The Hodge realization of the polylogarithm on the product of multiplicative groups

Multiplicative sub-Hodge structures of conjugate varieties

For any subfield K of the complex numbers which is not contained in an imaginary quadratic number field, we construct conjugate varieties whose algebras of K-rational (p,p)-classes are not isomorphic. This compares to the Hodge conjecture which predicts isomorphisms when K is contained in an imaginary quadratic number field; additionally, it shows that the complex Hodge structure on the complex cohomology algebra is not invariant under the Aut(\C)-action on varieties. In our proofs, we find simply connected conjugate varieties whose multilinear intersection forms on their second real cohomolog…

Middle multiplicative convolution and hypergeometric equations

Using a relation due to Katz linking up additive and multiplicative convolutions, we make explicit the behaviour of some Hodge invariants by middle multiplicative convolution, following [DS13] and [Mar18a] in the additive case. Moreover, the main theorem gives a new proof of a result of Fedorov computing the Hodge invariants of hypergeometric equations.

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。

Hodge correlators

Hodge correlators are complex numbers given by certain integrals assigned to a smooth complex curve. We show that they are correlators of a Feynman integral, and describe the real mixed Hodge structure on the pronilpotent completion of the fundamental group of the curve. We introduce motivic correlators, which are elements of the motivic Lie algebra and whose periods are the Hodge correlators. They describe the motivic fundamental group of the curve. We describe variations of real mixed Hodge structures on a variety by certain connections on the product of the variety by an afine line. We call…

Periods and mixed motives

We define motivic multiple polylogarithms and prove the double shuffle relations for them. We use this to study the motivic fundamental group of the multiplicative group - {N-th roots of unity} and relate it to geometry of modular varieties. In particular we get new information about the actionof the Galois group on the pro-l completion of the above fundamental group. This paper is the second part of "Multiple polylogarithms and mixed Tate motives" math.AG/0103059

Real mixed Hodge structures

We identify the category of real mixed Hodge structures with the category of vector bundles with connections (not necessarily flat) on C, equivariant with respect to C^*. Here C is the complex plane considered as a 2-dimensional real manifold, and C^* is the multiplicative group of complex numbers considered as a real group.

On Additive invariants of actions of additive and multiplicative groups

The additive invariants of an algebraic variety is calculated in terms of those of the fixed point set under the action of additive and multiplicative groups, by using Bialynicki-Birula's fixed point formula for a projective algebraicset with a G_m-action or G_a-action. The method is also generalized to calculate certain additive invariants for Chow varieties. As applications, we obtain the Hodge polynomial of Chow varieties in characteristic zero and the number of points for Chow varieties over finite fields. As applications, we obtain the l-adic Euler-Poincare characteristic for the Chow var…

Hodge-Witt decomposition of relative crystalline cohomology

For a smooth and proper scheme over an artinian local ring with ordinary reduction over the perfect residue field we prove - under some general assumptions - that the relative de Rham-Witt spectral sequence degenerates and the relative crystalline cohomology, equipped with its display structure arising from the Nygaard complexes, has a Hodge-Witt decomposition into a direct sum of (suitably Tate-Twisted) multiplicative displays. As examples our main results include the cases of abelian schemes, complete intersections, varieties of K3 type and some Calabi-Yau $n$-folds.

绝对Hodge上同调中的积与混合Hodge形式性

Products in absolute Hodge cohomology and mixed Hodge formality

本文证明混合Hodge形式性蕴含绝对Hodge上同调的平方零扩张分解，定义类似Massey积的阻碍积，并关联乘法Chow-Künneth分解。

A non-Archimedean analogue of the Hodge-D-conjecture for products of elliptic curves

In this paper we show that the map % $$\partial:CH^2(E_1 \times E_2,1)\otimes \Q \longrightarrow PCH^1(\XX_v)$$ % is surjective, where $E_1$ and $E_2$ are two non-isogenous semistable elliptic curves over a local field, $CH^2(E_1 \times E_2,1)$ is one of Bloch's higher Chow groups and $PCH^1(\XX_v)$ is a certain subquotient of a Chow group of the special fibre $\XX_{v}$ of a semi-stable model $\XX$ of $E_1 \times E_2$. On one hand, this can be viewed as a non-Archimedean analogue of the Hodge-$\D$-conjecture of Beilinson - which is known to be true in this case by the work of Chen and Lewis \c…

多对数变分与 $\mathbb{Q}$ 被 $\mathbb{Q}(m)$ 的动形扩张

Polylogarithm Variations and Motivic Extensions of $\mathbb{Q}$ by $\mathbb{Q}(m)$

本文通过在 Tate 椭圆曲线限制的幂单基本群 Lie 代数上实现导子，将 Deligne、Goncharov 等人关于动形 Galois 群深度 2 二次关系的结果从 $N=1$ 推广至所有 $N \ge 1$，并揭示了其与 $K$-群和 Eisenstein 级数的相容性。

The pure cohomology of multiplicative quiver varieties

To a quiver $Q$ and choices of nonzero scalars $q_i$, non-negative integers $α_i$, and integers $θ_i$ labeling each vertex $i$, Crawley-Boevey--Shaw associate a "multiplicative quiver variety" $\mathcal{M}_θ^q(α)$, a trigonometric analogue of the Nakajima quiver variety associated to $Q$, $α$, and $θ$. We prove that the pure cohomology, in the Hodge-theoretic sense, of the stable locus $\mathcal{M}_θ^q(α)^s$ is generated as a $\mathbb{Q}$-algebra by the tautological characteristic classes. In particular, the pure cohomology of genus $g$ twisted character varieties of $GL_n$ is generated by tau…

Motivic fundamental groups of CM elliptic curves and geometry of Bianchi hyperbolic threefolds

In this paper we describe a connection between realizations of the action of the motivic Galois group on the motivic fundamental groups of Gaussian and Eisenstein elliptic curves punctured at the $\mathfrak{p}$-torsion points, $π_1^{\rm Mot}(E-E[\mathfrak{p}],v_0)$, and the geometry of the Bianchi hyperbolic threefolds $Γ_1(\mathfrak{p})\setminus\mathbb{H}^3$, where $Γ_1(\mathfrak{p})$ is a congruence subgroup of ${\rm GL}_2({\rm End}(E))$. The first instance of such a connection was found by A.Goncharov (arXiv:math/0510310). In particular, we study the Hodge realization of the image of the ab…

A Cohomological Non Abelian Hodge Theorem in Positive Characteristic

We start with a curve over an algebraically closed ground field of positive characteristic $p>0$. By using specialization techniques, under suitable natural coprimality conditions, we prove a cohomological Simpson Correspondence between the moduli space of Higgs bundles and the one of connections on the curve. We also prove a new $p$-multiplicative periodicity concerning the cohomology rings of Dolbeault moduli spaces of degrees differing by a factor of $p$. By coupling this $p$-periodicity in characteristic $p$ with lifting/specialization techniques in mixed characteristic, we find, in arbitr…

Nonabelian Mixed Hodge Structure on Brill-Noether Stacks

A Brill-Noether stack is an algebraic very presentable stack whose homotopy type has two nontrivial homotopy groups. We consider one with a fundamental group --- a reductive algebraic group-scheme S and one higher homotopy group, represented by a vector space V. The homotopy type also defines an action of S on V. This stack is used as coefficient space for nonabelian cohomological space on a smooth algebraic variety X. We define nonabelian MHS on cohomological spaces of this type in the context of the work of C. Simpson related to MHS on the space of local systems. It is defined via an action …

Global Weyl groups and a new theory of multiplicative quiver varieties

In previous work a relation between a large class of Kac-Moody algebras and meromorphic connections on global curves was established---notably the Weyl group gives isomorphisms between different moduli spaces of connections, and the root system is also seen to play a role. This involved a modular interpretation of many Nakajima quiver varieties, as moduli spaces of connections, whenever the underlying graph was a complete k-partite graph (or more generally a supernova graph). However in the isomonodromy story, or wild nonabelian Hodge theory, slightly larger moduli spaces of connections are co…

4-穿孔球面上非阿贝尔 Hodge 对应的 Loop group 方法

Loop group methods for the non-abelian Hodge correspondence on a 4-punctured sphere

本文利用复分析和 twistor 方法，构造了4-穿孔球面上强抛物Higgs场的自对偶解，计算了极限模空间及其全纯辛形式的泰勒展开，并导出了多重多对数的恒等式。

Mochizuki环结构的anabelian变形、perfectoid几何与形式群的代数化

Algebraization of Mochizuki's anabelian variation of ring structures, perfectoid geometry and formal groups

本文证明存在配备乘法幺半群作用的泛一维形式群规则，并揭示其在p-adic域、perfectoid环及Mochizuki anabelian思想中的应用，为anabelian几何与perfectoid几何现象提供统一途径。

高Chow群与不一定可容许的环

Higher Chow groups and not necessarily admissible cycles

我们构造了高Chow群的类似物，通过指定分式域乘法群外幂元素定义环，证明其良定义并研究基本性质，最后证明其与多对数复形上同调同构。