Salim Tayou
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.
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$p$-adic Equidistribution of Special Loci in a Product of Modular Curves
该论文研究定义在ℂₚ上的光滑曲线C与模子簇Y₀(n)交轨迹的p进等分布性，给出等分布成立的条件，证明对应交集合非离散，并给出等分布不成立的典型例子。
A polarizable variation of Hodge structure over a smooth complex quasi projective variety $S$ is said to be defined over a number field $L$ if $S$ and the algebraic connection associated to the variation are both defined over $L$. Conjecturally any special subvariety (also called "an irreducible component of the Hodge locus) for such variations is defined over $\overline{\mathbb{Q}}$, and its Galois conjugates are also special subvarieties. We prove this conjecture for special subvarieties satisfying a simple monodromy condition. As a corollary we reduce the conjecture that special subvarietie…
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.
Let $\mathbb V$ be a polarized variation of Hodge structure over a smooth complex quasi-projective variety $S$. In this paper, we give a complete description of the typical Hodge locus for such variations. We prove that it is either empty or equidistributed with respect to a natural differential form, \emph{the pull-push form}. In particular, it is always analytically dense when the pull-push form does not vanish. When the weight is $2$, the Hodge numbers are $(q,p,q)$ and the dimension of $S$ is least $rq$, we prove that the typical locus where the Picard rank is at least $r$ is equidistribut…
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).
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。
Existence and density of typical Hodge loci
受Baldi-Klingler-Ullmo问题启发，提出可极化Z-霍奇结构变分下典型霍奇轨迹存在与解析稠密的一般充分判据，重证Noether-Lefschetz轨迹稠密性结果，证明g≥4时A_g的维数≥g的子簇有解析稠密典型霍奇轨迹，可应用于Torelli轨迹。
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…
We consider the loci of d-elliptic curves in $M_2$, and corresponding loci of d-elliptic surfaces in $A_2$. We show how a description of these loci as quotients of a product of modular curves can be used to calculate cohomology of natural local systems on them, both as mixed Hodge structures and $\ell$-adic Galois representations. We study in particular the case d=2, and compute the Euler characteristic of the moduli space of n-pointed bi-elliptic genus 2 curves in the Grothendieck group of Hodge structures.
We show that closures of families of unitary local systems on quasiprojective varieties for which the dimension of a graded component of Hodge filtration has a constant value can be identified with a finite union of polytopes. We also present a local version of the theorem. This yields the "Hodge decomposition" of the set of unitary local systems with a non-vanishing cohomology extending Hodge decomposition of characteristic varieties of links of plane curves studied by the author earlier. We consider a twisted version of the characteristic varieties generalizing the twisted Alexander polynomi…
This paper contains two results on Hodge loci in the moduli space of curves. The first concerns fibrations over curves with a non-trivial flat part in the Fujita decomposition. If local Torelli theorem holds for the fibres and the fibration is non-trivial, an appropriate exterior power of the cohomology of the fiber admits a Hodge substructure. In the case of curves it follows that the moduli image of the fibers is contained in a proper Hodge locus. The second result deals with divisors in the moduli space of curves. It is proved that the image of a divisor in the moduli of principally polariz…
Leaf schemes and Hodge loci
本文提出发展叶方案理论，赋予叶方案结构，并提出局部-全局猜想。
Flat Vector Bundles on Very General Curves and Codimension of Non-Abelian Hodge Loci
本文基于复Hodge结构的变体秩和水平，界定了非阿贝尔Hodge领域在相对de Rham模空间上的码维，并通过Landesman和Litt的成果，研究了等monodromy叶状结构及代数monodromy群的李代数秩。
Atypical Hodge Loci
本文为综述性工作，阐释并证明Baldi、Klingler和Ullmo关于霍奇结构算术性质的核心成果，通过周期域交表述Noether-Lefschetz轨迹，利用李代数分解对齐性与Pfaffian系统可积性解释超额相交性质。
We prove that the uniformizing map of any arithmetic quotient, as well as the period map associated to any pure polarized $\mathbb{Z}$-variation of Hodge structure $\mathbb{V}$ on a smooth complex quasi-projective variety $S$, are topologically tame. As an easy corollary of these results and of Peterzil-Starchenko's o-minimal GAGA theorem we obtain that the Hodge locus of $(S, \mathbb{V})$ is a countable union of algebraic subvarieties of $S$ (a result originally due to Cattani-Deligne-Kaplan).
On a conjecture on Hodge loci of linear combinations of linear subvarieties
本文针对 Movasati 关于包含相交 $k$-平面的三次和四次超曲面 Hodge 轨迹维数的猜想，在 $k \geq 5$ 时给出反例，并在 $k=3,4$ 时提供支持该猜想的新证据。
Algebraic Cycle Loci at the Integral Level
本研究针对数域上光滑射影族的代数闭链轨迹问题，提出整水平下统一研究的技术，给出非平凡常义代数闭链轨迹并集的非Zariski稠密性准则，推广了一般纤维的Hodge轨迹研究。
Hodge theory for local systems and cohomological support loci
本文以 $\partial\bar{\partial}$-引理为统一框架，重新建立一般消失定理与上同调跳跃轨迹理论，并发展拟紧凯勒流形上一阶局部系统的 $L^2$-霍奇理论。
We define two tau functions, $τ$ and $\hatτ$ , on moduli spaces of spectral covers of $GL(n)$ Hitchin's systems. Analyzing the properties of $τ$, we express the divisor class of the universal Hitchin's discriminant in terms of standard generators of the rational Picard group of the moduli spaces of spectral covers with variable base. The function $\hatτ$ is used to compute the divisor of canonical 1-forms with multiple zeros.
In this paper we prove the following results: $1)$ We show that any arithmetic quotient of a homogeneous space admits a natural real semi-algebraic structure for which its Hecke correspondences are semi-algebraic. A particularly important example is given by Hodge varieties, which parametrize pure polarized integral Hodge structures. $2)$ We prove that the period map associated to any pure polarized variation of integral Hodge structures $\mathbb{V}$ on a smooth complex quasi-projective variety $S$ is definable with respect to an o-minimal structure on the relevant Hodge variety induced by the…
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…
Hodge loci associated with linear subspaces intersecting in codimension one
本文研究光滑超曲面中两个余维一相交线性子空间所定义的Hodge轨迹是否重合的问题，证明在一般情形下二者局部重合，并构造出二者光滑性不同的反例，揭示其差异源于嵌入分量。
We announce the construction of toroidal partial compactifications of the moduli spaces of mixed Hodge structures with polarized graded quotients. They are moduli spaces of log mixed Hodge structures with polarized graded quotients. We include an application to the analyticity of zero loci of some functions.
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。
Differential complexes and Hodge theory on log-symplectic manifolds
本文研究对数辛流形上的微分形式复形（含反向de Rham复形），将其与Poisson双矢量的退化除子及秩轨迹关联，并在Kähler情形下建立重数轨迹与Hodge数的关系，同时揭示Hodge数消失与无阻碍形变的联系。
We describe the Hodge theory of brilliant families of K3 surfaces. Their characteristic feature is a close link between the Hodge structures of any two fibres over points in the Noether-Lefschetz locus. Twistor deformations, the analytic Tate-Safarevic group, and one-dimensional Shimura special cycles are covered by the theory. In this setting, the Brauer group is viewed as the Noether-Lefschetz locus of the Brauer family or as the specialization of the Noether-Lefschetz loci in a family of approaching twistor spaces. Passing from one algebraic twistor fibre to another, which by construction i…
In this paper, we prove a general theorem concerning the analyticity of the closure of a subspace defined by a family of variations of mixed Hodge structures, which includes the analyticity of the zero loci of degenerating normal functions. For the proof, we use a moduli of the valuative version of log mixed Hodge structures.
Non-semi-stable loci in Hecke stacks and Fargues' conjecture
该研究在Hodge-Newton可约条件下证明了非基本Rapoport-Zink空间推广形式的Harris-Viehmann猜想，得到非半稳定轨迹上同调消失的结论，并应用于GL(2)情形下Fargues猜想的Hecke本征层性质证明。
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。
We prove a few cases of a conjecture on the invariance of cohomological support loci under derived equivalence by establishing a concrete connection with the related problem of the invariance of Hodge numbers. We use the main case in order to study the derived behavior of fibrations over curves.
Density of Noether-Lefschetz loci for surfaces in Fano and Calabi-Yau threefolds
该研究利用 Hodge 轨迹的一般定理，考察 Fano 与 Calabi–Yau 三维簇中充分丰富线性系统曲面族的 Noether–Lefschetz 轨迹，并比较一般分支与例外分支并集的不同密度行为。
Local-global principle for leaf schemes
本文研究叶概型（如Hodge轨迹）的局部-全局原理，利用Gauss-Manin联络矩阵，并探讨定义环猜想及其代数性推论。
A closed formula is obtained for the integral $\int_{\mathcal{\bar{H}}_g^1}κ_{1}ψ^{2g-2}$ of tautological classes over the locus of hyperelliptic Weierstraß points in the moduli space of curves. As a corollary, a relation between Hodge integrals is obtained. The calculation utilizes the homeomorphism between the moduli space of curves $\mathcal{M}_{g,1}$ and the combinatorial moduli space $\mathcal{M}^{comb}_{g,1}$, a PL-orbifold whose cells are enumerated by fatgraphs. This cell decomposition can be used to naturally construct combinatorial PL-cycles $W_a\subset\mathcal{M}^{comb}_{g,1}$ whose…
A characteristic p analogue of the André--Pink--Zannier conjecture
该研究针对特征p下的André--Pink--Zannier猜想，证明了Hodge型Shimura簇中具大单值群的常值函数域点情形及相应Hecke等分布代数类似物，通过Hecke对应分析与Weyl特殊点充足性证明完成论证。
Non-abelian Hodge correspondence over singular Kähler spaces
本文在奇异凯勒空间上建立了非阿贝尔霍奇对应，并通过调和丛和降维定理扩展了射影klt品种的结果，应用于具有大canonical除数的品种的准统一化定理。
The generalized roof F(1,2,n): Hodge structures and derived categories
本文研究广义齐次屋顶的零轨迹的 Hodge 结构，并针对旗簇 F(1,2,n) 利用 B-膜范畴方法构造导出嵌入。
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…
Periods of join algebraic cycles
本文证明在特定维数和次数下，光滑超曲面模空间包含无穷多个与线性周期同余维数的 Hodge 轨迹，并构造了由 $0$ 维周期连接而成的 Hodge 周期，建立了连接周期类公式及 Artin Gorenstein 代数张量积刻画。
We prove a criterion for the existence of harmonic metrics on Higgs bundles that are defined on smooth loci of klt varieties. As one application, we resolve the quasi-etale uniformisation problem for minimal varieties of general type to obtain a complete numerical characterisation of singular quotients of the unit ball by discrete, co-compact groups of automorphisms that act freely in codimension one. As a further application, we establish a nonabelian Hodge correspondence on smooth loci of klt varieties.
On the algebraicity of non-abelian Noether-Lefschetz loci of rank-two real local systems
本文在曲线模空间上研究非阿贝尔Noether-Lefschetz轨迹的代数性，证明其等价于表示像的离散性，并通过狭缝手术构造两个显式有理族，其中离散单值族回答了Baldi-Lam的问题。
In this note we survey recent results on the extrinsic geometry of the Jacobian locus inside $\mathsf{A}_g$. We describe the second fundamental form of the Torelli map as a multiplication map, recall the relation between totally geodesic subvarieties and Hodge loci and survey various results related to totally geodesic subvarieties and the Jacobian locus.
Fixed-locus relations and purely non-symplectic automorphisms of order 6 on K3 surfaces
本文研究K3曲面上6阶纯非辛自同构的固定轨迹关系，通过比较弦性欧拉特征与霍奇数推导关系，并利用局部条件将候选数从817减少到204，且给出20种固定轨迹的显式模型。
The de Rham stack and the variety of very good splittings of a curve
本文定义并研究了特殊 Azumaya 代数中“非常好的分裂”构成的开放子栈，证明其具有拟射影驯顺模空间，且使用该分裂时非阿贝尔 Hodge 同构能保持半稳定轨迹，进而导出交截复形前推及反常层的同构性。
Analytically dense Mordell-Weil rank jumps on elliptic surfaces via transverse realization
本文研究椭圆面的高度模数，通过动机高度ζ函数的有理性与Kudla-Millson模性，探讨Mordell-Weil秩跃迁问题。
Intermediate hyperbolicity of varieties supporting a variation of Hodge structure
本文研究承载可极化Hodge结构变分的光滑复射影簇的对数余切丛的中间双曲性，证明其各次外幂为L-丰富，给出Viehweg-丰富的判定方法及局部对称簇、五次三维簇模空间覆盖的具体结果。
Maximal Variation in the Moduli of Curves
本文研究了曲线模空间中最大变化区域的性质，揭示了非Gorenstein奇点对退化的影响，并建立了退化区域的分类方法。
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