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

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。

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 …

The zero locus of the infinitesimal invariant

Let ν be a normal function on a complex manifold X. The infinitesimal invariant of ν has a well-defined zero locus inside the tangent bundle TX. When X is quasi-projective, and ν is admissible, we show that this zero locus is constructible in the Zariski topology.

On the Geometric Zilber-Pink Theorem and the Lawrence-Venkatesh method

Using our recent results on the algebraicity of the Hodge locus for variations of Hodge structures of level at least $3$, we improve the results of Lawrence-Venkatesh in direction of the refined Bombieri-Lang conjecture.

具有挠奇异性的容许正规函数的零轨迹

Zero loci of admissible normal functions with torsion singularities

证明了光滑复代数簇S上的正规函数的零轨迹是代数的，只要该正规函数扩张到S的具有扭奇异的光滑紧化上的容许正规函数.这个结果推广了我们以前关于曲线上容许正规函数的结果[arxiv：math/0604345 [math.AG]]。M. Saito在最近的预印本中使用了不同的方法[arXiv：0803.2771v2]。

An Exponential History of Functions with Logarithmic Growth

We survey recent work on normal functions, including limits and singularities of admissible normal functions, the Griffiths-Green approach to the Hodge conjecture, algebraicity of the zero-locus of a normal function, Neron models, and Mumford-Tate groups. Some of the material and many of the examples, esp. in sec. 5-6, are original. The first two sections should be easily accessible to graduate students.

Tame topology of arithmetic quotients and algebraicity of Hodge loci

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…

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.

具有有限0-位势的实调节映射

Real regulator maps with finite 0-locus

研究双变量拉格朗日多项式在条件下的调节映射，证明其扩展的调节函数和A多项式位势均为有限集。

代数卵形外Cauchy变换的代数性：同调表述

Algebraicity of exterior Cauchy transforms of algebraic ovals: a homological formulation

研究有界域边界为实代数曲线卵形时，外Cauchy变换为代数的条件，通过同调论建立留数判据，证明有理实代数曲线的光滑卵形和正亏格分离卵形的代数性。

除正常交叉定位外的解析性及伽罗瓦作用

Resolution Except for the Normal-Crossing Locus and Galois actions

本文在零特征下构造了一个基于加权吹脱的通用解析算法，严格保持正常交叉定位，解决Kollar问题。通过加权吹脱框架和对数细化，实现鲁棒解析算法，应用于正常交叉Deligne-Mumford堆的canonical compactification和子簇与堆的functorial解析。

关于 Ceresa 正规函数的挠轨迹

On the torsion locus of the Ceresa normal function

本文证明 $g\geq3$ 时模空间中 Ceresa 正规函数挠轨迹的正维数部分非 Zariski 稠密，并刻画其一般 Mumford–Tate 群为 $\mathrm{GSp}_{2g}$ 的有限分支及其 Galois 封闭性。

Complex-analytic Neron models for arbitrary families of intermediate Jacobians

Given a family of intermediate Jacobians (for a polarized variation of Hodge structure of weight -1) on a Zariski-open subset of a complex manifold, we construct an analytic space that naturally extends the family. Its two main properties are: (a) the horizontal and holomorphic sections are precisely the admissible normal functions without singularities; (b) the graph of any admissible normal function has an analytic closure inside our space. As a consequence, we obtain a new proof for the zero locus conjecture of M. Green and P. Griffiths. The construction uses filtered D-modules and M. Saito…

Schur乘子在$q$-永久函数的Pólya转换问题中的极限

The limits of Schur multipliers in Pólya conversion problems for the $q$-permanent function

本文研究$q$-永久函数的广义Pólya转换问题，建立了从低维代数灵活性到高维组合刚性的尖锐阈值，并分类了Schur乘子保持子空间。