Hodge realizations of 1-motives and the derived Albanese

Motivic interpretation of Albanese varieties of smooth varieties

For every noetherian smooth and separated scheme over an algebraically closed field, we define its derived Albanese in Voevodsky's triangulated category of effective Nisnevich motives. To justify our definition, we relate the derived Albanese with the Albanese scheme. We also prove that the derived Albanese satisfies the Nisnevich descent property.

On algebraic 1-motives related to Hodge cycles

The goal of this paper is to introduce Hodge 1-motives of algebraic varieties and to state a corresponding cohomological Grothendieck-Hodge conjecture, generalizing the classical Hodge conjecture to arbitrarily singular proper schemes.

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.

The De Rham, complex Hodge and $p$-adic Hodge realization functors on the derived category of relative motives over a field of characteristic zero

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 $k\subset K\subset\mathbb C_p$.

Weights and t-structures: in general triangulated categories, for 1-motives, mixed motives, and for mixed Hodge complexes and modules

We study certain 'weights' for triangulated categories endowed with $t$-structures. Our results axiomatize and describe in detail the relations between the Chow weight structure (introduced in a preceding paper), the (conjectural) motivic t-structure for Voevodsky's motives, and the conjectural weight filtration for them. This picture becomes non-conjectural when restricted to the derived categories of 1-motives (over a smooth base) and of Artin-Tate motives over number fields; other examples include Beilinson's derived category of graded polarizable mixed Hodge complexes and Saito's derived c…

1-Motives Associated to the Limiting Mixed Hodge Structures of Degenerations of Curves

In this article, we will give the Deligne 1-motives up to isogeny corresponding to the $\mathbb{Q}$-limiting mixed Hodge structures of semi-stable degenerations of curves, by using logarithmic structures and Steenbrink's cohomological mixed Hodge complexes associated to semi-stable degenerations of curves.

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。

Derived invariance of the Albanese relative canonical ring

We show the derived invariance of various geometric invariants of smooth complex projective varieties governed by the Albanese map, including the relative canonical ring and the class of the relative canonical model in a suitable variant of the Grothendieck ring of varieties. Then we derive some applications to the derived invariance of Hodge numbers.

Birational motives, I: pure birational motives

This is a considerably expanded version of the "pure" part of our 2002 preprint. We define a category of pure birational motives over a field, depending on the choice of an adequate equivalence relation on algebraic cycles. It is obtained by "killing" the Lefschetz motive in the corresponding category of effective motives. For rational equivalence, it encompasses Bloch's decomposition of the diagonal. We study the induced Chow-Künneth decompositions in this category, and establish relationships with Rost's cycle modules and the Albanese functor for smooth projective varieties.

关于Voevodsky动机的Nori实现与Hodge实现

On the Nori and Hodge realisations of Voevodsky motives

本文通过证明反常Nori动机与混合Hodge模导出范畴的可构造心性质，构造了六运算的∞-范畴提升，建立了Voevodsky étale动机到两者的实现函子，并证明了Artin动机范畴等价及motivic t-结构存在时的等价性。

The Hodge realization functor on the derived category of relative motives

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…

The fundamental group and extensions of motives of Jacobians of curves

In this paper we construct extensions of mixed Hodge structures coming from the mixed Hodge structure on the graded quotients of the group ring of the fundamental group of a smooth, projective, pointed curve. These extensions correspond to the regulators of certain motivic cycles in the Jacobian of the curve which were constructed by Beilinson and Bloch. This leads to a new iterated integral expression for the regulator. This is a generalisation of a theorem of Colombo where she constructed the extension corresponding to Collino's cycles in the Jacobian of a hyperelliptic curve.

Functional transcendence for the unipotent Albanese map

We prove a certain transcendence property of the unipotent Albanese map of a smooth variety, conditional on the Ax-Schanuel conjecture for variations of mixed Hodge structure. We show that this property allows the Chabauty-Kim method to be generalized to higher-dimensional varieties. In particular, we conditionally generalize several of the main Diophantine finiteness results in Chabauty-Kim theory to arbitrary number fields.

形式霍奇结构的扩张

Extensions of Formal Hodge Structures

本文将形式霍奇结构范畴从层级≤1推广到层级≤n，并利用其中的Ext¹群刻画广义Albanese簇，从而把经典公式推广到紧但未必光滑的复代数簇。

Motives and algebraic de rham cohomology

In this paper, we define a certain Hodge-theoretic structure for an arbitrary variety X over the complex number field by using the theory of mixed Hodge module due to Morihiko Saito. We call it an arithmetic Hodge structure of X. It is shown that extension groups of arithmetic Hodge structure do not vanish even for degree $\geq2$. Moreover, we define higher Abel-Jacobi maps from Bloch's higher Chow groups of X to these extension groups. These maps essentially involve the classical Abel-Jacobi maps by Weil and Griffiths, and Mumford's infinitesimal invariants of 0-cycles on surfaces.

Derived invariants arising from the Albanese map

Let $a_X:X\rightarrow \mathrm{Alb}\, X$ be the Albanese map of a smooth complex projective variety. Roughly speaking in this note we prove that for all $i \geq 0$ and $α\in \mathrm{Pic}^0\, X$, the cohomology ranks $h^i(\mathrm{Alb}\, X, \,{a_X}_* ω_X\otimes P_α)$ are derived invariants. In the case of varieties of maximal Albanese dimension this proves conjectures of Popa and Lombardi-Popa -including the derived invariance of the Hodge numbers $h^{0,j}$ -- and a weaker version of them for arbitrary varieties. Finally we provide an application to derived invariance of certain irregular fibrati…

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…

一般化θ除子上的混合Hodge结构与图动机

Mixed Hodge Structures on Generalized Theta Divisors and Graph Motives

本文研究一般化θ除子上的混合Hodge结构，并探讨其与图动机的可能联系。

1-动机与混合Hodge结构的容许变差

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。

附属于Kloosterman和Airy矩的动机的Hodge数

Hodge numbers of motives attached to Kloosterman and Airy moments

本文提出了一种计算$\mathrm{M}_{n+1}^k(\mathrm{Kl})$在$n=2$或$\gcd(k,n+1)=1$时的（非正则）Hodge数的方法，并发现大多数情况下Hodge数大于$1$。

Motives and the Hodge Conjecture for moduli spaces of pairs

Let $C$ be a smooth projective curve of genus $g\geq 2$ over $\mathbb C$. Fix $n\geq 1$, $d\in {\mathbb Z}$. A pair $(E,ϕ)$ over $C$ consists of an algebraic vector bundle $E$ of rank $n$ and degree $d$ over $C$ and a section $ϕ\in H^0(E)$. There is a concept of stability for pairs which depends on a real parameter $τ$. Let ${\mathfrak M}_τ(n,d)$ be the moduli space of $τ$-polystable pairs of rank $n$ and degree $d$ over $C$. Here we prove that for a generic curve $C$, the moduli space ${\mathfrak M}_τ(n,d)$ satisfies the Hodge Conjecture for $n \leq 4$. For obtaining this, we prove first that…

Hypergeometric Motives

Survey of hypergeometric motives, with a focus on their source varieties, Hodge numbers, and L-functions.

通过弱一般 Nakano 消没定理研究 L²-Betti 数与规范化 Hodge 数的收敛性

L^2-Betti Numbers and Convergence of Normalized Hodge Numbers via the Weak Generic Nakano Vanishing Theorem

本文利用弱一般 Nakano 消没定理，研究 Abel 覆盖塔中规范化 Hodge 数与多重亏格的增长及收敛，并计算相关非正则簇的 L²-Betti 数，推广 Kollár 的可乘性结果。

从 étale 动机视角看 Hodge 结构

Hodge structures through an étale motivic point of view

本研究定义了 étale Chow 动机范畴，证明其可嵌入 $\ ext{DM}_{\ ext{ét}}(k)$，并通过该构造用 étale 版本的对应猜想刻画了广义 Hodge 猜想。

高阶页雅可比与阿尔巴内塞环面

Higher-Page Jacobian and Albanese Tori

本文通过霍奇理论为紧致复流形构造高阶页雅可比环面、阿尔巴内塞环面及映射，并应用于结构定理、代数维数恒等式及六维球面的复结构研究，证明其无强高登度量。

A note on 1-motives

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.

The Chow motive of semismall resolutions

We consider proper, algebraic semismall maps f from a complex algebraic manifold X. We show that the topological Decomposition Theorem implies a "motivic" decomposition theorem for the rational algebraic cycles of X and, in the case X is compact, for the Chow motive of X.The result is a Chow-theoretic analogue of Borho-MacPherson's observation concerning the cohomology of the fibers and their relation to the relevant strata for f. Under suitable assumptions on the stratification, we prove an explicit version of the motivic decomposition theorem. The assumptions are fulfilled in many cases of i…

Formal Hodge Theory

Formal (mixed) Hodge structures FHS are introduced in such a way that the Hodge realization of Deligne's 1-motives extends to a realization from Laumon's 1-motives to formal Hodge structures of level 1, providing an equivalence of categories. For the sake of exposition we here confine our study to level 1 mixed Hodge structures. However, it is conceivable and suitable to consider formal mixed Hodge structures with arbitrary Hodge numbers: generalizing our definition herebelow it's not that difficult and we will treat such a matter nextly.

Mixed objects are embedded into log pure objects

We prove that a variation of mixed Hodge structure is embedded in a logarithmic variation of pure Hodge structure, and a generalized version of this result. These results suggest some simple construction of the category of mixed motives by using log pure motives.

Line bundles on rigid varieties and Hodge symmetry

We prove several related results on the low-degree Hodge numbers of proper smooth rigid analytic varieties over non-archimedean fields. Our arguments rely on known structure theorems for the relevant Picard varieties, together with recent advances in p-adic Hodge theory. We also define a rigid analytic Albanese naturally associated with any smooth proper rigid space.

Deligne's duality for de Rham realizations of 1-motives

We show that the pairing on de Rham realizations of 1-motives in "Theorie di Hodge III", IHES 44, can be defined over any base scheme and we prove that it gives rise to a perfect duality if one is working with a 1-motive and its Cartier dual. Furthermore we study universal extensions of 1-motives and their relation with $\natural$-extensions.

具有有限维 motive 的9维K3曲面族

A 9-dimensional family of K3 surfaces with finite dimensional motive

该研究利用与四次三维立方体的关联，证明了一类二次曲面三重覆盖K3曲面的Kimura有限维motive，并确立其Kuga-Satake Hodge猜想与Voisin零闭链猜想，进而得到带3阶非辛自同构K3曲面模空间2个9维不可约分支的对应结果。

Réalisations de Hodge des motifs de Voevodsky

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".

Triangulated categories of logarithmic motives over a field

In this work we develop a theory of motives for logarithmic schemes over fields in the sense of Fontaine, Illusie, and Kato. Our construction is based on the notion of finite log correspondences, the dividing Nisnevich topology on log schemes, and the basic idea of parameterizing homotopies by $\overline{\square}$, i.e. the projective line with respect to its compactifying logarithmic structure at infinity. We show that Hodge cohomology of log schemes is a $\overline{\square}$-invariant theory that is representable in the category of logarithmic motives. Our category is closely related to Voev…

Chow motives of universal families over some Shimura surfaces

We prove an absolute Chow-Kuenneth decomposition for the motive of universal families A of abelian varieties over some compact Shimura surface. We furthermore prove the Hodge conjecture for general fibres of A, extending results of Ribet.

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.

Deligne's Conjecture on 1-Motives

We reformulate a conjecture of Deligne on 1-motives by using the integral weight filtration of Gillet and Soulé on cohomology, and prove it. This implies the original conjecture up to isogeny. If the degree of cohomology is at most two, we can prove the conjecture for the Hodge realization without isogeny, and even for 1-motives with torsion.

Multiple polylogarithms and mixed Tate motives

We develop the theory of multiple polylogarithms from analytic, Hodge and motivic point of view. Define the category of mixed Tate motives over a ring of integers in a number field. Describe explicitly the multiple polylogarithm Hopf algebra.

Malcev 完成、Hodge 理论与动机

Malcev Completions, Hodge Theory, and Motives

该研究通过Tannakian方法证明几何起源局部系统在所有局部系统的范畴中稳定，并加强Hain关于单束表示Malcev完成的动机定理。

On the motive of codimension 2 linear sections of $\hbox{Gr}(3,6)$

We consider Fano sevenfolds $Y$ obtained by intersecting the Grassmannian $\hbox{Gr}(3,6)$ with a codimension 2 linear subspace (with respect to the Plücker embedding). We prove that the motive of $Y$ is Kimura finite-dimensional. We also prove the generalized Hodge conjecture for all powers of $Y$.

Derived invariants and motives, Part II integral derived invariants and some applications

In this paper we construct new derived invariants with integral coefficients using the theory of motifs, and give several applications. Specifically, we obtain the following results: For complex algebraic surfaces, we prove that certain torsion in the abelianized fundamental group is a derived invariant. We prove that the collection of Hodge-Witt cohomology groups is a derived invariant. In particular, Hodge-Witt reduction and ordinary reduction are preserved by derived equivalence when the characteristic is sufficiently large. Finally, using the techniques of non-commutative algebraic geometr…

Laumon 1-动机在等基因意义下的上同调维数

Cohomological dimension of Laumon 1-motives up to isogenies

本文证明了特征零域上Laumon 1-动机等基因范畴的上同调维数不超过1，并由此推出层级不超过1的形式霍奇结构范畴具有相同结果。

On the rationality and the finite dimensionality of a cubic fourfold

Let $X$ be a cubic fourfold in $P^5_{C}$. We prove that, assuming the Hodge conjecture for the product $S \times S$, where $S$ is a complex surface, and the finite dimensionality of the Chow motive $h(S)$, there are at most a countable number of decomposable integral polarized Hodge structures, arising from the fibers of a family of smooth projective surfaces. According to the results in [ABB] this is related to a conjecture proving the irrationality of a very general $X$. If $X$ is special, in the sense of B.Hasset, and $F(X) \simeq S^{[2]}$, with $S$ a K3 surface associated to $X$, then we s…

Gerbes and abelian motives

Assuming the Hodge conjecture for abelian varieties of CM-type, one obtains a good category of abelian motives over the algebraic closure of a finite field and a reduction functor to it from the category of CM-motives. Consequentely, one obtains a morphism of gerbes of fibre functors with certain properties. We prove unconditionally that there exists a morphism of gerbes with these properties, and we classify them.

On the invariant cycle theorem for families of Nori motives

In this paper, we prove a motivic enhancement of the theorem of the fixed part in Hodge theory due to Deligne. In the pure motivic case, this was done for the first time by André in [And96]. Our main result is an extension to the mixed case, which strengthens a result by Arapura [Ara13] and also provides an alternative and simpler proof, in the framework of Nori motives, of a result by Ayoub [Ayo14].

Hodge theory of Kloosterman connections

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.

多对数动机的构造

A construction of the polylogarithm motive

本文通过显式的相对上同调动机构造了多对数动机，即仿射空间 $\mathbb{A}^n_S$ 中超曲面 $\{1-zt_1\cdots t_n=0\}$ 的补集相对于超平面 $\{t_i=0\}$ 和 $\{t_i=1\}$ 的并集的相对上同调动机。

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.

卡拉比-丘 motives 的算术与移动高阶调节子

The arithmetic of Calabi-Yau motives and mobile higher regulators

本研究结合三种思路构造了特定Calabi–Yau motives的动机上同调元素，显式计算了两类情形的正规函数与对应闭链，证明了Beilinson Hodge型猜想并完成了相关$L$-函数特殊值猜想的首次数值验证。

SL_2-orbits and degenerations of mixed Hodge structure

We prove an analog of Schmid's $\text{\rm SL}_2$-orbit theorem for a class of variations of mixed Hodge structure which includes logarithmic deformations, degenerations of 1-motives and archimedean heights. In particular, as consequence this theorem, we obtain a simple formula for the asymptotic behavior of the archimedean height of a flat family of algebraic cycles which depends only on the weight filtration and local monodromy.

On the Voevodsky motive of the moduli space of Higgs bundles on a curve

We study the motive of the moduli space of semistable Higgs bundles of coprime rank and degree on a smooth projective curve C over a field k under the assumption that C has a rational point. We show this motive is contained in the thick tensor subcategory of Voevodsky's triangulated category of motives with rational coefficients generated by the motive of C. Moreover, over a field of characteristic zero, we prove a motivic non-abelian Hodge correspondence: the integral motives of the Higgs and de Rham moduli spaces are isomorphic.

带有界极点的Hochschild和循环同调

Hochschild and cyclic Homologies with bounded poles

本文通过为模对构造Hochschild与循环同调，证明了其在带模动机范畴中的可表示性，并建立了模版HKR定理及模循环同调的可表示性。

A Construction of a Differential Graded Lie Algebra in the Category of Effective Homological Motives

This text gives a construction of a differential graded Lie algebra in Nori's category of effective homological motives. In fact the construction works in more a general setting than that of an Abelian category. This allows us to give the rational homotopy Lie algebra of a 1-connected space a motivic structure. As a consequence the rational homotopy Lie algebra inherits a mixed Hodge structure and Galois module structure.

Irregular varieites with geometric genus one, theta divisors, and fake tori

We study the Albanese image of a compact Kähler manifold whose geometric genus is one. We prove that if the Albanese map is not surjective, then the manifold maps surjectively onto an ample divisor in some abelian variety, and in many cases the ample divisor is a theta divisor. With a further natural assumption on the topology of the manifold, we prove that the manifold is an algebraic fiber space over a genus two curve. Finally we apply these results to study the geometry of a compact Kähler manifold which has the same Hodge numbers as those of an abelian variety of the same dimension.