivic partition function. We obtain new examples of higher rank (motivic) Donaldson-Th…
The motivic nearby fiber is an invariant obtained from degenerating a complex variety over a disc. It specializes to the Euler characteristic of the original variety but also contains information on the variation of Hodge structure associated to the degeneration which is encoded as a limit mixed Hodge structure. However, this invariant is difficult to compute in practice. Using the techniques of tropical geometry we present a new formula for the motivic nearby fiber. Moreover, since there is a range of available software implementing the main algorithms in tropical geometry, our formula can be…
Fix a base field F, a finite field K and consider a sequence of central simple F-algebras A_1,...,A_n. In this note we provide some results toward a classification of the indecomposable motives lying in the motivic decompositions of projective homogeneous varieties under the action of PGL(A_1)x...xPGL(A_n) with coefficients in K. We give a complete classification of those motives if n=1 and derive from it the motivic dichotomy of projective linear groups. We then provide several classification results as well as counterexamples for arbitrary n showing that the situation is less rigid. These re…
We study the motivic Serre invariant of a smoothly bounded algebraic or rigid variety $X$ over a complete discretely valued field $K$ with perfect residue field $k$. If $K$ has characteristic zero, we extend the definition to arbitrary $K$-varieties using Bittner's presentation of the Grothendieck ring and a process of Néron smoothening of pairs of varieties. The motivic Serre invariant can be considered as a measure for the set of unramified points on $X$. Under certain tameness conditions, it admits a cohomological interpretation by means of a trace formula. In the curve case, we use T. Sait…
Efficient Algorithms for Maximal Matroid Degenerations and Irreducible Decompositions of Circuit Varieties
本文提出高效算法用于分解 matroid 的电路品种，通过识别极大衰减来实现显式且可计算的不可约分解。
On the fixed locus of the antisymplectic involution of an EPW cube
本文研究了 EPW 立方体的反辛酉变换固定点集，证明其为刚性的原子拉格朗日子流形，并基于奇异退化分析与退化方法进行了证明。
The modular properties of some higher dimensional varieties of special Fano type are analyzed by computing the L-function of their $Ω-$motives. It is shown that the emerging modular forms are string theoretic in origin, derived from the characters of the underlying rational conformal field theory. The definition of the class of Fano varieties of special type is motivated by the goal to find candidates for a geometric realization of the mirrors of rigid Calabi-Yau varieties. We consider explicitly the cubic sevenfold and the quartic fivefold, and show that their motivic L-functions agree with t…
An irreducible integrable connection $(E,\nabla)$ on a smooth projective complex variety $X$ is called rigid if it gives rise to an isolated point of the corresponding moduli space $\mathcal{M}_{dR}(X)$. According to Simpson's motivicity conjecture, irreducible rigid flat connections are of geometric origin, that is, arise as subquotients of a Gauß-Manin connection of a family of smooth projective varieties defined on an open dense subvariety of $X$. In this article we study mod $p$ reductions of irreducible rigid connections and establish results which confirm Simpson's prediction. In particu…
Tyurin Degenerations, Derived Lagrangians and Categorification of DT Invariants
本文研究 Calabi-Yau 三维流形的 Tyurin 退化，证明全模空间具有相对 Lagrangian 叶状结构，从而通过平坦 Gauss-Manin 联络建立范畴化 DT 不变量的形变不变性，并将其表示为特殊纤维上 Fano 模空间的导出交上同调。
Rigid analytic 1-motives and conjugate uniformization of abeloid varieties
本文针对p进域上的阿贝尔簇，研究刚性解析1-动机理论及阿贝尔簇的共轭单值化，建立了形式与刚性解析1-动机的等价性等结果，还构造了相关p可除群。
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。