Albanese and Picard 1-motives

Generalized Albanese and its dual

We consider categories of rational maps to algebraic groups and study existence and construction of universal objects for such categories, using the duality theory of Laumon 1-motives. In particular, we obtain functorial descriptions of the Albanese of a singular projective variety X over an algebraically closed field of characteristic zero, which is the universal regular quotient of a relative Chow group of points on X, and of its dual.

On Albanese torsors and the elementary obstruction

We show that the elementary obstruction to the existence of 0-cycles of degree 1 on an arbitrary variety X (over an arbitrary field) can be expressed in terms of the Albanese 1-motives associated with dense open subsets of X. Arithmetic applications are given.

On the interior motive of certain Shimura varieties: the case of Picard surfaces

The purpose of this article is to construct a Hecke-equivariant Chow motive whose realizations equal interior (or intersection) cohomology of Picard surfaces with regular algebraic coefficients. As a consequence, we are able to define Grothendieck motives for Picard modular forms.

Picard groups, weight structures, and (noncommutative) mixed motives

We develop a general theory which enables the computation of the Picard group of a symmetric monoidal triangulated category, equipped with a weight structure, in terms of the Picard group of the associated heart. As an application, we compute the Picard group of several categories of motivic nature - mixed Artin motives, mixed Artin-Tate motives, motivic spectra, noncommutative mixed Artin motives, noncommutative mixed motives of central simple algebras, noncommutative mixed motives of separable algebras - as well as the Picard group of the derived categories of symmetric ring spectra.

The Galois module structure of l-adic realizations of Picard 1-motives and applications

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…

Deligne's conjecture on extensions of 1-motives

We introduce the notion of extension of 1-motives. Using the dictionary between strictly commutative Picard stacks and complexes of abelian sheaves concentrated in degrees -1 and 0, we check that an extension of 1-motives induces an extension of the corresponding strictly commutative Picard stacks. We compute the Hodge, the de Rham and the l-adic realizations of an extention of 1-motives. Using these results we can prove Deligne's conjecture on extensions of 1-motives.

Affine quadrics and the Picard group of the motivic category

In this article we study the subgroup of the Picard group of Voevodsky's category of geometric motives generated by the reduced motives of affine quadrics. Our main tools here are the functors of Bachmann, but we also provide an alternative method. We show that the group in question can be described in terms of indecomposable direct summands in the motives of projective quadrics. In particular, we describe all the relations among the reduced motives of affine quadrics. We also extend the Criterion of motivic equivalence of projective quadrics.

Duality for commutative group stacks

We study in this article the dual of a (strictly) commutative group stack $G$ and give some applications. Using the Picard functor and the Picard stack of $G$, we first give some sufficient conditions for $G$ to be dualizable. Then, for an algebraic stack $X$ with suitable assumptions, we define an Albanese morphism $a_X : X\to A^1(X)$ where $A^1(X)$ is a torsor under the dual commutative group stack $A^0(X)$ of $Pic_{X/S}$. We prove that $a_X$ satisfies a natural universal property. We give two applications of our Albanese morphism. On the one hand, we give a geometric description of the elem…

Para-abelian 簇与 Albanese 映射

Para-abelian varieties and Albanese maps

本研究为基域上满足 Stein 分解条件的本征代数空间构造了到 para-abelian 簇的唯一 Albanese 映射，该映射在族情形及群概形作用下保持等变，依赖 para-abelian 簇族与相对 Picard 群 tau 部分可表性等理论。

Albanese varieties with modulus over a perfect field

Let X be a smooth proper variety over a perfect field k of arbitrary characteristic. Let D be an effective divisor on X with multiplicity. We introduce an Albanese variety Alb(X, D) of X of modulus D as a higher dimensional analogon of the generalized Jacobian of Rosenlicht-Serre with modulus for smooth proper curves. Basing on duality of 1-motives with unipotent part (which are introduced here), we obtain explicit and functorial descriptions of these generalized Albanese varieties and their dual functors. We define a relative Chow group of zero cycles w.r.t. the modulus D and show that Alb(X,…

Motives and representability of algebraic cycles on threefolds over a field

We study links between algebraic cycles on threefolds and finite-dimensionality of their motives with coefficients in Q. We decompose the motive of a non-singular projective threefold X with representable algebraic part of CH_0(X) into Lefschetz motives and the Picard motive of a certain abelian variety, isogenous to the corresponding intermediate Jacobian J^2(X) when the ground field is C. In particular, it implies motivic finite-dimensionality of Fano threefolds over a field. We also prove representability of zero-cycles on several classes of threefolds fibered by surfaces with algebraic H^2…

From Jacobians to one-motives: exposition of a conjecture of Deligne

Deligne has conjectured that certain mixed Hodge theoretic invariants of complex algebraic invariants are motivic. This conjecture specializes to an algebraic construction of the Jacobian for smooth projective curves, which was done by A. Weil. The conjecture (and one-motives) are motivated by means of Jacobians, generalized Jacobians of Rosenlicht, and Serre's generalized Albanese varieties. We discuss the connections with the Hodge and the generalized Hodge conjecture. We end with some applications to number theory by providing partial answers to questions of Serre, Katz and Jannsen.

On the geometry of algebraic groups and homogeneous spaces

Given a connected algebraic group G over an algebraically closed field and a G-homogeneous space X, we describe the Chow ring of G and the rational Chow ring of X, with special attention to the Picard group. Also, we investigate the Albanese and the "anti-affine" fibrations of G and X.

Morphisms of 1-motives defined by line bundles

Let $S$ be a normal base scheme. The aim of this paper is to study the line bundles on 1-motives defined over $S$. We first compute a dévissage of the Picard group of a 1-motive $M$ according to the weight filtration of $M$. This dévissage allows us to associate, to each line bundle $L$ on $M$, a linear morphism $φ_{L}: M \rightarrow M^*$ from $M$ to its Cartier dual. This yields a group homomorphism $Φ: Pic(M) / Pic(S) \to Hom(M,M^*)$. We also prove the Theorem of the Cube for 1-motives, which furnishes another construction of the group homomorphism $Φ: Pic(M) / Pic(S) \to Hom(M,M^*)$. Finall…

Hecke特征标与Picard曲面的Eisenstein上同调的Bloch-Beilinson猜想

Bloch-Beilinson conjectures for Hecke characters and Eisenstein cohomology of Picard surfaces

本文研究Hecke特征标与Picard曲面的Eisenstein上同调之间的Bloch-Beilinson猜想，通过构造Hodge结构延伸并计算双延伸高度来验证其非平凡性。

Triangulated categories of relative 1-motives

We construct and study a candidate for the standard motivic t-structure on the triangulated category of relative cohomological 1-motives with rational coefficients over a noetherian finite dimensional scheme S. This t-structure is defined as a generated t-structure, and we show it is non-degenerate. We relate its heart MM^1(S) with Deligne 1-motives over S; in particular, when S is regular, the category of Deligne 1-motives embeds in MM^1(S) fully faithfully. We also study the inclusion of DA^1(S) into the larger category DA^{coh}(S) of relative cohomological motives on S, and prove that its r…

Voevodsky's mixed motives versus Kontsevich's noncommutative mixed motives

Following an insight of Kontsevich, we prove that the quotient of Voevodsky's category of geometric mixed motives DM by the endofunctor -Q(1)[2] embeds fully-faithfully into Kontsevich's category of noncommutative mixed motives KMM. We show also that this embedding is compatible with the one between pure motives. As an application, we obtain a precise relation between the Picard groups Pic(-), the Grothendieck groups, the Schur-finitenss, and the Kimura-finitenss of the categories DM and KMM. In particular, the quotient of Pic(DM) by the subgroup of Tate twists Q(i)[2i] injects into Pic(KMM). …

紧支撑上同调的代数部分

Algebraic part of motivic cohomology with compact supports

本文将代数等价关系与正则同态推广至域上Voevodsky motives，证明了特定上同调群存在泛正则同态，恢复Murre定理等经典结论，并涵盖高阶Chow群与Milnor K-群。

数域上 K3 曲面约化的 Picard 秩的例外跳跃

Exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields

本文证明了数域上处处具有潜在好约化的 K3 曲面在无限多个素数处几何 Picard 秩发生跳跃，并基于 GSpin Shimura 簇的例外类理论探讨了其在有理曲线与阿贝尔曲面同构中的应用。

双椭圆Picard曲线的Ceresa圈

Ceresa cycles of bielliptic Picard curves

本文证明了双椭圆Picard曲线的Ceresa圈为挠群当且仅当对应椭圆曲线上的点为挠点，揭示了其高度的Northcott性质，并利用Chow motive同构完成了证明。