The locus of Hodge classes in an admissible variation of mixed Hodge structure

The zero locus of an admissible normal function

We prove that the zero locus of an admissible normal function over an algebraic parameter space S is algebraic in the case where S is a curve.

Fields of definition of Hodge loci

We show that an irreducible component of the Hodge locus of a polarizable variation of Hodge structure of weight 0 on a smooth complex variety X is defined over an algebraically closed subfield k of finite transcendence degree if X is defined over k and the component contains a k-rational point. We also prove a similar assertion for the Hodge locus inside the Hodge bundle if the Hodge bundle together with the connection is defined over k. This is closely related with the theory of absolute Hodge classes. The proof uses the spread of the Hodge locus, and is quite similar to the case of the zero…

Small codimension components of the Hodge locus containing the Fermat variety

We characterize the smallest codimension components of the Hodge locus of smooth degree $d$ hypersurfaces of the projective space $\mathbb{P}^{n+1}$ of even dimension $n$, passing through the Fermat variety (with $d\neq 3,4,6$). They correspond to the locus of hypersurfaces containing a linear algebraic cycle of dimension $\frac{n}{2}$. Furthermore, we prove that among all the local Hodge loci associated to a non-linear cycle passing through Fermat, the ones associated to a complete intersection cycle of type $(1,1,\ldots,1,2)$ attain the minimal possible codimension of their Zariski tangent s…

On the algebraicity of the zero locus of an admissible normal function

We show that the zero locus of an admissible normal function on a smooth complex algebraic variety is algebraic.

Hodge轨迹的次数

Degrees of Hodge Loci

本研究证明了Hodge轨迹次数增长的渐近估计，方法通用且适用于多数Hodge轨迹稠密的Hodge结构变分，并给出了控制Noether-Lefschetz轨迹次数与Torelli轨迹子簇次数的渐近公式。

Hodge-type integrals on moduli spaces of admissible covers

We study Hodge Integrals on Moduli Spaces of Admissible Covers. Motivation for this work comes from Bryan and Pandharipande's recent work on the local GW theory of curves, where analogouos intersection numbers, computed on Moduli Spaces of Relative Stable Maps, are the structure coefficients for a Topological Quantum Field Theory. Admissible Covers provide an alternative compactification of the Moduli Space of Maps, that is smooth and doesn't contain boundary components of excessive dimension. A parallel, yet different, TQFT, can then be constructed. In this paper we compute, using localizatio…

Hodge locus and Brill-Noether type locus

Given a family $π:\mc{X} \rightarrow B$ of smooth projective varieties, a closed fiber $\mc{X}_o$ and an invertible sheaf $\mc{L}$ on $\mc{X}_o$, we compare the Hodge locus in $B$ corresponding to the Hodge class $c_1(\mc{L})$ with the locus of points $b\,\in\, B$ such that $\mc{L}$ deforms to an invertible sheaf $\mc{L}_b$ on $\mc{X}_b$ with at least $h^0(\mc{L})$--dimensional space of global sections (it is a Brill-Noether type locus associated to $\mc{L}$). We finally give an application by comparing the Brill-Noether locus to a family of curves on a surface passing through a fixed set of p…

Weakly admissible locus and Newton stratification in p-adic Hodge theory

The basic admissible locus $\mathcal{F}(G, μ, b)^a$ inside the flag variety $\mathcal{F}(G, μ)$, attached to a reductive group $G$ with a minuscule cocharacter $μ$ of $G$, is a $p$-adic analogue of the complex analytic period spaces. It has an algebraic approximation $\mathcal{F}(G, μ, b)^{wa}$ inside the flag variety, called the weakly admissible locus. On the flag variety $\mathcal{F}(G, μ)$, we have the Newton stratification which has the admissible locus as its unique open stratum. In this paper, we study the relation between the Newton strata and the weakly admissible locus. We show that …

Linear Hyperelliptic Hodge Integrals

We provide a closed form expression for linear Hodge integrals on the hyperelliptic locus. Specifically, we find a succinct combinatorial formula for all intersection numbers on the hyperelliptic locus with one $λ$-class, and powers of a $ψ$-class pulled back along the branch map. This is achieved by using Atiyah-Bott localization on a stack of stable maps into the orbifold $[\mathbb{P}^1/\mathbb{Z}_2]$.

On the Locus of Hodge Classes

Let $f: X \rightarrow S$ be a family of non singular projective varieties parametrized by a complex algebraic variety $S$. Fix $s \in S$, an integer $p$, and a class $h \in {\rm H}^{2p}(X_s,\Z)$ of Hodge type $(p,p)$. We show that the locus, on $S$, where $h$ remains of type $(p,p)$ is algebraic. This result, which in the geometric case would follow from the rational Hodge conjecture, is obtained in the setting of variations of Hodge structures.

关于非阿贝尔霍奇位形I

On the non-abelian Hodge locus I

本文研究了准射影变种上整数局部系统背后的极化Hodge结构，证明了非阿贝尔霍奇位形的代数性及有限性定理。

关于混合Hodge域的分布

On the distribution of mixed Hodge loci

研究混合Hodge域在Zariski拓扑下的稠密性及其在大monodromy条件下的空集性质，改进了Brosnan-Pearlstein-Schnell的经典结果。

Calabi-Yau attractor varieties and degeneration of Hodge structure

We study the structure of string theory flux compactification for a general family of elliptic CY 3-folds. We investigate the locus of the attractor points of the flux compactification in type IIB string theory on the boundary components of period domains. Specifically we give equations describing this locus through the asymptotic of nilpotent orbits on period domains. our approach is a mixture of techniques of asymptotic Hodge theory and the numerical period vectors used in physics.

可容许覆盖上的Hodge积分与Prym类的tautological投影

Hodge integrals on admissible covers and tautological projections of Prym classes

本文提出一种算法，利用可容许覆盖模空间上三个Hodge丛的比较公式，计算Prym轨迹类的tautological投影，并应用于计算Prym映射次数及证明低亏格下雅可比类的不可整除性。

Sur les varietes de Hodge

Let $Y$ be a smooth complex projective variety of dimension $N+1$, $L$ an invertible sufficiently ample sheaf, $X\in |L|$ a smooth hypersurface and $λ\in F^kH^N(X,C)$ a vanishing cohomology class, where $F^{*}$ is the Hodge filtration and $k\in\{1,...,[N/2]\}$. Assume that $L$ is sufficiently ample and that the codimension in $|L|$ of the Hodge variety associated to $λ$ (locally defined as the locus where the image of $λ$ by flat transport over $|L|$ remains in $F^k$) is sufficiently small. I show that this forces $N$ to be even and $k=[N/2]$, and that the class $λ$ is a linear combination wit…

Singularities of admissible normal functions

In a recent paper, M. Green and P. Griffiths used R. Thomas' works on nodal hypersurfaces to establish the equivalence of the Hodge conjecture and the existence of certain singular admissible normal functions. Inspired by their work, we study normal functions using M. Saito's mixed Hodge modules and prove that the existence of singularities of the type considered by Griffiths and Green is equivalent to the Hodge conjecture. Several of the intermediate results, including a relative version of the weak Lefschetz theorem for perverse sheaves, are of independent interest.

关于 Hodge 轨迹的分布

On the distribution of the Hodge locus

本文证明 level 至少为 3 的 Hodge 轨迹（正周期维数）是有限并集因而是代数的，而 level 1 或 2 时若含典型特殊子簇则解析稠密，并完整阐明了特殊子簇的典型/非典型分布。

Hyperbolicity of the base of an admissible family of log canonical stable minimal models

We investigate the stratified hyperbolicity properties of Birkar's moduli stack of log canonical (lc) stable minimal models. The main technical result is a construction of Viehweg-Zuo's system of Higgs sheaves associated with an admissible family of lc stable minimal models, using the theory of degenerations of Hodge structure and non-abelian Hodge theory.

The μ-ordinary locus for Shimura varieties of Hodge type

We review the Newton stratification and Ekedahl-Oort stratification on the special fiber of a smooth integral model for a Shimura variety of Hodge type at a prime of good reduction. We show that the μ-ordinary locus coincides with the generic Ekedahl-Oort stratum, and that for any two geometric points in the μ-ordinary locus there is an isomorphism of the attached Dieudonne modules with additional structure. As a consequence, we proof that the μ-ordinary locus is open and dense, thus generalizing the results which were already known in the PEL-case. To prove our results we provide a method whi…

Hodge loci and absolute Hodge classes

We combine Deligne's global invariant cycle theorem, and the algebraicity theorem of Cattani, Deligne and Kaplan, for the connected components of the locus of Hodge classes, to conclude that under simple assumptions these components are defined over number fields (assuming the initial family is), as expected from the Hodge conjecture. We also show that the Hodge conjecture for (weakly) absolute Hodge classes reduces to the Hodge conjecture for (weakly) absolute Hodge classes on varieties defined over number fields.

Abelian Hurwitz-Hodge integrals

Hodge classes on the moduli space of admissible covers with monodromy group G are associated to irreducible representations of G. We evaluate all linear Hodge integrals over moduli spaces of admissible covers with abelian monodromy in terms of multiplication in an associated wreath group algebra. In case G is cyclic and the representation is faithful, the evaluation is in terms of double Hurwitz numbers. In case G is trivial, the formula specializes to the well-known result of Ekedahl-Lando-Shapiro-Vainshtein for linear Hodge integrals over the moduli space of curves in terms of single Hurwitz…

例外极值处的允许子范畴

All Quiet on the Exceptional Locus

研究光滑射影表面有理映射的有界导出范畴中支持于例外极值处的允许子范畴，证明若f:X→Y为光滑射影表面的有理映射，则所有支持于Exc(f)的允许子范畴由有限的例外集合生成。若K_Y为 nef，则所有支持于X闭合子集的允许子范畴同样成立。

Generating Functions for Hurwitz-Hodge Integrals

In this paper we describe explicit generating functions for a large class of Hurwitz-Hodge integrals. These are integrals of tautological classes on moduli spaces of admissible covers, a (stackily) smooth compactification of the Hurwitz schemes. Admissible covers and their tautological classes are interesting mathematical objects on their own, but recently they have proved to be a useful tool for the study of the tautological ring of the moduli space of curves, and the orbifold Gromov-Witten theory of DM stacks. Our main tool is Atiyah-Bott localization: its underlying philosophy is to transla…

关于实阿贝尔三维流形的整霍奇猜想

On the integral Hodge conjecture for real abelian threefolds

本文证明了若干类实阿贝尔三维流形（如实轨迹连通或可分解为阿贝尔曲面与椭圆曲线之积的情形）的实整霍奇猜想，并证明所有实阿贝尔三维流形在模去挠群时均满足该猜想，将一般情形归约为雅可比情形。

On the equidistribution of some Hodge loci

We prove the equidistribution of the Hodge locus for certain non-isotrivial, polarized variations of Hodge structure of weight $2$ with $h^{2,0}=1$ over complex, quasi-projective curves. Given some norm condition, we also give an asymptotic on the growth of the Hodge locus. In particular, this implies the equidistribution of elliptic fibrations in quasi-polarized, non-isotrivial families of $K3$ surfaces.

Hodge loci and atypical intersections: conjectures

We present a conjecture on the geometry of the Hodge locus of a (graded polarizable, admissible) variation of mixed Hodge structure over a complex smooth quasi-projective base, generalizing to this context the Zilber-Pink Conjecture for mixed Shimura varieties (in particular the André-Oort conjecture).

Variations of Hodge structures of a Teichmueller curve

Teichmueller curves are geodesic discs in Teichmueller space that project to an algebraic curve in the moduli space $M_g$. We show that for all $g \geq 2$ Teichmueller curves map to the locus of real multiplication in the moduli space of abelian varieties. Remark that McMullen has shown that precisely for $g=2$ the locus of real multiplication is stable under the $\SL_2(\RR)$-action on the tautological bundle $Ω\M_g$. We also show that Teichmueller curves are defined over number fields and we provide a completely algebraic description of Teichmueller curves in terms of Higgs bundles. As a cons…

Normal functions and spread of zero locus

If there is a topologically locally constant family of smooth algebraic varieties together with an admissible normal function on the total space, then the latter is constant on any fiber if this holds on some fiber. Combined with spreading out, it implies for instance that an irreducible component of the zero locus of an admissible normal function is defined over k if it has a k-rational point where k is an algebraically closed subfield of the complex number field with finite transcendence degree. This generalizes a result of F. Charles that was shown in case the normal function is associated …

关于一个Hodge轨迹

On a Hodge locus

本文通过三次超曲面的实例，研究代数循环同调类作为Hodge循环与代数循环的形变空间差异，并提议验证（变分）Hodge猜想或寻找其反例。

Period integrals and Hodge modules

We define a map $\mathcal{P}_M$ attached to any polarized Hodge module $M$ such that the restriction of $\mathcal{P}_M$ to a locus on which $M$ is a variation of Hodge structures induces the usual period integral pairing for this variation of Hodge structures. In the case that $M$ is the minimal extension of a simple polarized variation of Hodge structures $V$, we show that the homotopy image of $\mathcal{P}_M$ is the minimal extension of the graph morphism of the usual period integral map for $V$.

可接受对和p进Hodge结构 I：de Rham格的超越性

Admissible pairs and $p$-adic Hodge structures I: Transcendence of the de Rham lattice

本文定义了p进Hodge结构的Tannakian范畴，并展示了其与可接受对之间的等价关系，通过研究p进周期的超越性来推广一维形式群的结果。

Asymptotics of degenerations of mixed Hodge structures

We construct a hermitian metric on the classifying spaces of graded-polarized mixed Hodge structures and prove analogs of the strong distance estimate between an admissible period map and the approximating nilpotent orbit. We also consider the asymptotic behavior of the biextension metric, the norm estimates and the asymptotics of the reduced limit Hodge filtration.

Gauss-Manin connection in disguise: Noether-Lefschetz and Hodge loci

We give a classification of components of the Hodge locus in any parameter space of smooth projective varieties. This is done using determinantal varieties constructed from the infinitesimal variation of Hodge structures (IVHS) of the underlying family. As a corollary we prove that the minimum codimension for the components of the Hodge locus in the parameter space of $m$-dimensional hypersurfaces of degree $d$ with $d\geq 2+\frac{4}{m}$ and in a Zariski neighborhood of the point representing the Fermat variety, is obtained by the locus of hypersurfaces passing through an $\frac{m}{2}$-dimensi…

Variations of mixed Hodge structure and semi-positivity theorems

We discuss the variations of mixed Hodge structure for cohomology with compact support of quasi-projective simple normal crossing pairs. We show that they are graded polarizable admissible variations of mixed Hodge structure. Then we prove a generalization of the Fujita--Kawamata semi-positivity theorem.

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。

A representation-valued relative Riemann-Hurwitz theorem and the Hurwitz-Hodge bundle

We provide a formula describing the G-module structure of the Hurwitz-Hodge bundle for admissible G-covers in terms of the Hodge bundle of the base curve, and more generally, for describing the G-module structure of the push-forward to the base of any sheaf on a family of admissible G-covers. This formula can be interpreted as a representation-ring-valued relative Riemann-Hurwitz formula for families of admissible G-covers.

可容许对与$p$-adic Hodge结构 II：双解析Ax-Lindemann定理

Admissible pairs and $p$-adic Hodge structures II: The bi-analytic Ax-Lindemann theorem

本文通过可容许对的模空间重新解释局部Shimura簇，证明双解析Ax-Lindemann定理，表明基本情形下双解析子菱形仅为特殊子簇，并建立与全局双代数几何平行的局部p-adic双解析几何理论。

Structures de Hodge-Pink pour les $φ/\mathfrak S$-modules de Breuil et Kisin

In this article, we apply the methods of our work on Fontaine's theory in equal characteristics to the $φ/\mathfrak S$-modules of Breuil and Kisin. Thanks to a previous article of Kisin, this yields a new and rather elementary proof of the theorem "weakly admissible implies admissible" of Colmez-Fontaine.

关于超越 Hodge-Tate 轨迹的 $O_K$ 棱柱化

On the prismatization of $O_K$ beyond the Hodge-Tate locus

本文通过研究Spf(O_K)的n-截断棱柱化上的完美复形，对n≤1+(p-1)/e时的n-截断棱柱晶体完美复形进行分类，并借助向量丛有理化描述G_K在B_dR,n^+中的连续半线性表示及其上同调，同时分类了若干de Rham棱柱晶体的整模型。

关于具有半稳定退化的紧复流形的Hodge结构

On Hodge structures of compact complex manifolds with semistable degenerations

研究半稳定退化紧复流形上Hodge理论性质的保持条件，证明在特定单值性条件下，Hodge对称性和Hodge-Riemann关系成立。

Variation Of Mixed Hodge Structures

Variation of mixed Hodge structures(VMHS), introduced by P. Deligne, is a linear structure reflecting the geometry on cohomology of the fibers of an algebraic family, generalizing variation of Hodge structures for smooth proper families, introduced by P. Griffiths. Hence, it is a strong tool to study the variation of the geometric structure of fibers of a morphism. We describe here the degenerating properties of a VMHS of geometric origin and the existence of a relative monodromy filtration, as well the definition and properties of abstract admissible VMHS.

Period Spaces for Hodge Structures in Equal Characteristic

We develop the analog in equal positive characteristic of Fontaine's theory for crystalline Galois representations of a p-adic field. In particular we describe the analog of Fontaine's functor which assigns to a crystalline Galois representation an isocrystal with a Hodge filtration. In equal characteristic the role of isocrystals and Hodge filtrations is played by z-isocrystals and Hodge-Pink structures. The latter were invented by Pink. Our first main result in this article is the analog of the Colmez-Fontaine Theorem that "weakly admissible implies admissible". Next we construct period spac…

Singularities of variations of mixed Hodge structure

We prove that a variation of graded-polarizable mixed Hodge structure over a punctured disk with unipotent monodromy, has a limiting mixed Hodge structure at the puncture (i.e., it is admissible in the sense of [SZ]) which splits over $\R$, if and only if certain grading of the complexified weight filtration, depending smoothly on the Hodge filtration, extends across the puncture. In particular, the result exactly supplements Schmid's Theorem for pure structures, which holds for the graded variation, and gives a Hodge-theoretic condition for the relative monodromy weight filtration to exist.

Limits of Hodge structures in several variables, II

The aim of this article is to study degeneration of the variations of Hodge structure associated to a proper Kähler semistable morphism. We prove that the weight filtrations constructed in the author's previous paper coincide with the monodromy weight filtrations on the relative log de Rham cohomology groups of a proper Kähler semistable morphism. Moreover, we show that the limiting mixed Hodge structures form admissible variations of mixed Hodge structure.

$L^2$-Hodge理论构造Calabi-Yau光滑射影超曲面的Frobenius流形

$L^2$-Hodge theoretic construction of Frobenius manifolds for Calabi-Yau smooth projective hypersurfaces

本文通过Li-Wen的$L^2$-Hodge理论，为Calabi-Yau光滑射影超曲面的上同调构造了Frobenius流形结构，并与Barannikov-Kontsevich的构造进行了精确比较。

On the monodromy of the moduli space of Calabi-Yau threefolds coming from eight planes in $\mathbb{P}^3$

It is a fundamental problem in geometry to decide which moduli spaces of polarized algebraic varieties are embedded by their period maps as Zariski open subsets of locally Hermitian symmetric domains. In the present work we prove that the moduli space of Calabi-Yau threefolds coming from eight planes in $\mathbb{P}^3$ does {\em not} have this property. We show furthermore that the monodromy group of a good family is Zariski dense in the corresponding symplectic group. Moreover, we study a natural sublocus which we call hyperelliptic locus, over which the variation of Hodge structures is natura…

Progrès récents sur les fonctions normales (d'après Green-Griffiths, Brosnan-Pearlstein, M. Saito, Schnell...)

Given a family of smooth complex projective varieties, the Hodge conjecture predicts the algebraicity of the locus of Hodge classes. This was proven unconditionnally by Cattani, Deligne and Kaplan in 1995. In a similar way, conjectures on algebraic cycles have led Green and Griffiths to conjecture the algebraicity of the zero locus of normal functions. This corresponds to a mixed version of the theorem of Cattani, Deligne and Kaplan. This result has been proven recently by Brosnan-Pearlstein, Kato-Nakayama-Usui, and Schnell building on work of M. Saito. We will present some of the ideas around…

Hurwitz-Hodge Integrals, the E6 and D4 root systems, and the Crepant Resolution Conjecture

Let G be the group A_4 or Z_2xZ_2. We compute the integral of λ_g on the Hurwitz locus H_G\subset M_g of curves admitting a degree 4 cover of P^1 having monodromy group G. We compute the generating functions for these integrals and write them as a trigonometric expression summed over the positive roots of the E_6 and D_4 root systems respectively. As an application, we prove the Crepant Resolution Conjecture for the orbifolds [C^3/A_4] and [C^3/(Z_2xZ_2)].

The Ax-Schanuel conjecture for variations of mixed Hodge structures

We prove in this paper the Ax-Schanuel conjecture for all admissible variations of mixed Hodge structures.

Disproof of modularity of moduli space of CY 3-folds of double covers of P3 ramified along eight planes in general positions

We prove that the moduli space of Calabi-Yau 3-folds coming from eight planes of $P^3$ in general positions is not modular. In fact we show the stronger statement that the Zariski closure of the monodromy group is actually the whole $Sp(20,R)$. We construct an interesting submoduli, which we call \emph{hyperelliptic locus}, over which the weight 3 $Q$-Hodge structure is the third wedge product of the weight 1 $Q$-Hodge structure on the corresponding hyperelliptic curve. The non-extendibility of the hyperelliptic locus inside the moduli space of a genuine Shimura subvariety is proved.

All the $λ_1$'s on cyclic admissible covers

We compute the degree of Hurwitz-Hodge classes $λ_1^e$ on one dimensional moduli spaces of cyclic admissible covers of the projective line. We also compute the degree of the the first Chern class of the Hodge bundle $λ_1$ for all one dimensional moduli spaces. In higher dimension, we express the divisor class $λ_1$ as a linear combination of $ψ$ classes and boundary strata.

Fargues-Rapoport conjecture for p-adic period domains in the non-basic case

We prove the Fargues-Rapoport conjecture for p-adic period domains in the non-basic case with minuscule cocharacter. More precisely, we give a group theoretical criterion for the cases when the admissible locus and weakly admissible locus coincide.

Extending holomorphic forms from the regular locus of a complex space to a resolution of singularities

We investigate under what conditions holomorphic forms defined on the regular locus of a reduced complex space extend to holomorphic (or logarithmic) forms on a resolution of singularities. We give a simple necessary and sufficient condition for this, whose proof relies on the Decomposition Theorem and Saito's theory of mixed Hodge modules. We use it to generalize the theorem of Greb-Kebekus-Kovács-Peternell to complex spaces with rational singularities, and to prove the existence of a functorial pull-back for reflexive differentials on such spaces. We also use our methods to settle the "local…

约化曲线的 Hodge-奇异对应

Hodge-to-singular correspondence for reduced curves

本文研究了约化谱曲线上 $\mathrm{GL}_n$ 的 Hitchin 系统分解定理的直和项，建立了其与超环面箭图簇拓扑的新对应关系，并描述了正则 Higgs 丛模空间交叉上同调群对度数的依赖性。