Cristiana Bertolin
In this note we prove the geometrical origin of pairings of abelian schemes. According to Deligne's philosophy of motives, this means that these pairings are motivic. We make also explicit the link between pairings and linear morphisms. We give a geometrical explanation of the main properties of the Weil pairings and of the Riemann forms of line bundles. We generalize our remarks to pairings of 1-motives.
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For a smooth proper variety over a $p$-adic field, the Brauer group and abelian fundamental group are related to the higher Chow groups by the Brauer-Manin pairing and the class field theory. We generalize this relation to smooth (possibly non-proper) varieties, using the motivic homology and the tame version of Wiesend's ideal class group. Several examples are discussed.
We show that the constructions done in part I generalize their classical counterparts: firstly, the classical Beilinson regulator is induced by the abstract Chern class map from $BGL$ to the Deligne cohomology spectrum. Secondly, Arakelov motivic cohomology is a generalization of arithmetic $K$-theory and arithmetic Chow groups. Finally, we give a conceptual explanation of the height pairing: it is given by the natural pairing of motivic homology and Arakelov motivic cohomology.
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…
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 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.
We collect some properties of the motivic zeta functions and the motivic nearby fiber defined by Denef and Loeser. In particular, we calculate the relative dual of the motivic nearby fiber. We give a candidate for a nearby cycle morphism on the level of Grothendieck groups of varieties using the motivic nearby fiber.
We use an analogue of Karoubi's construction in the motivic situation to give some cohomology operations in motivic cohomology. We prove many properties of these operations, and we show that they coincide, up to some nonzero constants, with the reduced power operations in motivic cohomology originally constructed by Voevodsky. The relation of our construction to Voevodsky's is, roughly speaking, that of a fixed point set to its associated homotopy fixed point set.
We use the theory of motivic integration in order to give a geometric explanation of the behavior of some p-adic integrals.
We study the moduli space of rank 0 semistable sheaves on some rational surfaces. We show the irreducibility and stable rationality of them under some conditions. We also compute several (virtual) Betti numbers of those moduli spaces by computing their motivic measures.
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…
This note addresses the motivic nature of some classical cohomological results due to Lefschetz, namely the primitive decomposition (for the cohomology of smooth projective varieties), and, secondly, the splitting of the cohomology of a complete intersection into the "fixed" and "variable part".
On The Motivic Leray-Hirsch Theorem For Pure Tate Fibre Bundles
本文在Chow动机范畴中证明了纯塔利纤维束的motivic Leray-Hirsch定理，并探讨了其应用。
We extend Ayoub's formalism of motivic nearby cycle functor to the $\infty$-categorical level, and prove some desired cohomological properties by relating the motivic nearby cycle functor to the notion of local acyclicity in motivic homotopy.
Some Motivic Properties of Gushel-Mukai Sixfolds
本文研究了Gushel-Mukai六维流形的动机性质，证明其在代数等价意义下允许乘性Chow-Künneth分解并具有Franchetta性质，同时附带证明了双重EPW六维簇及相关立方的Franchetta性质与Chow环的消没结果。
We prove that the class of the classifying stack $B PGL_n$ is the multiplicative inverse of the class of the projective linear group $PGL_n$ in the Grothendieck ring of stacks for $n = 2$ and $n = 3$ under mild conditions on the base field $k$. In particular, although it is known that the multiplicativity relation $\{T\} = \{S\}\cdot\{PGL_n\}$ does not hold for all $PGL_n$-torsors $T \to S$, it holds for the universal $PGL_n$-torsors for said $n$.
We explicitly calculate some Gromov--Witten correspondences determined by maps of labeled curves of genus zero to the moduli spaces of labeled curves of genus zero. We consider these calculations as the first step towards studying the self--referential nature of motivic quantum cohomology.
We generalize some classical results on Chow group of an abelian variety to semiabelian varieties and to motivic (co)homology, using a result of Ancona--Enright-Ward--Huber on a decomposition of the motive of a semiabelian variety in the Voevodsky's category.
A motivic approach to rational $p$-adic cohomologies
本文综述母题同伦理论在p进上同调理论中的应用，并利用母题框架重新证明光滑射影超曲面的p进权-单值猜想。
We review some recent results and conjectures saying that, roughly speaking, periodic cyclic homology of a smooth non-commutative algebraic variety should carry all the additional "motivic" structures possessed by the usual de Rham cohomology of a smooth algebraic variety (specifically, an R-Hodge structure for varieties over R, and a filtered Dieudonne module structure for varieties over Z_p). To appear in Proc. ICM 2010.
Motivic Toda brackets
本文在不稳定motivic同伦论中构造Toda brackets，证明其基本性质，并给出若干具体例子。
The aim of this work is to construct certain homotopy t-structures on various categories of motivic homotopy theory, extending works of Voevodsky, Morel, Déglise and Ayoub. We prove these $t$-structures possess many good properties, some analogous to that of the perverse $t$-structure of Beilinson, Bernstein and Deligne. We compute the homology of certain motives, notably in the case of relative curves. We also show that their hearts provide convenient extensions of the theory of homotopy invariant sheaves with transfers, extending some of the main results of Voevodsky. These t-structures are …
For $k$ a perfect field of characteristic $p>0$ and $G/k$ a split reductive group with $p$ a non-torsion prime for $G,$ we compute the mod $p$ motivic cohomology of the geometric classifying space $BG_{(r)}$, where $G_{(r)}$ is the $r$th Frobenius kernel of $G.$ Our main tool is a motivic version of the Eilenberg-Moore spectral sequence, due to Krishna. For a flat affine group scheme $G/k$ of finite type, we define a cycle class map from the mod $p$ motivic cohomology of the classifying space $BG$ to the mod $p$ étale motivic cohomology of the classifying stack $\mathcal{B}G.$ This also gives …
Motivic cohomology and algebraic $K$-theory of some surfaces over finite fields
本文通过先计算动机上同调群、再分析动机 Atiyah-Hirzebruch 谱序列，得到有限域上若干类曲面的代数 $K$-理论，并在附录中扩展了 Parshin 猜想的已知适用范围。
We prove that if $X$ is a smooth projective variety of dimension greater than 1 over a field $K$ of characteristic zero such that $\operatorname{Pic}(X_{\bar{K}}) = \mathbb{Z}$ and $X_{\bar{K}}$ is simply connected, then the natural map $ρ: \operatorname{Aut}(X) \to \operatorname{Aut}(\operatorname{Sym}^d(X))$ is an isomorphism for every $d > 0$. We also partially compute the motivic zeta function of a Severi-Brauer surface and explain some relations between the classes of Severi-Brauer varieties in the Grothendieck ring of varieties.
The local motivic monodromy conjecture for simplicial nondegenerate singularities
本文证明了针对非退化奇点的局部动机monodromy猜想，并展示了其与局部拓扑zeta函数和Igusa zeta函数极点之间的联系。
We apply Wildeshaus's theory of motivic intermediate extensions to the motivic decomposition conjecture, formulated by Deninger-Murre and Corti-Hanamura. We first obtain a general motivic decomposition for the Chow motive of an arbitrary smooth projective family $f:X \rightarrow S$ whose geometric fibers are Tate. Using Voevodsky's motives with rational coefficients, the formula is valid for an arbitrary regular base $S$, without assuming the existence of a base field or even of a prime integer $\ell$ invertible on $S$. This result, and some of Bondarko' ideas, lead us to a generalized formula…
On Some Unramified Families of Motivic Euler Sums
本文应用 Glanois 的无分歧判据构造两类无分歧动机 Euler 和族，并在解析恒等式成立的假设下证明其中一族与动机 MZV 的具体关系。
The motivic homotopy categories can be defined with respect to different topologies and different underlying categories of schemes. For a number of reasons (mainly because of the Gluing Theorem) the motivic homotopy category built out of smooth schemes with respect to the Nisnevich topology plays a distinguished role but in some cases it is very desirable to be able to work with all schemes instead of the smooth ones. In this paper we prove that, under the resolution of singularities assumption, the unstable motivic homotopy category of all schemes over a field with respect to the cdh-topology…
We give a brief introduction to tensor triangulated geometry, a brief introduction to various motivic categories, and then make some observations about the conjectural structure of the tensor triangulated spectrum of the Morel-Voevodsky stable homotopy category over a finite field.
Donaldson-Thomas invariant is expressed as the weighted Euler characteristic of the so-called Behrend (constructible) function. In \cite{Behrend} Behrend introduced a DT-type invariant for a morphism. Motivated by this invariant, we extend the motivic Hirzebruch class to naive Donaldson-Thomas type analogues. We also discuss a categorification of the DT-type invariant for a morphism from a bivariant-theoretic viewpoint, and we finally pose some related questions for further investigations.
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 present some data on the cohomology of the motivic Steenrod algebra over an algebraically closed field. We discuss several features of the associated Adams spectral sequence, including the basic construction and convergence properties. The paper also deals with motivic versions of the May and Adams-Novikov spectral sequences. It is shown how these tools can be used to give new proofs of some classical results in algebraic topology. Also, the considerations reveal the existence of certain "exotic" motivic homotopy classes which have no classical analogues.
Old and new motivic cycles on Abelian surfaces
本文证明早先构造的、推广 $g=2$ 时 Collino 循环的新动机循环与更高亏格的 Collino 循环相关，从而说明新循环难寻且既有研究工具可沿用。
Odd rank vector bundles in eta-periodic motivic homotopy theory
本文研究eta-周期motivic稳定同伦范畴中奇秩向量丛的类截面行为，探讨了motivic环谱的SLc-定向与代数群étale分类空间的推论，并计算了对角化群的分类空间。
The term "motivic Moore spectrum" refers to a cone of an element in the motivic stable homotopy groups of spheres. This article discusses some properties of motivic Moore spectra, among them the question whether the ring structure on the motivic sphere spectrum descends to a ring structure on a motivic Moore spectrum. This discussion requires an understanding of some Toda brackets in the motivic stable homotopy groups of spheres.
The purpose of this survey is to explain some recent results about analogies between characteristic 0 and characteristic $p>0$ geometry, and to discuss an infinitesimal variant of motivic cohomology. More specifically, we review results showing that the big de Rham Witt complex of a field is a complex of 0-cycles (section 2), as well as results showing analogies between char. $p>0$ and char. 0 geometry. Some of those results determine local cohomology invariants of singular varieties (section 3) and others are concerned with congruences modulo $q$-powers for the number of rational points of va…
We prove that under some assumptions on an algebraic group $G$, indecomposable direct summands of the motive of a projective $G$-homogeneous variety with coefficients in $\mathbb{F}_p$ remain indecomposable if the ring of coefficients is any field of characteristic $p$. In particular for any projective $G$-homogeneous variety $X$, the decomposition of the motive of $X$ in a direct sum of indecomposable motives with coefficients in any finite field of characteristic $p$ corresponds to the decomposition of the motive of $X$ with coefficients in $\mathbb{F}_p$. We also construct a counterexample …
Algorithm for motivic Hilbert zeta function of some curve singularities
本文提出算法计算特定曲线奇点的两种动机Hilbertζ函数，并提供Python实现。
This paper has two aims. The former is to give an introduction to our earlier work on the Hodge theory of algebraic maps and more generally to some of the main themes of the theory of perverse sheaves and to some of its geometric applications. Particular emphasis is put on the topological properties of algebraic maps. The latter is to prove a motivic version of the decomposition theorem for the resolution of a threefold Y. This result allows to define a pure motive whose Betti realization is the intersection cohomology of Y. We provide a few explicit computations of perverse truncations and in…
Effective Bialynicki-Birula-Brosnan motivic decompositions
本文针对内型等径约化群对应的射影齐性簇，实现了Brosnan提出的动机分解的SageMath算法，将其推广到特征2的经典群并给出例外群的新分解。
These are notes of a series of talks about motivic integration I gave on the Münster Model Theory Month. Readers are assumed to have some basic knowledge of model theory and of valued fields. The notes are closest to the Cluckers-Loeser style of motivic integration, though mostly they are about doing p-adic integration uniformly in all $\mathbb{Q}_p$.
Motivic homological stability of configuration spaces
本文证明了流形中点的构型空间的经典同调稳定性结果可被提升至motivic上同调。
Let k be a field, let G be a finite group and let T be a split k-torus on which G acts multiplicatively, and for every m greater than 1 denote by T[m] the m-torsion subgroup of T. Under a suitable assumption on m, we show that the motivic class of the classifying stack of the semi-direct product of T[m] and G in K_0(Stacks_k) is trivial. As a consequence, we prove that the motivic class of BW is trivial for a large class of complex reflection groups W.
Some time ago the first author suggested two different constructions of Grassmannian n-logarithms. One of them works for all n, but its motivic nature is hidden. The other is a motivic but so far available only for n=2,3,4. We derived some general results about the first construction of Grassmannian polylogarithms, and found an explicit relation between the two constructions of the Grassmannian trilogarithms.
These notes grew out of several introductory talks I gave during the years 2003--2005 on motivic integration. They give a short but thorough introduction to the flavor of motivic integration which nowadays goes by the name of geometric motivic integration. As an illustration of the theory some applications to birational geometry are also included.
An object in motivic homotopy theory is called cellular if it can be built out of motivic spheres using homotopy colimit constructions. We explore some examples and consequences of cellularity. We explain why the algebraic K-theory and algebraic cobordism spectra are both cellular, and prove some Kunneth theorems for cellular objects.
These notes give an exposition of the theory of arithmetic motivic integration, as developed by J. Denef and F. Loeser. An appendix by M. Fried gives some historical comments on Galois stratifications.
The paper studies categories of definable subassignments with some category equivalences to semi-algebraic and constructible subsets of arc spaces of algebraic varieties. These materials allow us to compare the motivic measure of Cluckers-Loeser and the one of Denef-Loeser in certain classes of definable subassignments.
We show that the cellular objects in the module category over a motivic E infinity ring spectrum E can be described as the module category over a graded topological spectrum if E is strongly periodizable in our language. A similar statement is proven for triangulated categories of motives. Since MGL is strongly periodizable we obtain topological incarnations of motivic Landweber spectra. Under some categorical assumptions the unit object of the model category for triangulated motives is as well strongly periodizable giving motivic cochains whose module category models integral triangulated cat…
It oftens occurs that Taylor coefficients of (dimensionally regularized) Feynman amplitudes $I$ with rational parameters, expanded at an integral dimension $D= D_0$, are not only periods (Belkale, Brosnan, Bogner, Weinzierl) but actually multiple zeta values (Broadhurst, Kreimer). In order to determine, at least heuristically, whether this is the case in concrete instances, the philosophy of motives - more specifically, the theory of mixed Tate motives - suggests an arithmetic approach (Kontsevich): counting points of algebraic varieties related to $I$ modulo sufficiently many primes $p$ and c…
We derive some combinatorial consequences from the positivity of Donaldson-Thomas invariants for symmetric quivers conjectured by Kontsevich and Soibelman and proved recently by Efimov. These results are used to prove the Kac conjecture for quivers having at least one loop at every vertex.
We give an introduction to unstable motivic homotopy theory of Morel and Voevodsky, and survey some results.
Artin shapes
本文引入并研究了与射影齐次簇相关的动机的 Artin shape 概念，将其应用于求解该簇的完整动机分解，所得结果推翻了先前对射影齐次簇动机分解形态的预期。
In this short note we establish some properties of all those motivic measures which can be exponentiated. As a first application, we show that the rationality of Kapranov's zeta function is stable under products. As a second application, we give an elementary proof of a result of Totaro.
This paper is motivated by the question of how motivic Donaldson--Thomas invariants behave in families. We compute the invariants for some simple families of noncommutative Calabi--Yau threefolds, defined by quivers with homogeneous potentials. These families give deformation quantizations of affine three-space, the resolved conifold, and the resolution of the transversal $A_n$-singularity. It turns out that their invariants are generically constant, but jump at special values of the deformation parameter, such as roots of unity. The corresponding generating series are written in closed form, …
We develop a theory of local densities and tangent cones in a motivic framework, extending work by Cluckers-Comte-Loeser about $p$-adic local density. We prove some results about geometry of definable sets in Henselian valued fields of characteristic zero, both in semi-algebraic and subanalytic languages, and study Lipschitz continuous maps between such sets. We prove existence of regular stratifications satisfying analogous of Verdier condition $(w_f)$. Using Cluckers-Loeser theory of motivic integration, we define a notion of motivic local density with values in the Grothendieck ring of the …
We review Deitmar's theory of monoidal schemes to start with, and have a detailed look at the standard examples. It is explained how one can combinatorially study such schemes through a generalization of graph theory. In a more general setting we then introduce the author's version of $\mathbb{F}_1$-schemes (called $Υ$-schemes here), after which we study Grothendieck's motives in some detail in order to pass to "absolute motives". Throughout several considerations about absolute zeta functions are written. In a final part of the chapter, we describe the approach of Connes and Consani to unders…
On the motivic cohomology of some singular rings
本研究利用非$\text{A}^1$不变的动机上同调与整$p$进霍奇理论，对奇异环如$\text{Z}/p^n$等的代数K理论计算得到了动机细化结果，为相关领域提供了新的理论工具。
Motivic invariants of moduli stacks of Higgs bundles and bundles with connections: results and speculations
本文探讨了模堆栈Higgs丛和带连接丛的动机不变量，提出了未来研究方向及P=W猜想的推广。
In the paper M. Somekawa, {\it{On Milnor $K$-groups attached at semi-Abelian varieties}}, K-theory, \textbf{4} (1990) p.105, Somekawa conjectures that his Milnor K-group $K(k,G_1,...,G_r)$ attached to semi-abelian varieties $G_1$,...,$G_r$ over a field $k$ is isomorphic to ${\rm Ext}_{\mathcal{M}_k}^r(\mathbb{Z},G_1[-1] \otimes ... \otimes G_r[-1])$ where $\mathcal{M}_k$ is a certain category of motives over $k$. The purpose of this note is to give remarks on this conjecture, when we take $\mathcal{M}_k$ as Voevodsky's category of motives ${\rm DM}^{eff}_{-}(k)$ .
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