Giuseppe Ancona
We show that the l-adic realization functor is conservative when restricted to the Chow motives of abelian type over a finite field. A weak version of this conservativity result extends to mixed motives of abelian type.
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We prove: if the (étale or de Rham) realization functor is conservative on the category $DM_{gm}$ of Voevodsky motives with rational coefficients then motivic zeta functions of arbitrary varieties are rational and numerical motives are Kimura-finite. The latter statement immediately implies that the category of numerical motives is (essentially) Tannakian. This observation becomes actual due to the recent announcement of J. Ayoub that the De Rham cohomology realization is conservative on $DM_{gm}(k)$ whenever $\operatorname{char} k=0$. We apply this statement to exterior powers of motives comi…
Given a 0-connective motivic spectrum $E \in SH(k)$ over a perfect field k, we determine $h_0$ of the associated motive $M E \in DM(k)$ in terms of $π_0 (E)$. Using this we show that if k has finite 2-étale cohomological dimension, then the functor M is conservative when restricted to the subcategory of compact spectra, and induces an injection on Picard groups. We extend the conservativity result to fields of finite virtual 2-étale cohomological dimension by considering what we call "real motives". As a by-product we reprove a variant of a rigidity Theorem of Röndings-Østvær.
We study mixed versions of the classical quotient functor from Chow motives to numerical motives. We compare two natural definitions, which turn out to be very different. We investigate fullness, conservativity and exactness of these two functors.
Deligne's duality for de Rham realizations of 1-motives
我们表明，配对德拉姆实现的1-动机在“理论的霍奇III”，IHES 44，可以定义在任何基础计划，我们证明，它会产生一个完美的对偶，如果一个是工作与1-动机和卡地亚双。进一步研究了1-模的泛扩张及其与$\natural$ -扩张的关系。
We give an alternative construction of the Betti realization functor on the derived category of motives of complex algebraic varieties via the category of CW complexes instead of the category of complex analytic spaces. In particular we show that the functor we define via the category of CW complexes coincide with Ayoub's one. We deduce from this construction that Ayoub's realization functor on geometric motives factors trough Nori motives and that the image of this functor on the morphisms between the motive of a point and a shift of a Tate twist of the motive with compact support of a comple…
We introduce the categories of geometric mixed Hodge modules on algebraic varieties over a subfield $k\subset\mathbb C$, and for a prime number $p$, the categories of geometric $p$-adic mixed Hodge modules on algebraic varieties over a subfield $k\subset K\subset\mathbb C_p$ of a p-adic field $K$. We then give an Hodge realization functor on the de derived category of relative motives over $k\subset\mathbb C$, and for a prime number $p$ a $p$-adic Hodge realization functor on the derived category of relative motives over $k\subset K\subset\mathbb C_p$.
We give, for a complex algebraic variety $S$, a Hodge realization functor $\mathcal F_S^{Hdg}$ from the derived category of constructible motives $DA_c(S)$ to the derived category $D(MHM(S))$ of algebraic mixed Hodge modules over $S$. Moreover, for $f:T\to S$ a morphism of complex quasi-projective algebraic varieties, $\mathcal F_{-}^{Hdg}$ commutes with the four operation $f^*$,$f_*$,$f_!$,$f^!$ on $DA_c(-)$ and $D(MHM(-))$, making the Hodge realization functor a morphism of 2-functor wich for a given $S$ sends $DA_c(S)$ to $D(MHM(S)$, moreover $\mathcal F_S^{Hdg}$ commutes with tensor produc…
We prove that the embedding of the derived category of 1-motives into the triangulated category of effective Voevodsky motives, as well as its left adjoint functor $LAlb$, commute with the Hodge realization.
We introduce the motivic coniveau exact couple of a smooth scheme, in the framework of mixed motives, whose property is to universally give rise to coniveau spectral sequences through realizations. The main result is a computation of its differentials in terms of residues and transfers of mixed motives, with a formula analog to the one defining the Weil divisor of a rational function. We then show how to recover and extend classical results of Bloch and Ogus for motivic realizations.
We prove that for $1$-motives defined over an algebraically closed subfield of $\C$, viewed as Nori motives, the motivic Galois group is the Mumford-Tate group. In particular, the Hodge realization of the tannakian category of (Nori) motives generated by $1$-motives is fully faithful.
Crystalline realizations of 1-motives
我们考虑Deligne的1-动机的正特征的晶体实现，并证明了一个比较定理与De Rham实现的提升到零特征。然后，我们表明，一维晶体上同调的代数品种，定义通过德容的变更普遍上同调下降，符合晶体实现（上同调）皮卡1动机，在完满域。
Assuming the Künneth type standard conjecture, we propose a way to describe objects of mixed motives explicitly. We study their formal properties, and we associate mixed motives to schemes smooth and separated over a field. This serves as a universal cohomology theory. We also discuss $\ell$-adic realizations, and we discuss an unconditional construction of 2-motives and their properties.
Let EHM be Nori's category of effective homological mixed motives. In this paper, we consider the thick abelian subcategory EHM_1 generated by the i-th relative homology of pairs of varieties for i = 0,1. We show that EHM_1 is naturally equivalent to the abelian category M_1 of Deligne 1-motives with torsion; this is our main theorem. Along the way, we obtain several interesting results. Firstly, we realize M_1 as the universal abelian category obtained, using Nori's formalism, from the Betti representation of an explicit diagram of curves. Secondly, we obtain a conceptual proof of a theorem o…
Pro-étale motives and solid rigidity
本文引入了概率-étale 动力学与谱，并展示其赋予六种运算。通过凝聚范畴理论定义固有过程，证明其与Fargues-Scholze的固有sheaves接近，从而获得固有动力学实现函子，扩展了ℓ-adic实现函子。
We introduce the notion of Artin motives and cohomological motives over a scheme X. Given a cohomological motive M over X, we construct its punctual weight zero part $ω^0_X(M)$ as the universal Artin motive mapping to M. We use this to define a motive E_X over X which is an invariant of the singularities of X. The first half of the paper is devoted to the study of the functors $ω^0_X$ and the computation of the motives E_X. In the second half of the paper, we develop the application to locally symmetric varieties. Specifically, let Y be a locally symmetric variety and denote by p:W-->Z the pro…
Solid realization of motives with modulus
本文构造了带模动机的固体实现函子，并利用pro-固体比较定理计算了特定开浸入锥的固体实现，通过形式完备化给出显式描述。
After introducing the Ogus realization of 1-motives we prove that it is a fully faithful functor. More precisely, following a framework introduced by Ogus, considering an enriched structure on the de Rham realization of 1-motives over a number field, we show that it yields a full functor by making use of an algebraicity theorem of Bost.
We prove that the functor associating to a rigid analytic variety the singular complex of the underlying Berkovich topological space is motivic, and defines the maximal Artin quotient of a motive. We use this to generalize Berkovich's results on the weight-zero part of the étale cohomology of a variety defined over a non-archimedean valued field.
Albanese and Picard 1-motives
本文用纯代数的方法定义了特征为零的代数闭域上任意代数模式$X$的1-基序$Alb^{+}(X)$，$Alb^{-}(X)$，$Pic^{+}(X)$和$Pic^{-}(X)$.对于$n$维的$X$ over $\C$，Hodge实现分别是$H^{2n-1}(X)(n)$、$H_{1}(X)$、$H^{1}(X)(1)$和$H_{2n-1}(X)(1-n)$。
Categorical Künneth formulas for cohomological motives
本文研究范畴化 Künneth 公式，给出抽象六函子形式体系的判定准则，并证明 adic 层与上同调动机的相应公式以支持 étale 动机范畴猜想。
We show that the l-adic realizations of certain Picard 1-motives associated to a G-Galois cover of smooth, projective curves defined over an algebraically closed field are G-cohomologically trivial, for all primes l. In the process, we generalize a well-known theorem of Nakajima on the Galois module structure of certain spaces of Kahler differentials associated to the top curve of the cover, assuming that the field of definition is of characteristic p. If the cover and the Picard 1-motive are defined over a finite field and if G is abelian, we compute the first Fitting ideal of each l-adic rea…
This paper is concerned with an interpretation of f-cohomology, a modification of motivic cohomology of motives over number fields, in terms of motives over number rings. Under standard assumptions on mixed motives over finite fields, number fields and number rings, we show that the two extant definitions of f-cohomology of mixed motives $M_η$ over F--one via ramification conditions on $\ell$-adic realizations, another one via the K-theory of proper regular models--both agree with motivic cohomology of $η_{!*} M_η[1]$. Here $η_{!*}$ is constructed by a limiting process in terms of intermediate…
We prove an effective Hilbert Irreducibility result for residual realizations of a family of motives with motivic Galois group G2.
Crystalline Realizations of 1-Motives: Integral Structure and Classification via Linear Frobenius
本文在1-动机范围内证明了p进版本的塔猜想，通过Deligne 1-动机的全忠实性定理，将Barsotti-Tate晶体函子与滤过Dieudonné模的Hom群联系起来，从而得出Frobenius兼容的晶体类本质上是动机的，进而给出动机映射的线性代数描述。
A Voevodsky motive associated to a log scheme
本文为域上 fs 对数概形构造几何 Voevodsky 动机，证明复数域上其 Betti 实现等于对数 Betti 上同调，并应用于管状邻域、极限动机和单值滤子。
We describe the Voevodsky's category $DM^{eff}_{gm}$ of motives in terms of Suslin complexes of smooth projective varieties. This shows that Voeovodsky's $DM_{gm}$ is anti-equivalent to Hanamura's one. We give a description of any triangulated subcategory of $DM^{eff}_{gm}$ (including the category of effective mixed Tate motives). We descibe 'truncation' functors $t_N$ for $N>0$. $t=t_0$ generalizes the weight complex of Soule and Gillet; its target is $K^b(Chow_{eff})$; it calculates $K_0(DM^{eff}_{gm})$, and checks whether a motive is a mixed Tate one. $t_N$ give a weight filtration and a 'm…
Relative motives and the theory of pseudo-finite fields
我们推广的动机化身态射从算术积分理论的相对情况下，我们工作在一个基品种S的特征为零的字段K。我们发展了一个关于S的可构建有效Chow动机的理论，并展示了如何将动机与任何S-簇联系起来。给出了伪有限域相对量词消去的一个几何证明，并构造了S上伪有限域理论的Grothendieck环到Q与可构造有效Chow模的Grothendieck环的张量积的一个态射.这种态射产生了一个动机实现的参数化的算术motivic积分的Cluckers-Loeser。最后，定义了相对弧空间和相对喷流空间，以及三个相对运动生成序列。
We define a theory of etale motives over a noetherian scheme. This provides a system of categories of complexes of motivic sheaves with integral coefficients which is closed under the six operations of Grothendieck. The rational part of these categories coincides with the triangulated categories of Beilinson motives (and is thus strongly related to algebraic $K$-theory). We extend the rigity theorem of Suslin and Voevodsky over a general base scheme. This can be reformulated by saying that torsion etale motives essentially coincide with the usual complexes of torsion etale sheaves (at least if…
We define 1-motives of a variety X over a perfect field of positive characteristic which realize the etale cohomology groups of X in dimension and codimension one. This is the analogue in positive characteristic of previous results of Barbieri-Viale and Srinivas, except that we only consider the etale realization but also consider compactly supported cohomology. The dimension-1 case (called the Picard 1-motives) can be done by standard techniques, and indeed this case is probably well known. But the codimension-one case (Albanese 1-motive) requires stronger tools, namely a strong version of de…
Using the localization property, we construct a triangulated category of motives over quasi-projective T-schemes for any coefficient where T is a noetherian separated scheme, and we prove the Grothendieck six operations formalism. We also construct integral étale realization of motives.
The Grothendieck ring of varieties has well-known realization maps to, say, mixed Hodge structures or compactly supported $\ell$-adic cohomology. Zakharevich and\ Campbell have developed {a spectral refinement} of the Grothendieck ring of varieties. We develop a realization map to Voevodsky mixed motives, and this lifts the standard realizations of motives to this setting, at least over perfect fields which have resolution of singularities.
Hypercomplete étale framed motives and comparison of stable homotopy groups of motivic spectra and étale realizations over a field
本文在 motivic 同伦论框架下，构造$l$-进完备超完备平展标架motive理论，证明相关函子的交换性与识别原理，推导平展局部同构，并给出Levine比较同构及其代数闭域推广的新证明，结果适用于任意基域。
In this thesis we compare V. Voevodsky's geometric motives to the derived category of M. Nori's abelian category of mixed motives by constructing a triangulated tensor functor between them. It will be compatible with the Betti realizations on both sides. We allow an arbitrary noetherian ring of coefficients, but require it to be a field or a Dedekind domain for the tensor structure on derived Nori motives to exist. There are three key ingredients: we present a theory of Nisnevich covers on finite acyclic diagrams of finite correspondences, explain, following D. Rydh, how to interpret finite co…
We study genus 2 Hilbert-Siegel varieties, i.e. Shimura varieties $S_K$ corresponding to the group $\mbox{GSp}_{4,F}$ over a totally real field $F$, along with the relative Chow motives $^λ\mathcal{V}$ of abelian type over $S_K$ obtained from irreducible representations $V_λ$ of $\mbox{GSp}_{4,F}$. We analyse the weight filtration on the degeneration of such motives at the boundary of the Baily-Borel compactification and we find a criterion on the highest weight $λ$ which characterises the absence of the middle weights 0 and 1 in the corresponding degeneration. Thanks to Wildeshaus' theory, th…
A Criterion for Phantomness of dg-categories
本文研究了光滑proper dg范畴的幻性是否由加法不变量的消失所刻画，构造了非紧致动机并证明其l-adic实现在代数K-理论中的作用，同时给出了在Chow动机为Kimura有限时幻性的判别准则。
We construct the dagger realization functor for analytic motives over non-archimedean fields of mixed characteristic, as well as the Monsky-Washnitzer realization functor for algebraic motives over a discrete field of positive characteristic. In particular, the motivic language on the classic étale site provides a new direct definition of the overconvergent de Rham cohomology and rigid cohomology and shows that their finite dimensionality follows formally from the one of Betti cohomology for smooth projective complex varieties.
Sharp de Rham realization
我们介绍了“尖锐”（普遍）的扩展1动机（添加剂的因素和扭转）的领域的特征为零。我们通过传递到李代数来定义“夏普德拉姆实现”。在复数上，我们证明了形式Hodge结构范畴中的一个（sharp de Rham）比较定理.对于一个自由的1-动机及其Cartier对偶，我们得到了一个关于它们的尖锐扩张的正则联系，从而产生了一个关于尖锐实现的完美配对。因此，我们提供了“一维sharp de Rham上同调”的代数簇。
We first study the weight structure on the triangulated category of Artin-Tate motives over a perfect base field k, building on results of Bondarko's. We then study the t-structure on the triangulated category of Artin-Tate motives, when k is algebraic over the rationals, generalizing a result of Levine's. We finally study the interaction of the weight structure and the t-structure. When k is a number field, this will give a useful criterion identifying the weight structure via realizations.
K-motives, Springer Theory and the Local Langlands Correspondence
本文建立K-动机式Springer理论，几何实现仿射Hecke代数与分裂约化p进群表示范畴，并通过范畴化Chern特征连接凝聚Springer理论与Langlands纲领。
Berkovich $2$-motives and normed ring stacks
本文提出并证明了动机同调理论核范畴与同调平凡的光滑缝合环堆栈的关系，并探讨了其在分析几何中的应用。
This book discusses the construction of triangulated categories of mixed motives over a noetherian scheme of finite dimension, extending Voevodsky's definition of motives over a field. In particular, it is shown that motives with rational coefficients satisfy the formalism of the six operations of Grothendieck. This is achieved by studying descent properties of motives, as well as by comparing different presentations of these categories, following and extending insights and constructions of Deligne, Beilinson, Bloch, Thomason, Gabber, Levine, Morel, Voevodsky, Ayoub, Spitzweck, Röndigs, Østvær…
The motive of the moduli stack of G-bundles over the universal curve
一个相对纯动机的阿贝尔范畴是沿着André的思路构造的（在特征为0的域上）。一个代数栈被证明拥有在这个意义上的动机。对适当的G，研究了泛曲线上G-丛的模叠加的动力，得到了动力的Atiyah-Bott公式.通过引入Hodge实现，给出了M_g上Hodge结构的相应变化.特别是，它遵循这些谎言在张量类别所产生的变化霍奇结构相关的曲线本身。对于与稳定G-丛的模空间相关的Hodge结构的低次变分也给出了类似的结果。
A Weight Structure on Rigid Analytic Motives over a Field
本文通过伽罗瓦下降在局部域上刚性解析母题的稳定 ∞-范畴上构造幺半权结构，推广了 Binda-Gallauer-Vezzani 的好约化情形，并证明 Hyodo-Kato 实现经由权复形函子分解，从而得到收敛于 Hyodo-Kato 上同调的谱序列与权滤过。
Formal Hodge Theory
正式（混合）霍奇结构FHS介绍这样一种方式，霍奇实现德利涅的1-动机延伸到实现从劳蒙的1-动机正式霍奇结构的1级，提供一个等价的类别。为了便于说明，我们在这里将我们的研究限制在1级混合Hodge结构。然而，考虑具有任意Hodge数的形式混合Hodge结构是可以想象和合适的：推广我们的定义并不困难，我们将在下一节中处理这样的问题。
1-motives and admissible variations of mixed Hodge structures
本文证明，在复数域上光滑有限型连通概形S上，每个类型为(0,0),(-1,0),(0,-1),(-1,-1)的容许混合Hodge结构变差（至多相差同源）都来自S上的一个1-动机，从而肯定回答了André关于此类变差几何起源的问题。
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…
P-adic Period Conjectures for 1-motives: Integration and Linear Relations
本文为1-动机建立p进周期理论，构造p进积分配对与分层周期空间，提出并证明深度1和2的周期猜想，并发展p进解析子群定理，为研究p进周期线性关系及超越性开辟新途径。
In this paper we establish a precise comparison between vanishing cycles and the singularity category of Landau-Ginzburg models over a complete discrete valuation ring. By using noncommutative motives, we first construct a motivic $\ell$-adic realization functor for dg-categories. Our main result, then asserts that, given a Landau-Ginzburg model over a complete discrete valuation ring with potential induced by a uniformizer, the $\ell$-adic realization of its singularity category is given by the inertia-invariant part of vanishing cohomology. We also prove a functorial and $\infty$-categorical…
Over a subfield of the field of complex numbers, the Hodge realization of a geometrical motive is defined and represented as the cohomology of a mixed Hodge DG-complex in the sense of Deligne. Both filtrations are represented by truncation functors, on a Bondarko weight complex for the weight filtration and on the De Rham motivic complex for the Hodge one. The Deligne-Beilinson realization is also constructed. This preprint partially replaces the former "Réalisation des complexes motiviques de Voevodsky".
Unstable étale motives
本文在有界 étale 上同调维数的基概形上证明 anima 的 p-完备 étale A¹-不变层刚性，并用于构造不稳定 étale 实现函子。
We construct motives over the rational numbers associated with symmetric power moments of Kloosterman sums, and prove that their L-functions extend meromorphically to the complex plane and satisfy a functional equation conjectured by Broadhurst and Roberts. Although the motives in question turn out to be "classical", we compute their Hodge numbers by means of the irregular Hodge filtration on their realizations as exponential mixed Hodge structures. We show that all Hodge numbers are either zero or one, which implies potential automorphy thanks to recent results of Patrikis and Taylor.
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 perform Hochschild homology calculations in the algebro-geometric setting of motives. The motivic Hochschild homology coefficient ring contains torsion classes which arise from the mod-$p$ motivic Steenrod algebra and from generating functions on the natural numbers with finite non-empty support. Under the Betti realization, we recover Bökstedt's calculation of the topological Hochschild homology of finite prime fields.
This thesis establishes a geometric approach to the de Rham realization of the polylogarithm. As a central result we construct the logarithm sheaves of rational abelian schemes in terms of the birigidified Poincaré bundle with universal integrable connection on the product of the abelian scheme and the universal vectorial extension of its dual. This is achieved essentially by restricting the mentioned data of the Poincaré bundle along the infinitesimal neighborhoods of the zero section of the universal extension. We also clarify how these constructions naturally express within the language of …
We give an algebro-geometric interpretation of $C_2$-equivariant stable homotopy theory by means of the $b$-topology introduced by Claus Scheiderer in his study of $2$-torsion phenomena in étale cohomology. To accomplish this, we first revisit and extend work of Scheiderer on equivariant topos theory by functorially associating to a $\infty$-topos $\mathscr{X}$ with $G$-action a presentable stable $\infty$-category $\mathrm{Sp}^G(\mathscr{X})$, which recovers the $\infty$-category $\mathrm{Sp}^G$ of genuine $G$-spectra when $\mathscr{X}$ is the terminal $G$-$\infty$-topos. Given a scheme $X$ w…
We give a closed expression for the number of points over finite fields (or the motive) of the Lusztig nilpotent variety associated to any quiver, in terms of Kac's A-polynomials. When the quiver has 1-loops or oriented cycles, there are several possible variants of the Lusztig nilpotent variety, and we provide formulas for the point count of each. This involves nilpotent versions of the Kac A-polynomial, which we introduce and for which we give a closed formula similar to Hua's formula for the usual Kac A-polynomial. Finally we compute the number of points over a finite field of the various s…
Coarse moduli of motivic augmentations
本文提出动机增广的粗模空间，建立可表示性准则，计算实例并与Selmer簇比较，为Chabauty-Kim理论和整点界提供新工具。
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