Gerbes and abelian motives

Mixed motives and quotient stacks: Abelian varieties

We prove that the symmetric monoidal category of mixed motives generated by an abelian variety (more generally, an abelian scheme) can be described as a certain module category. More precisely, we describe it as the category of quasi-coherent complexes over a derived quotient stack constructed from a motivic algebra of the abelian variety. We then study the structure of the motivic Galois groups of their mixed motives. We prove that the motivic Galois group is decomposed into a unipotent part constructed from the motivic algebra, and the reductive quotient which is the Tannaka dual of Grothend…

Conservativity of realizations on motives of abelian type over finite fields

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.

A family of special cubic fourfolds with motive of abelian type

In this short note, we show that there exist one dimensional families of cubic fourfolds with Chow motive of abelian type and finite dimensional inside every Hassett divisor of special cubic fourfolds. This also implies abelianity and finite dimensionality of the motive of related Hyperkähler varieties, such as the Fano variety of lines and the LLSvS 8fold.

Intermediate extension of Chow motives of Abelian type

In this article, we give an unconditional construction of a motivic analogue of the intermediate extension in the context of Chow motives of Abelian type. Our main application concerns intermediate extensions of Chow motives associated to Kuga families to the Baily--Borel compactification of a Shimura variety.

具有阿贝尔动机的K3曲面与三次四维流形

K3 surfaces and cubic fourfolds with Abelian motive

本文证明高代数2-圈秩的三次四维流形及其相关超凯勒簇具有阿贝尔且有限维的Chow动机，并从四维K3曲面族构造出两族动机有限维的Fano簇。

A Candidate for the abelian category of mixed elliptic motives

In this paper we suggest a definition for the category of mixed motives generated by the motive h^1(E) for E an elliptic curve without complex multiplication. We then compute the cohomology of this category. Modulo a strengthening of the Beilinson-Soule conjecture, we show that the cohomology of our category agrees with the expected motivic cohomology groups. Finally for each pure motive (Sym^{n}h^1(E))(-1) we construct families of nontrivial motives whose highest associated weight graded piece is $Sym^{n}h^1(E))(-1). This paper was essentially written in the late 1990's whilst the author was …

Remarks on motives of abelian type

A motive over a field $k$ is of abelian type if it belongs to the thick and rigid subcategory of Chow motives spanned by the motives of abelian varieties over $k$. This paper contains three sections of independent interest. First, we show that a motive which becomes of abelian type after a base field extension of algebraically closed fields is of abelian type. Given a field extension $K/k$ and a motive $M$ over $k$, we also show that $M$ is finite-dimensional if and only if $M_K$ is finite-dimensional. As a corollary, we obtain Chow--Kuenneth decompositions for varieties that become isomorphic…

阿贝尔簇上 $\mathbb{G}_m$-gerbes 的最小平凡化同源与周期-指数问题

Minimal Trivializing Isogenies of $\mathbb{G}_m$-gerbes over Abelian Varieties and Period-Index Problem

本文通过引入新不变量 $Ind_{SH}(\alpha)$ 并研究 $\mathbb{G}_m$-gerbe 的 $\mu_n$-提升和 $1$-扭曲半齐次向量丛，改进了周期-指数关系，并证明了周期-指数猜想对任意维椭圆曲线乘积成立。

Higgs bundles, abelian gerbes and cameral data

We study the Hitchin map for $G_{\mathbb{R}}$-Higgs bundles on a smooth curve, where $G_{\mathbb{R}}$ is a quasi-split real form of a complex reductive algebraic group $G$. By looking at the moduli stack of regular $G_{\mathbb{R}}$-Higgs bundles, we prove it induces a banded gerbe structure on a slightly larger stack, whose band is given by sheaves of tori. This characterization yields a cocyclic description of the fibres of the corresponding Hitchin map by means of cameral data. According to this, fibres of the Hitchin map are categories of principal torus bundles on the cameral cover. The co…

The conjugate uniformization via 1-motives

We use the $p$-divisible group attached to a 1-motive to generalize the conjugate $p$-adic uniformization of Iovita--Morrow--Zaharescu to arbitrary $p$-adic formal semi-abelian schemes or $p$-divisible groups over the ring of integers in a $p$-adic field. This mirrors a mixed Hodge theory construction of the inverse uniformization map for complex semi-abelian varieties. We also highlight the geometric structure of the target of the conjugate uniformization map, which is an étale cover of a negative Banach--Colmez space in the sense of Fargues--Scholze.

具有半稳定约化的 Motive 同构类中高度的有限性

Finiteness of Heights in Isogeny Classes of Motives with Semistable Reduction

本文利用积分 $p$-adic Hodge 理论推广了阿贝尔簇的 Faltings 高度，并在假设 adelic Mumford-Tate 猜想下，证明了 motive 同构类中高度的有限性，扩展了 Kisin 和 Mocz 的相关结果。

非阿贝尔雷斯构造与纯动机

Non-abelian Rees construction and pure motives

本文基于非阿贝尔雷斯构造，建立准齐性空间与单调范畴之间的伽罗瓦对应关系，并应用于具有具体应用的动机单调范畴，推导了关于有限域上阿贝尔簇数值等价的新的证明与推广。

Distinguished cycles on varieties with motive of abelian type and the Section Property

A remarkable result of Peter O'Sullivan asserts that the algebra epimorphism from the rational Chow ring of an abelian variety to its rational Chow ring modulo numerical equivalence admits a (canonical) section. Motivated by Beauville's splitting principle, we formulate a conjectural Section Property which predicts that for smooth projective holomorphic symplectic varieties there exists such a section of algebra whose image contains all the Chern classes of the variety. In this paper, we investigate this property for (not necessarily symplectic) varieties with Chow motive of abelian type. We i…

On deformations of $1$-motives

According to a well-known theorem of Serre and Tate, the infinitesimal deformation theory of an abelian variety in positive characteristic is equivalent to the infinitesimal deformation theory of its Barsotti-Tate group. We extend this result to $1$-motives.

The gerbe of Higgs bundles

The purpose of this work is to describe the (category of) Higgs bundles on a complex scheme X having a given cameral cover X~. We show that this category is a T_{X~}-gerbe, where T_{X~} is a certain sheaf of abelian groups on X, and we describe the class of this gerbe precisely. In particular, it follows that the set of isomorphism classes of Higgs bundles with a fixed cameral cover X~ is a torsor over the group H^1(X, T_{X~}), which itself parametrizes T_{X~}-torsors on X. This underlying group can be described as a generalized Prym variety, whose connected component is either an abelian vari…

Neron models of 1-motives and duality

In this paper, we propose a definition of Neron models of arbitrary Deligne 1-motives over Dedekind schemes, extending Neron models of semi-abelian varieties. The key property of our Neron models is that they satisfy a generalization of Grothendieck's duality conjecture in SGA 7 when the residue fields of the base scheme at closed points are perfect. The assumption on the residue fields is unnecessary for the class of 1-motives with semistable reduction everywhere. In general, this duality holds after inverting the residual characteristics. The definition of Neron models involves careful treat…

Galois descent for motives: the K3 case

A theorem of Grothendieck tells us that if the Galois action on the Tate module of an abelian variety factors through a smaller field, then the abelian variety, up to isogeny and finite extension of the base, is itself defined over the smaller field. Inspired by this, we give a Galois descent datum for a motive $H$ over a field by asking that the Galois action on an $\ell$-adic realisation factor through a smaller field. We conjecture that this descent datum is effective, that is if a motive $H$ satisfies the above criterion, then it must itself descend to the smaller field. We prove this conj…

Product Formulas for Periods of CM Abelian Varieties and the Function Field Analog

We survey Colmez's theory and conjecture about the Faltings height and a product formula for the periods of abelian varieties with complex multiplication, along with the function field analog developed by the authors. In this analog, abelian varieties are replaced by Drinfeld modules and $A$-motives. We also explain the necessary background on abelian varieties, Drinfeld modules and $A$-motives, including their cohomology theories and comparison isomorphisms and their theory of complex multiplication.

On the motive of some hyperKaehler varieties

We show that the motive of the Hilbert scheme of length-$n$ subschemes on a K3 surface or on an abelian surface admits a decomposition similar to the decomposition of the motive of an abelian variety obtained by Shermenev, Beauville, and Deninger and Murre.

Deformation principle and André motives of projective hyperkähler manifolds

Let $X_1$ and $X_2$ be deformation equivalent projective hyperkähler manifolds. We prove that the André motive of $X_1$ is abelian if and only if the André motive of $X_2$ is abelian. Applying this to manifolds of $\mbox{K3}^{[n]}$, generalized Kummer and OG6 deformation types, we deduce that their André motives are abelian. As a consequence, we prove that all Hodge classes in arbitrary degree on such manifolds are absolute. We discuss applications to the Mumford-Tate conjecture, showing in particular that it holds for even degree cohomology of such manifolds.

On compatibility of the $\ell$-adic realisations of an abelian motive

In this article we introduce the notion of a quasi-compatible system of Galois representations. The quasi-compatibility condition is a slight relaxation of the classical compatibility condition in the sense of Serre. The main theorem that we prove is the following: Let $M$ be an abelian motive, in the sense of Yves André. Then the $\ell$-adic realisations of $M$ form a quasi-compatible system of Galois representations. (In fact, we actually prove something stronger. See theorem 5.1.) As an application, we deduce that the absolute rank of the $\ell$-adic monodromy groups of $M$ does not depend …

Integral Motives and Special Values of Zeta Functions

For each field k, we define an abelian category of rationally decomposed mixed motives with integer coefficients. When k is finite, we show that the category is Tannakian, and we prove formulas relating the behaviour of zeta functions near integers to certain Ext groups. This is the submitted version, with minor corrections and additions from the first version.

The Voevodsky motive of a rank one semiabelian variety

Over a perfect field k, let G be an extension of an abelian variety by the multiplicative group $\G_m$. We compute the motive of G in Voevodsky's category of etale motivic complexes with rational coefficients. The result is a decomposition similar to the classical decomposition of the Chow motive of an abelian variety obtained by Deninger/Murre, Kuennemann, Shermenev and Scholl, and other authors.

Chow motives associated to certain algebraic Hecke characters

Shimura and Taniyama proved that if $A$ is a potentially CM abelian variety over a number field $F$ with CM by a field $K$ linearly disjoint from F, then there is an algebraic Hecke character $λ_A$ of $K$ such that $L(A/F,s)=L(λ_A,s)$. We consider a certain converse to their result. Namely, let $A$ be a potentially CM abelian variety appearing as a factor of the Jacobian of a curve of the form $y^e=γx^f+δ$. Fix positive integers $a$ and $n$ such that $n/2 < a \leq n$. Under mild conditions on $e, f, γ, δ$, we construct a Chow motive $M$, defined over $F=\mathbb{Q}(γ,δ)$, such that $L(M/F,s)$ a…

$p$-进数域上 $1$-motives 的凝聚 Weil-étale 实现的对偶性

Duality for the condensed Weil-étale realisation of $1$-motives over $p$-adic fields

本文以凝聚 Weil 群上同调替代 Galois 上同调，引入 $1$-motives 的凝聚 Weil-étale 实现，从而将 Tate 对偶推广到 $p$-进数域上的 $1$-motives。

Nori 1-motives

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…

Fundamental gerbes

For a class of affine algebraic groups $\mathcal C$ over a field, we define the notions of $\mathcal C$-fundamental gerbe of a fibered category, generalizing what we had done in arXiv:1204.1260 for finite group schemes. We give sufficient conditions on $\mathcal C$ implying that a fibered category $X$ over $κ$ satisfying mild hypotheses admits a Nori $\mathcal C$-fundamental gerbe. We show that these are verified in particular by the classes of virtually abelian and virtually unipotent group schemes. In the second situation, under a properness condition on $X$, we give a tannakian interpretati…

Gerbes for the Chow

Finding coherent relations to define non Abelian cohomology is a thriller which entertains the mathematical community since fifty one years. The purpose of this paper is to simplify the attempt to beat it defined by the author which used the notion of sequences of fibred categories and to apply the resulting theory to higher divisors and Chow theory.

On the Chow motive of an abelian scheme with non-trivial endomorphisms

Let X be an abelian scheme over a base variety S with endomorphism algebra D. We prove that the relative Chow motive R(X/S) has a canonical decomposition as a direct sum of motives R^(ξ)$ where ξruns over an explicitly determined finite set of irreducible representations of the group D^{opp,*}, such that R^(ξ), seen as a functor from Chow motives to D^{\opp,*}-representations, is ξ-isotypic. Our decomposition refines the motivic decomposition of Deninger and Murre, as well as Beauville's decomposition of the Chow group. The second main result is that we construct a canonical generalized motivi…

阿贝尔动机模空间的$p$进双曲性

$p$-adic hyperbolicity for moduli spaces of abelian motives

本研究证明阿贝尔型Shimura簇在离散赋值域上满足$p$进Borel延拓性质，推导了其在多领域的应用，并证明了Rapoport-Zink空间的延拓结果。

滤过阿贝尔范畴中的混合扩张理论与具有极大幂单根的 motive

On blended extensions in filtered abelian categories and motives with maximal unipotent radicals

本文推广 Grothendieck 混合扩张理论至任意有限步滤过，用于分类具有固定相伴分次的对象，并应用于混合 motive，在分次独立条件下证明同调分类结果。

The motive of the moduli stack of G-bundles over the universal curve

An abelian category of relative pure motives is constructed along the lines of André (over a field of characteristic 0). An algebraic stack is shown to possess a motive in this sense. This motive is studied for the moduli stack of G-bundles over the universal curve for appropriate G, and a motivic Atiyah-Bott formula is obtained. By taking Hodge realization, a description of the corresponding variations of Hodge structures over M_g is given. In particular, it follows that these lie in the tensor category generated by the variation of Hodge structure associated to the curve itself. Similar resu…

Homotopy classification of gerbes

Gerbes are locally connected presheaves of groupoids. They are classified up to local weak equivalence by path components in a 2-cocycle category taking values in all sheaves of groups, their isomorphisms and homotopies. If F is a full presheaf of sheaves of groups, isomorphisms and homotopies, then [*,BF] is isomorphic to equivalence classes of gerbes locally equivalent to groups appearing in F. Giraud's non-abelian cohomology object of equivalence classes of gerbes with band L is isomorphic to morphisms in the homotopy category from the point * to the homotopy fibre over L for a map defined …

Definable categories and T-motives

Making use of Freyd's free abelian category on a preadditive category we show that if $T:D\rightarrow \mathcal{A}$ is a representation of a quiver $D$ in an abelian category $\mathcal{A}$ then there is an abelian category $\mathcal{A} (T)$, a faithful exact functor $F_T: \mathcal{A} (T) \to \mathcal{A}$ and an induced representation $\tilde T: D \to \mathcal{A} (T)$ such that $F_T\tilde T= T$ universally. We then can show that $\mathbb{T}$-motives as well as Nori's motives are given by a certain category of functors on definable categories.

Moduli Problems in Abelian Categories and the Reconstruction Theorem

We give a moduli-theoretic proof of the classical theorem of Gabriel, stating that a scheme can be reconstructed from the abelian category of quasi-coherent sheaves over it. The methods employed are elementary and allow us to extend the theorem to (quasi-compact and separated) algebraic spaces. Using more advanced technology (and assuming flatness) we also give a proof of the folklore result that the group of autoequivalences of the category of quasi-coherent sheaves consists of automorphisms of the underlying space and twists by line bundles. We apply our strategy to prove analogous statement…

1-动机的Chow迹与Lang-Néron群

Chow trace of 1-motives and the Lang-Néron groups

本文推广Chow关于阿贝尔簇的结果，构造Deligne 1-动机标量扩张的Chow像与Chow迹，并在Voevodsky动机范畴中证明阿贝尔簇的第零次直接像为Chow迹、第一次直接像由Lang-Néron群给出。

Nori motives of curves with modulus and Laumon 1-motives

Let $k$ be a number field. We describe the category of Laumon 1-isomotives over $k$ as the universal category in the sense of Nori associated with a quiver representation built out of smooth proper $k$-curves with two disjoint effective divisors and a notion of $H^1_\dR$ for such "curves with modulus". This result extends and relies on the theorem of J. Ayoub and L. Barbieri-Viale that describes Deligne's category of 1-isomotives in terms of Nori's Abelian category of motives.

Cohomology and André motives of hyperkähler orbifolds

One of the main tools for the study of compact hyperkähler manifolds is the natural action of the Looijenga-Lunts-Verbitsky Lie algebra on the cohomology of such manifolds. This also applies to the mildly singular holomorphic symplectic varieties - hyperkähler orbifolds, allowing us to prove that André motives of such orbifolds tend to be abelian.

LSV超Kähler manifold的Chow动机

The Chow motive of LSV hyper-Kälher manifolds

本文研究了LSV超Kähler manifold的Chow动机，证明其为X^5的动机的直接和，并构造了一个10维的三次体族，使得其紧致化具有不可约纤维和abelian类型的Chow动机。

混合动机

Mixed Motives

本文证明了取值于阿贝尔刚性张量范畴的泛混合韦伊上同调存在，揭示诺里动机可通过局部化作为贝蒂上同调的泛富集恢复，并为任意特征混合动机理论提供新图景及结果。

On log motives

We define the categories of log motives and log mixed motives. The latter gives a new formulation for the category of mixed motives. We prove that the former is a semisimple abelian category if and only if the numerical equivalence and homological equivalence coincide, and that it is also equivalent to that the latter is a Tannakian category. We discuss various realizations, formulate Tate and Hodge conjectures, and verify them in curve case.

Relative DGA and mixed elliptic motives

Bloch and Kriz construct an abelian category of mixed Tate motives as the category of comodules over a Hopf algebra obtained by the bar construction of the DGA of cycle complexes. In this paper we generalize their construction to give the definition of a category of mixed elliptic motives, i.e. a Tannakian category of mixed motives generated by an elliptic curve. We introduce the notion of a relative DGA over a reductive group. Then the category of mixed elliptic motives is defined as the category of comodules over the relative bar construction of a certain DGA $A_{EM}$ which is constructed fr…

Local and stable homological algebra in Grothendieck abelian categories

We define model category structures on the category of chain complexes over a Grothendieck abelian category depending on the choice of a generating family, and we study their behaviour with respect to tensor products and stabilization. This gives convenient tools to construct and understand triangulated categories of motives and we consider here the case of mixed motives over a regular base scheme.

On topological motives

Following Eilenberg-Steenrod axiomatic approach we construct the universal ordinary homology theory for any homological structure on a given category by representing ordinary theories with values in abelian categories. For a convenient category of spaces we then obtain a universal abelian category which can be actually described for CW-complexes as the category of hieratic modules.

Algebraic cycles on a generalized Kummer variety

We compute explicitly the Chow motive of any generalized Kummer variety associated to any abelian surface. In fact, it lies in the rigid tensor subcategory of the category of Chow motives generated by the Chow motive of the underlying abelian surface. One application of this calculation is to show that the Hodge conjecture holds for arbitrary products of generalized Kummer varieties. As another application, all numerically trivial 1-cycles on arbitrary products of generalized Kummer varieties are smash-nipotent.

Mixed Tate Motives and Multiple Zeta Values

In this paper, we construct an object of the abelian category of mixed Tate motive associated to multiple zeta values. as a consequence, we prove the inequality of the dimension of the vector space generated by multiple zeta values, which is conjectured by Zagier. (Some errors was corrected from the first version. Namely, we changed the definition of the boundary $\bold B^0$.)

Syntactic categories for Nori motives

We give a new construction, based on categorical logic, of Nori's $\mathbb Q$-linear abelian category of mixed motives associated to a cohomology or homology functor with values in finite-dimensional vector spaces over $\mathbb Q$. This new construction makes sense for infinite-dimensional vector spaces as well, so that it associates a $\mathbb Q$-linear abelian category of mixed motives to any (co)homology functor, not only Betti homology (as Nori had done) but also, for instance, $\ell$-adic, $p$-adic or motivic cohomology. We prove that the $\mathbb Q$-linear abelian categories of mixed mot…

通过 Frobenius 自同态中的形式留数配对 Anderson 动机

Pairing Anderson motives via formal residues in the Frobenius endomorphism

本工作通过取 Frobenius 自同态中形式幂级数的留数，构建了 Anderson 模块动机与对偶动机间的完美配对，并以此回答了 Hartl-Juschka 问题，将“abelian 等于 A-finite”定理推广至更广泛的代数类。

带挠的Deligne 1-动机与平展动机

Deligne 1-motives with torsion and étale motives

该研究在特征零概形或戴德金概形上构造整系数1-动机的动机t-结构，倒置基剩余特征指数后诱导光滑1-动机范畴的t-结构，证明其心带挠的Deligne 1-动机范畴是阿贝尔的，依赖正规概形上该动机由一般点纤维决定及好约化情形的显式描述。

Tensor structure for Nori motives

We construct a tensor product on Freyd's universal abelian category attached to an additive tensor category or a tensor quiver and establish a universal property. This is used to give an alternative construction for the tensor product on Nori motives.

Kontsevich's noncommutative numerical motives

In this note we prove that Kontsevich's category NCnum of noncommutative numerical motives is equivalent to the one constructed by the authors. As a consequence, we conclude that NCnum is abelian semi-simple as conjectured by Kontsevich.

Jacobians of noncommutative motives

In this article one extends the classical theory of (intermediate) Jacobians to the "noncommutative world". Concretely, one constructs a Q-linear additive Jacobian functor J(-) from the category of noncommutative Chow motives to the category of abelian varieties up to isogeny, with the following properties: (i) the first de Rham cohomology group of J(N) agrees with the subspace of the odd periodic cyclic homology of N which is generated by algebraic curves; (ii) the abelian variety J(perf(X)) (associated to the derived dg category perf(X) of a smooth projective scheme X) identifies with the un…

Voevodsky's motives and Weil reciprocity

We describe Somekawa's K-group associated to a finite collection of semi-abelian varieties (or more general sheaves) in terms of the tensor product in Voevodsky's category of motives. While Somekawa's definition is based on Weil reciprocity, Voevodsky's category is based on homotopy invariance. We apply this to explicit descriptions of certain algebraic cycles.

Nori motives（和混合Hodge模）具有整数系数

Nori motives (and mixed Hodge modules) with integral coefficients

本文构建了具有整数系数的Nori动机sheaves的abelian范畴，研究了Nori动机的派生范畴，并构造了étale motives的代数，提供了混合Hodge模在实数方案上的类似构造。