On Algebraic Hyperbolicity of Log Varieties

正性、plethysm与正特征下Siegel簇的双曲性

Positivity, plethysm and hyperbolicity of Siegel varieties in positive characteristic

本文研究了特征 $p$ 下Siegel modular variety的双曲性，利用Schur函子的plethysm运算和新的 $(φ,D)$-ample正性概念，证明了多种自守向量丛满足该正性，推广了Hodge线丛的已知结论。

Hyperbolicity, automorphic forms and Siegel modular varieties

We study the hyperbolicity of compactifications of quotients of bounded symmetric domains by arithmetic groups. We prove that, up to an étale cover, they are Kobayashi hyperbolic modulo the boundary. Applying our techniques to Siegel modular varieties, we improve some former results of Nadel on the non-existence of certain level structures on abelian varieties over complex function fields.

Hyperbolicity of varieties supporting a variation of Hodge structure

We generalize former results of Zuo and the first author showing some hyperbolicity properties of varieties supporting a variation of Hodge structure. Our proof only uses the special curvature properties of period domains. In particular, in contrast to the former approaches, it does not use any result on the asymptotic behaviour of the Hodge metric.

皮卡数为1的法诺三重折叠中曲面的代数双曲性

Algebraic hyperbolicity of surfaces in Fano threefolds with Picard number one

本文研究皮卡数为1的一般法诺三重折叠中极一般曲面的代数双曲性，除加权超曲面中的曲面外，对其代数双曲性完成了完整分类。

齐性簇子簇的代数双曲性

Algebraic hyperbolicity of subvarieties of homogeneous varieties

本文将超曲面的代数双曲性结果推广到高余维完全交，证明在射影空间中非常一般的完全交在次数和满足特定阈值条件时具有或不具有代数双曲性。

伴随线系统在球面变种上的代数双曲性

Algebraic hyperbolicity of adjoint linear systems on spherical varieties

论文证明了在球面变种上，伴随线系统非常一般的元素具有代数双曲性，并推导出该猜想在horospherical变种和toroidal球面变种上成立。

Hyperbolicity of varieties of log general type

These notes provide an overview of various notions of hyperbolicity for varieties of log general type from the viewpoint of both arithmetic and birational geometry. The main results are based on our paper entitled "Hyperbolicity and uniformity of varieties of log general type." They are expanded notes from a minicourse the authors gave as part of the Geometry and arithmetic of orbifolds workshop at UQÁM.

阿贝尔簇上完全线性系统的小舟超曲率

Kobayashi hyperbolicity of complete linear systems on abelian varieties

研究阿贝尔簇上完全线性系统中非常一般元素的柯拜ashi超曲率条件，并提出相关猜想。

Hyperbolicity of varieties with big linear representation of $π_1$

We show the following algebraicity result for a complex projective variety $X$ with big representation of $π_1$ into a semi-simple algebraic group: There exists a proper subvariety $Z \subset X$ such that for any algebraic curve $C$, any holomorphic map $γ:\, C \to X$ with $γ(C) \not\subset Z$ is induced from an algebraic morphism. As an application, we prove pseudo-Brody hyperbolicity of certain varieties with big reductive representations of $π_1$.

余线性系列在具有正切包的变种上的超曲性

Hyperbolicity of adjoint linear series on varieties with positive tangent bundle

本文研究了在具有正切包的变种上，余线性系列的超曲性，证明了对于不同类别的极化对，当m足够大时，线性系统具有超曲性。

Algebraic Hyperbolicity for Surfaces in Toric Threefolds

Adapting focal loci techniques used by Chiantini and Lopez, we provide lower bounds on the genera of curves contained in very general surfaces in Gorenstein toric threefolds. We illustrate the utility of these bounds by obtaining results on algebraic hyperbolicity of very general surfaces in toric threefolds.

$\bar{M}_{0,n}$ 的对数代数双曲性

Log algebraic hyperbolicity of $\overline{M}_{0,n}$

本文证明了带边界 Δ 的稳定 n 点有理曲线模空间 $\bar{M}_{0,n}$ 具有代数双曲性。

拓扑、双曲性与复代数簇的沙法列夫猜想

Topology, Hyperbolicity, and the Shafarevich Conjecture for Complex Algebraic Varieties

本文探讨复代数簇的双曲性、拓扑结构及沙法列夫猜想等核心问题，通过非阿贝尔霍奇理论和基本群表示等方法，揭示这些领域间的深刻联系与最新进展。

加权射影空间中非常一般超曲面的代数双曲性

Algebraic hyperbolicity of very general hypersurfaces in weighted projective spaces

本文研究加权射影空间中非常一般超曲面的代数双曲性，通过给出丰沛线丛倍数 $m$ 的界，证明了相应零点轨迹具有代数双曲性。

Viehweg hyperbolicity for Whitney equisingular families with Gorenstein rational singularities

We prove the analogue of Viehweg's hyperbolicity conjecture for Whitney equisingular families of projective varieties with Gorenstein rational singularities whose geometric generic fiber has a good minimal model. Namely, for such families with maximal variation, the base spaces are of log general type. The main new ingredient is the use of intersection complexes as Hodge modules in the construction of logarithmic Higgs sheaves by Viehweg-Zuo and Popa-Schnell. This construction suggests an equisingular stratification of the moduli space of varieties of general type, with each stratum being hype…

Brody hyperbolicity of base spaces of certain families of varieties

We prove that quasi-projective base spaces of smooth families of minimal varieties of general type with maximal variation do not admit Zariski dense entire curves. We deduce the fact that moduli stacks of polarized varieties of this sort are Brody hyperbolic, answering a special case of a question of Viehweg and Zuo. For two-dimensional bases, we show analogous results in the more general case of families of varieties admitting a good minimal model.

Algebraic hyperbolicity of very general surfaces

Recently, Haase and Ilten initiated the study of classifying algebraically hyperbolic surfaces in toric threefolds. We complete this classification for $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$, $\mathbb{P}^2 \times \mathbb{P}^1$, $\mathbb{F}_e \times \mathbb{P}^1$ and the blowup of $\mathbb{P}^3$ at a point, augmenting our earlier work on $\mathbb{P}^3$. In the process, we codify several different techniques for proving algebraic hyperbolicity, allowing us to prove similar results for hypersurface in any variety admitting a group action with dense orbit.

Algebraic hyperbolicity of the very general quintic surface in $\mathbb{P}^3$

We prove that a curve of degree $dk$ on a very general surface of degree $d \geq 5$ in $\mathbb{P}^3$ has geometric genus at least $\frac{dk(d-5)+k}{2} + 1$. This improves bounds given by G. Xu. As a corollary, we conclude that the very general quintic surface in $\mathbb{P}^3$ is algebraically hyperbolic.

射影空间中非常一般超曲面的代数双曲性

Algebraic hyperbolicity of very general hypersurfaces in projective spaces

本文证明了射影空间中非常一般超曲面代数双曲性的最后未解情形，即四次三维簇，完成了分类，方法基于Coskun--Riedl构造及稳定映射模空间和对数几何。

Algebraic non-hyperbolicity of hyperkahler manifolds with Picard rank greater than one

A projective manifold is algebraically hyperbolic if the degree of any curve is bounded from above by its genus times a constant, which is independent from the curve. This is a property which follows from Kobayashi hyperbolicity. We prove that hyperkahler manifolds are non algebraically hyperbolic when the Picard rank is at least 3, or if the Picard rank is 2 and the SYZ conjecture on existence of Lagrangian fibrations is true. We also prove that if the automorphism group of a hyperkahler manifold is infinite then it is algebraically non-hyperbolic.

Arakelov Type Inequalities and Deformation Boundedness of polarized varieties

We give two kinds of generalizations of Arakelov type inequalities for higher dimensional families. These results give higher dimensional generalizations (in both fibers and bases) of the weakly boundedness in Paršin-Arakelov's reformulation of the geometric Shafarevich conjecture. As a consequence, we deduce the following results. Hyperbolicity: We give an alternative proof (using the theory of degeneration of Hodge structure) to the hyperbolicity (in Paršin-Arakelov's reformulation, i.e. Viehweg's hyperbolicity conjecture) of the base of a family with maximal variation whose general fibers a…

Hyperbolicity and uniformity of varieties of log general type

Projective varieties with ample cotangent bundle satisfy many notions of hyperbolicity, and one goal of this paper is to discuss generalizations to quasi-projective varieties. A major hurdle is that the naive generalization fails, i.e. the log cotangent bundle is never ample. Instead, we define a notion called almost ample which roughly asks that the log cotangent is as positive as possible. We show that all subvarieties of a quasi-projective variety with almost ample log cotangent bundle are of log general type. In addition, if one assumes globally generated then we obtain that such varieties…

Picard秩为2和3的光滑射影环面三维体中曲面的代数双曲性

Algebraic hyperbolicity for surfaces in smooth projective toric threefolds with Picard rank 2 and 3

本文研究Picard秩为2和3的光滑射影环面三维体中曲面的代数双曲性，利用组合描述与Haase和Ilten的方法，在这些簇中确定了众多代数双曲性曲面。

Algebraic hyperbolicity of very general hypersurfaces in products of projective spaces

We study the algebraic hyperbolicity of very general hypersurfaces in $\mathbb{P}^m \times \mathbb{P}^n$ by using three techniques that build on past work by Ein, Voisin, Pacienza, Coskun and Riedl, and others. As a result, we completely answer the question of whether or not a very general hypersurface of bidegree $(a,b)$ in $\mathbb{P}^m \times \mathbb{P}^n$ is algebraically hyperbolic, except in $\mathbb{P}^3 \times \mathbb{P}^1$ for the bidegrees $(a,b)= (7,3), (6,3)$ and $(5,b)$ with $b\geq 3.$ As another application of these techniques, we improve the known result that very general hypers…

Increasing hyperbolicity of varieties supporting a variation of Hodge structures with level structures

Looking at the finite étale congruence covers $X(p)$ of a complex algebraic variety $X$ equipped with a variation of integral polarized Hodge structures whose period map is quasi-finite, we show that both the minimal gonality among all curves contained in $X(p)$ and the minimal volume among all subvarieties of $X(p)$ tend to infinity with $p$. This applies for example to Shimura varieties, moduli spaces of curves, moduli spaces of abelian varieties, moduli spaces of Calabi-Yau varieties, and can be made effective in many cases. The proof goes roughly as follows. We first prove a generalization…

On the hyperbolicity of base spaces for maximally variational families of smooth projective varieties

For smooth families with maximal variation, whose general fibers have semi-ample canonical bundle, the generalized Viehweg hyperbolicity conjecture states that the base spaces of such families are of log general type. This deep conjecture was recently proved by Popa-Schnell using the theory of Hodge modules and a theorem by Campana-Păun. In this paper we prove that those base spaces are pseudo Kobayashi hyperbolic, as predicted by the Lang conjecture: any complex quasi-projective manifold is pseudo Kobayashi hyperbolic if it is of log general type. As a consequence, we prove the Brody hyperbol…

Non-archimedean hyperbolicity and applications

Inspired by the work of Cherry, we introduce and study a new notion of Brody hyperbolicity for rigid analytic varieties over a non-archimedean field $K$ of characteristic zero. We use this notion of hyperbolicity to show the following algebraic statement: if a projective variety admits a non-constant morphism from an abelian variety, then so does any specialization of it. As an application of this result, we show that the moduli space of abelian varieties is $K$-analytically Brody hyperbolic in equal characteristic zero. These two results are predicted by the Green-Griffiths-Lang conjecture on…

Nevanlinna Pair and Algebraic Hyperbolicity

We introduce the notion of the $\textit{Nevanlinna pair}$ for a pair $(X, D)$, where $X$ is a projective variety and $D$ is an effective Cartier divisor on $X$. This notion links and unifies the Nevanlinna theory, the complex hyperbolicity (Brody and Kobayashi hyperbolicity), the big Picard type extension theorem (more generally the Borel hyperbolicity), as well as the algebraic hyperbolicity. The key is to use the Nevanlinna theory on parabolic Riemann surfaces recently developed by Pǎun and Sibony.

全纯辛簇的非双曲性

Non-hyperbolicity of holomorphic symplectic varieties

本文证明了满足有理SYZ猜想的第二贝蒂数至少为5的本原辛簇的非双曲性，并进一步在第二贝蒂数至少为7时建立了Kobayashi伪度量恒为零的结果，从而完成了Kamenova-Lu-Verbitsky的工作。

The Lang-Vojta conjectures on projective pseudo-hyperbolic varieties

These notes grew out of a mini-course given from May 13th to May 17th at UQAM in Montreal during a workshop on Diophantine Approximation and Value Distribution Theory. We start with an overview of Lang-Vojta's conjectures on pseudo-hyperbolic projective varieties. These conjectures relate various different notions of hyperbolicity. We begin with Brody hyperbolicity and discuss conjecturally related notions of hyperbolicity in arithmetic geometry and algebraic geometry in subsequent sections. We slowly work our way towards the most general version of Lang-Vojta's conjectures and provide a summa…

On Algebraic Hyperbolicity of Log Surfaces

We call a log variety (X, D) algebraically hyperbolic if there exists a positive number e such that 2g(C) - 2 + i(C, D) >= e deg(C) for all curves C on X, where i(C, D) is the number of the intersections between D and the normalization of C. Among other things, we proved that (P^2, D) is algebraically hyperbolic for a very general curve D of degree at least 5. More specifically, we showed that 2g(C) - 2 + i(C, D) >= (deg(D) - 4) deg C for all curves C on P^2. For example, fix a very general quintic curve D and then for any map f: C = P^1 --> P^2, there are at least deg(f) + 2 distinct points o…

大皮卡定理与代数超几何性对于允许Hodge结构变化的概形

Big Picard theorems and algebraic hyperbolicity for varieties admitting a variation of Hodge structures

本文研究了允许Hodge结构变化的拟紧Kähler流形的超几何性质，证明了其代数超几何性及大皮卡定理，并展示了有限étale覆盖下的超几何性结果。

存在大局部系统的背景下超曲面性

Hyperbolicity in presence of a large local system

本文证明了允许大复局部系统的射影复代数簇满足Green-Griffiths-Lang猜想的强版本。

单连通性与双曲性

Simply connectedness and hyperbolicity

该研究将单连通弱特殊而非特殊的簇推广到任意维，验证其满足Campana猜想的相关部分，并首次给出任意维下满足函数域Lang猜想的光滑单连通非迷向一般型射影簇例子。

双曲性与复准项目ive簇的基本群（二）：通过非阿贝尔霍奇理论

Hyperbolicity and fundamental groups of complex quasi-projective varieties (II): via non-abelian Hodge theories

本文通过非阿贝尔霍奇理论研究复准项目ive簇的双曲性，证明了在特定表示下，非双曲性位置的性质，并扩展了前人关于项目ive簇的结果。

射影空间中一般超曲面补集的代数双曲性

Algebraic Hyperbolicity of Complements of Generic Hypersurfaces in Projective Spaces

本文证明了射影空间中非常一般的 $2n$ 次超曲面补集满足代数 Green-Griffiths-Lang 猜想，并完全刻画了四次平面曲线补集的例外轨迹。

双曲性与复准项目ive簇的基本群（III）：应用

Hyperbolicity and fundamental groups of complex quasi-projective varieties (III): applications

本文研究了复准项目ive簇的双曲性和基本群性质，证明了特殊和$h$-特殊簇的基本群表示的虚nilpotent性，并修正了Campana的阿贝尔猜想，同时解决了Kollár的猜想。

Arithmetic hyperbolicity: automorphisms and persistence

We show that if the automorphism group of a projective variety is torsion, then it is finite. Motivated by Lang's conjecture on rational points of hyperbolic varieties, we use this to prove that a projective variety with only finitely many rational points has only finitely many automorphisms. Moreover, we investigate to what extent finiteness of $S$-integral points on a variety over a number field persists over finitely generated fields. To this end, we introduce the class of mildly bounded varieties and prove a general criterion for proving this persistence.

Algebraic intermediate hyperbolicities

We extend Lang's conjectures to the setting of intermediate hyperbolicity and prove two new results motivated by these conjectures. More precisely, we first extend the notion of algebraic hyperbolicity (originally introduced by Demailly) to the setting of intermediate hyperbolicity and show that this property holds if the appropriate exterior power of the cotangent bundle is ample. Then, we prove that this intermediate algebraic hyperbolicity implies the finiteness of the group of birational automorphisms and of the set of surjective maps from a given projective variety. Our work answers the a…

关于超凯勒簇的反双曲性

On anti-hyperbolicity for hyperkähler varieties

本文研究超凯勒流形被同维数复仿射空间全纯或亚纯支配的反双曲性判据与例子，推广了 K3 曲面的相关已知结果。

Hyperbolicity of the complement of plane algebraic curves

The paper is a contribution of the conjecture of Kobayashi that the complement of a generic plain curve of degree at least five is hyperbolic. The main result is that the complement of a generic configuration of three quadrics is hyperbolic and hyperbolically embedded as well as the complement of two quadrics and a line.

关于伴随丛的一个双曲性猜想

A hyperbolicity conjecture for adjoint bundles

本文提出了关于伴随丛的双曲性猜想，并在光滑射影环面簇情形下证实了该猜想，同时证明了 Gorenstein 环面簇的伪双曲性及三维簇的双曲性。

关于实曲线的双曲性轨迹

On the hyperbolicity locus of a real curve

本文研究射影3维空间中实代数曲线的双曲性轨迹，并构造光滑不可约曲线，说明其轨迹可不连通且各分量不能由环绕数区分。

复拟射影簇的双曲性与基本群（I）：由 Nevanlinna 理论给出的极大拟 Albanese 维数

Hyperbolicity and fundamental groups of complex quasi-projective varieties (I): Maximal quasi-Albanese dimension by Nevanlinna theory

本文利用 Nevanlinna 理论，对定义在穿孔圆盘小分歧覆盖上、取值于极大拟 Albanese 维数且为对数一般型的复拟射影簇的全纯映射建立大 Picard 型定理，并由此证明广义 Green-Griffiths-Lang 猜想。

代数线映射与对称幂的双曲性

Algebroid maps and hyperbolicity of symmetric powers

本文定义了复射影代数簇对称幂双曲性的不变量$h(X)$，利用Nevanlinna理论给出其非平凡下界，并探讨了与数域上有界度代数点有限性的关联及具体应用。

Hyperbolicity for log smooth families with maximal variation

We prove that the base space of a log smooth family of log canonical pairs of log general type is of log general type as well as algebraically degenerate, when the family admits a relative good minimal model over a Zariski open subset of the base and the relative log canonical model is of maximal variation.

Non-Degeneracy of Kobayashi Volume Measures for Singular Directed Varieties

In this note, we prove the generic Kobayashi volume measure hyperbolicity of singular directed varieties $(X, V)$, as soon as the canonical sheaf $\mathcal{K}\_V$ of $V$ is big in the sense of Demailly.

Demailly's notion of algebraic hyperbolicity: geometricity, boundedness, moduli of maps

Demailly's conjecture, which is a consequence of the Green-Griffiths-Lang conjecture on varieties of general type, states that an algebraically hyperbolic complex projective variety is Kobayashi hyperbolic. Our aim is to provide evidence for Demailly's conjecture by verifying several predictions it makes. We first define what an algebraically hyperbolic projective variety is, extending Demailly's definition to (not necessarily smooth) projective varieties over an arbitrary algebraically closed field of characteristic zero, and we prove that this property is stable under extensions of algebraic…

族中的有界性及其在算术双曲性中的应用

Boundedness in families with applications to arithmetic hyperbolicity

受多位学者猜想启发，研究族中双曲性相关概念的一致性，证明非零不规则射影正规曲面的算术双曲性在域扩张下保持，并提出推广Demailly概念的伪代数双曲性。

Orbifold hyperbolicity

We define and study jet bundles in the geometric orbifold category. We show that the usual arguments from the compact and the logarithmic settings do not all extend to this more general framework. This is illustrated by simple examples of orbifold pairs of general type that do not admit any global jet differential, even if some of these examples satisfy the Green-Griffiths-Lang conjecture. This contrasts with an important result of Demailly (2010) proving that compact varieties of general type always admit jet differentials. We illustrate the usefulness of the study of orbifold jets by establi…

Arithmetic hyperbolicity and a stacky Chevalley-Weil theorem

We prove an analogue for algebraic stacks of Hermite-Minkowski's finiteness theorem from algebraic number theory, and establish a Chevalley-Weil type theorem for integral points on stacks. As an application of our results, we prove analogues of the Shafarevich conjecture for some surfaces of general type.

Explicit computation of symmetric differentials and its application to quasi-hyperbolicity

We develop explicit techniques to investigate algebraic quasi-hyperbolicity of singular surfaces through the constraints imposed by symmetric differentials. We apply these methods to prove that rational curves on Barth's sextic surface, apart from some well-known ones, must pass through at least four singularities, and that genus 1 curves must pass through at least two. On the surface classifying perfect cuboids, our methods show that rational curves, again apart from some well-known ones, must pass through at least seven singularities, and that genus 1 curves must pass through at least two. W…

Algebraicity of analytic maps to a hyperbolic variety

Let $X$ be an algebraic variety over $\mathbb{C}$. We say that $X$ is Borel hyperbolic if, for every finite type reduced scheme $S$ over $\mathbb{C}$, every holomorphic map $S^{an}\to X^{an}$ is algebraic. We use a transcendental specialization technique to prove that $X$ is Borel hyperbolic if and only if, for every smooth affine curve $C$ over $\mathbb{C}$, every holomorphic map $C^{an}\to X^{an}$ is algebraic. We use the latter result to prove that Borel hyperbolicity shares many common features with other notions of hyperbolicity such as Kobayashi hyperbolicity.

Logarithmic Surfaces and Hyperbolicity

In 1981 J.Noguchi proved that in a logarithmic algebraic manifold, having logarithmic irregularity strictly bigger than its dimension, any entire curve is algebraically degenerate. In the present paper we are interested in the case of manifolds having logarithmic irregularity equal to its dimension. We restrict our attention to Brody curves, for which we resolve the problem completely in dimension 2: Theorem: In a logarithmic surface with logarithmic irregularity 2 and logarithmic Kodaira dimension 2, any Brody curve is algebraically degenerate. We also deal with the case of arbitrary logarith…