The Gysin map is compatible with mixed Hodge structures

Local Mixed Hodge Structure on Brill-Noether Stacks

On a smooth algebraic curve X with genus greater than 1 we consider a flat principal bundle with a reductive structure group S and a vector bundle associated with it. To this set of information we put in correspondence a pro-algebraic group on whose functional algebra we introduce a mixed Hodge structure. This construction, in fact, works for any smooth algebraic variety X which, considered as an analytic space, has a nonabelian first homotopy group, and the rest are trivial. The Hodge structure defined in this way can be expressed in terms of iterated integrals. Furthermore, considered in the…

Kuga-Satake construction and cohomology of hyperkahler manifolds

Let M be a simple hyperkahler manifold. Kuga-Satake construction gives an embedding of H^2(M,C) into the second cohomology of a torus, compatible with the Hodge structure. We construct a torus T and an embedding of the graded cohomology space H^*(M,C) \to H^{*+l}(T,C) for some l, which is compatible with the Hodge structures and the Poincare pairing. Moreover, this embedding is compatible with an action of the Lie algebra generated by all Lefschetz sl(2)-triples on M.

Some remarks on the semi-positivity theorems

We show that the dualizing sheaves of reduced simple normal crossings pairs have a canonical weight filtration in a compatible way with the one on the corresponding mixed Hodge modules by calculating the extension classes between the dualizing sheaves of smooth varieties. Using the weight spectral sequence of mixed Hodge modules, we then reduce the semi-positivity theorem for the higher direct images of dualizing sheaves to the smooth case where the assertion is well known. This may be used to simplify some constructions in a recent paper of Y. Kawamata. We also give a simple proof of the semi…

Mixed Hodge structure of affine hypersurfaces

In this article we introduce the mixed Hodge structure of the Brieskorn module of a polynomial $f$ in $\C^{n+1}$, where $f$ satisfies a certain regularity condition at infinity (and hence has isolated singularities). We give an algorithm which produces a basis of a localization of the Brieskorn module which is compatible with its mixed Hodge structure. As an application we show that the notion of a Hodge cycle in regular fibers of $f$ is given in terms of the vanishing of integrals of certain polynomial $n$-forms in $\C^{n+1}$ over topological $n$-cycles on the fibers of $f$. Since the $n$-th …

拓扑拉普拉斯变换与非交换霍奇结构的分解

Topological Laplace Transform and Decomposition of nc-Hodge Structures

本文构造了从指数型Stokes结构到可构造层的拓扑拉普拉斯变换函子，证明其与$D$-模傅里叶变换相容并诱导范畴等价，且应用于研究Landau-Ginzburg模型的B模型nc-Hodge结构及光滑射影簇的A模型nc-Hodge结构分解。

A purity theorem for configuration spaces of smooth compact algebraic varieties

B. Totaro showed \cite{totaro} that the rational cohomology of configuration spaces of smooth complex projective varieties is isomorphic as an algebra to the $E_2$ term of the Leray spectral sequence corresponding to the open embedding of the configuration space into the Cartesian power. In this note we show that the isomorphism can be chosen to be compatible with the mixed Hodge structures. In particular, we prove that the mixed Hodge structures on the configuration spaces of smooth complex projective varieties are direct sums of pure Hodge structures.

Alexander模上Hodge理论的兼容性

Compatibility of Hodge Theory on Alexander Modules

本文证明在Alexander模上通过挠子模和有限维商构建的两种混合Hodge结构是兼容的，其间的映射构成混合Hodge结构态射。

Hodge structure on the fundamental group and its application to p-adic integration

We study the unipotent completion $Π^{DR}_{un}(x_0, x_1, X_K)$ of the de Rham fundamental groupoid [De] of a smooth algebraic variety over a local non-archimedean field K of characteristic 0. We show that the vector space $Π^{DR}_{un}(x_0, x_1, X_K)$ possesses a distinguished element. In the other words, given a vector bundle E on $X_K$ together with a unipotent integrable connection, we have {\sf a canonical} isomorphism $E_{x_0}\simeq E_{x_1}$ between the fibers. The latter construction is a generalization of Colmez's p-adic integration (rk E=2) and Coleman's p-adic iterated integrals ($X_K$…

The Hodge structure of semiample hypersurfaces and a generalization of the monomial-divisor mirror map

We solved the long-standing problem of describing the cohomology ring of semiample hypersurfaces in complete simplicial toric varieties. Also, the monomial-divisor mirror map is generalized to a map between the whole Picard group and the space of infinitesimal deformations for a mirror pair of Calabi-Yau hypersurfaces. This map is compatible with certain vanishing limiting products of the subrings of the chiral rings, on which the ring structure is related to a product of the roots of $A$-type Lie algebra.

The Leray spectral sequence is motivic

Our goal is to prove that the Leray spectral sequence associated to a map of algebraic varieties is motivic in the following sense: If the singular cohomology groups of the category of quasiprojective varieties defined over a subfield of C can be canonically endowed with some additional structure, such as its mixed Hodge structure or its motive (in Nori's sense or as a product of realizations). Then the Leray spectral sequence for a projective map in this category is automatically compatible with this structure. As byproduct, we get a relatively cheap construction of a mixed Hodge structure on…

混合 Hodge 结构背景下的算术稀疏性

Arithmetic sparsity in mixed Hodge settings

本文研究了混合 Hodge 结构背景下的算术稀疏性，证明了配备 $p$-进平展局部系统的代数簇上的 $S$-整点可被次多项式数量的子簇覆盖，并以此回答了关于 $S$-整 Laurent 多项式数量的一个公开问题。

辛结构与射影小覆盖 over 多边形的积

Symplectic and projective small covers over products of polygons

本文研究了多边形积上辛和射影结构的性质，引入了因子兼容类，并证明了每个因子兼容的小覆盖都具有光滑射影模型作为曲线积的有限商，同时展示了格雷德模2上同调环决定了关联射影模型的Hodge钻石。

安德森模的德尔塔理论 II：霍奇-平克结构

Delta theory of Anderson Modules II: Hodge-Pink structure

本文利用δ-几何理论，对任意阿贝尔安德森模E构造了带霍奇-平克结构的典范z-等晶体，证明其与de Rham上同调的相容关系及同构，还验证了Drinfeld模、Carlitz模的相关性质。

Tropical Normal Functions -- Higher Abel-Jacobi Invariants of Tropical cycles

We consider the variation of tropical Hodge structure (TVHS) associated to families of tropical varieties. The family of the tropical intermediate Jacobians of the associated tropical Hodge structure defines a bundle of tropical Jacobians, whose sections we call the tropical normal functions. We define formal sequential derivatives of these functions on the base with respect to the natural Gauss-Manin connection as the Hodge theoretic invariants detecting tropical cycles in the fibers. The associated invariants which are defined inductively are the higher Abel-Jacobi invariants in the tropical…

从有限节点锥形几何到BPS结构 I：代数状态数据

From Finite-Node Conifold Geometry to BPS Structures I: Algebraic State Data

本文研究有限节点锥形退化几何，通过代数方法提取状态数据，揭示其与混合Hodge模和Schober实现的兼容性。

非阿贝尔Hodge类型的Lagrange对应与平移扭曲结构

Lagrangian correspondences of nonabelian Hodge type and shifted twistor structures

本文建立平坦与Higgs完美复形的导出栈之间的Lagrange对应，并证明光滑射影簇上的解析Deligne-Hitchin-Simpson模栈具有规范平移预扭曲结构。