Kenneth Ascher, Amos Turchet
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.
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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…
This is an expository article on the recent developments of Hodge theory on moduli spaces of smooth projective varieties with semi-ample canonical line bundles.
On Algebraic Hyperbolicity of Log Varieties
在P^n中固定一个次数至少为2n + 1的非常一般的超曲面D，我们证明补P^n-D不包含任何代数环面C^*。
The moduli stacks of Calabi-Yau varieties are known to enjoy several hyperbolicity properties. The best results have so far been proven using sophisticated analytic tools such as complex Hodge theory. Although the situation is very different in positive characteristic (e.g. the moduli stack of principally polarized abelian varieties of dimension at least 2 contains rational curves), we explain in this note how one can prove many hyperbolicity results by reduction to positive characteristic, relying ultimately on the nonabelian Hodge theory in positive characteristic developed by Ogus and Volog…
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…
We show that all subvarieties of a quotient of a bounded symmetric domain by a sufficiently small arithmetic discrete group of automorphisms are of general type. This result corresponds through the Green-Griffiths-Lang's conjecture to a well-known result of Nadel.
Algebraic hyperbolicity of very general hypersurfaces in homogeneous varieties
本文推广了Coskun、Riedl和Yeong的技巧，得到了有理齐次簇中非常一般超曲面的代数双曲性的几乎最优次数界，并给出了格拉斯曼簇、正交/辛格拉斯曼簇及旗簇等实例。
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…
Hyperbolicity, automorphic forms and Siegel modular varieties
利用算术群研究了有界对称整环的直积的紧化的双曲性。我们证明，直到一个tale覆盖，他们是小林双曲模的边界。将我们的技巧应用于Siegel模簇，改进了Nadel关于复函数域上阿贝尔簇不存在某种水平结构的一些结果.
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.
Firstly, we pursue the work of W. Cherry on the analogue of the Kobayashi semi distance dCK that he introduced for analytic spaces defined over a non-Archimedean metrized field k. We prove various characterizations of smooth projective varieties for which dCK is an actual distance. Secondly, we explore several notions of hyperbolicity for a smooth algebraic curve X defined over k. We prove a non-Archimedean analogue of the equivalence between having negative Euler characteristic and the normality of certain families of analytic maps taking values in X.
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。
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 $π_1$.
Hyperbolicity Related Problems for Complete Intersection Varieties
本文研究了复射影空间中关于高次完全交簇的不同问题。首先，我们展示了如何从超曲面的已知结果中推导出高重度和高余维的一般完全交的双曲性。然后我们证明了喷流微分的存在性定理，推广了S。迪韦里奥最后，基于O. Debarre，重点讨论了完备交的余切丛的正性，并证明了关于这一猜想的一些结果，其中证明了至少四维的射影空间中的高重度一般完备交曲面有充足的余切丛.
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.
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。
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。
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。
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.
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。
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。
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。
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。
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…
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。
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-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。
By implementing jet differential techniques in non-archimedean geometry, we obtain a big Picard type extension theorem, which generalizes a previous result of Cherry and Ru. As applications, we establish two hyperbolicity-related results. Firstly, we prove a non-archimedean Ax-Lindemann theorem for totally degenerate abelian varieties. Secondly, we show the pseudo-Borel hyperbolicity for subvarieties of general type in abelian varieties.
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…
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。
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 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。
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。
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 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。
About the hyperbolicity of complete intersections
本文是2011年9月12日至9月17日在博洛尼亚举行的“第十九届意大利数学联盟大会”上的一个30分钟演讲的加长版。这基本上是一个调查谈论小林双曲性质和积极性质之间的联系典范丛的射影代数簇。
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。
Xeric varieties
本文针对数域上光滑射影簇，引入“xeric”概念（有理点在所有有限扩张上均稀疏），并系统研究其与无有理曲线及p-adic双曲性的联系，回应Green-Griffiths-Lang猜想。
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 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…
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.
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.
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。
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。
We discuss the cone theorem for quasi-log schemes and the Mori hyperbolicity. In particular, we establish that the log canonical divisor of a Mori hyperbolic projective normal pair is nef if it is nef when restricted to the non-lc locus. This answers Svaldi's question completely. We also treat the uniruledness of the degenerate locus of an extremal contraction morphism for quasi-log schemes. Furthermore, we prove that every fiber of a relative quasi-log Fano scheme is rationally chain connected modulo the non-qlc locus.
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…
We prove several statements about arithmetic hyperbolicity of certain blow-up varieties. As a corollary we obtain multiple examples of simply connected quasi-projective varieties that are pseudo-arithmetically hyperbolic. This generalizes results of Corvaja and Zannier obtained in dimension 2 to arbitrary dimension. The key input is an application of the Ru-Vojta's strategy. We also obtain the analogue results for function fields and Nevanlinna theory with the goal to apply them in a future paper in the context of Campana's conjectures.
Families of varieties of general type over compact bases
设f：X -> Y是光滑基上正则极化复簇的光滑族. Viehweg推广了经典的Shafarevich双曲性猜想，证明了如果族具有最大变差，则Y必然是对数一般型的。作者在arXiv：math/0511378中证明了维韦格猜想的一个更强、更精确的版本，其中Y是一个准射影曲面。假设极小模型程序成立，这篇很短的论文证明了任意维数的射影基流形Y的相同结果。
Hyperbolicity and fundamental groups of complex quasi-projective varieties
本文利用基本群的线性表示研究复拟射影簇的双曲性，证明整曲线轨迹、广义 Green–Griffiths–Lang 猜想及特殊簇表示群的幂零性等结果。
In this paper, we prove that the quasi-projective base of any maximally variational smooth family of Calabi-Yau klt pairs is both of log general type, and pseudo Kobayashi hyperbolic. Moreover, such a base is Brody hyperbolic if the family is effectively parametrized.
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。
Logarithmic Jets and Hyperbolicity
我们证明了一个非常通用的曲线的程度$d$至少等于15在射影平面的补充是双曲的意义上的小林（这里，术语“非常通用”是指补充可数工会的参数空间的真代数子集）。我们首先考虑Dethloff和陆的推广Demailly的射流束的对数情况。我们研究他们的基地轨迹表面的日志一般类型的方式一样，它是在紧凑的情况下，Demailly和埃尔古尔。在log-Chern类上满足一定条件的情况下，曲面上的全纯映射可以提升为分歧覆盖上的叶理的叶子。然后我们得到了McQuillan关于全纯叶理的结果的一个对数模拟，从而可以得出结论。使用对数形式主义，我们甚至得到了一些简化的原始证明McQuillan的工作。
On Algebraic Hyperbolicity of Log Surfaces
我们称对数簇（X，D）是代数双曲的，如果存在一个正数e使得对X上的所有曲线C，2g（C）- 2 + i（C，D）>= e deg（C），其中i（C，D）是D与C的正规化的交点的个数.除此之外，我们还证明了对于次数至少为5的非常一般的曲线D，（P^2，D）是代数双曲的。更具体地说，我们证明了对于P^2上的所有曲线C，2g（C）- 2 + i（C，D）>=（deg（D）- 4）deg C。例如，固定一条非常一般的五次曲线D，然后对于任何映射f：C = P^1 --> P^2，C上至少有deg（f）+ 2个不同的点通过f映射到D上的点。
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。
We study (weak) log abelian varieties with constant degeneration in the log flat topology. If the base is a log point, we further study the endomorphism algebras of log abelian varieties. In particular, we prove the dual short exact sequence for isogenies, Poincaré complete reducibility theorem for log abelian varieties, and the semi-simplicity of the endomorphism algebras of log abelian varieties.
Proof of the Kobayashi conjecture on the hyperbolicity of very general hypersurfaces
Green-Griffiths-Lang猜想规定：对于${\mathbb C}$上的任何一般型投射簇$X$，存在$X$的一个真代数子簇，它包含所有非常数整曲线$f:{\mathbb C}\to X$ .使用的形式主义的有向品种，我们证明了在这里，这一论断成立的情况下$X$满足一个强的一般类型的条件，是有关的切丛$T\_X$的一定的喷射半稳定性。然后我们用这个事实来证实Kobayashi（1970）的一个长期存在的猜想，根据这个猜想，复射影空间${\mathbb P}^{n+1}$中维数为$n$且次数至少为$2n+2$的非常一般的代数超曲面是双曲的。
We relate the geometry of curves to the notion of hyperbolicity in real algebraic geometry. A hyperbolic variety is a real algebraic variety that (in particular) admits a real fibered morphism to a projective space whose dimension is equal to the dimension of the variety. We study hyperbolic varieties with a special interest in the case of hypersurfaces that admit a real algebraic ruling. The central part of the paper is concerned with secant varieties of real algebraic curves where the real locus has the maximal number of connected components, which is determined by the genus of the curve. Fo…
Moduli spaces of varieties of general type are naturally of log general type
该研究将Popa-Schnell的Viehweg双曲性成果推广到光滑Deligne-Mumford栈，证明参数化极大变差一般型代数簇族的光滑真DM栈的自然对数典范丛丰沛，并将结果应用于标准模空间。
On the logarithmic Kobayashi conjecture
我们研究了对数簇$(\mathbb{P}^n, X)$的双曲性，其中$X$是一个非常一般的度为$d\geq 2n+1$的超曲面（这是Kobayashi猜想所预言的界）.利用一个关于（扭曲的）对数向量场层的正性结果，证明了$(\mathbb{P}^n, X)$的任何对数-子簇都是对数-一般型的，给出了$(\mathbb{P}^n, X)$的代数双曲性的一个新证明，并排除了泛度$d$超曲面的补曲面中整曲线的最大秩族的存在性.此外，我们证明了，在紧的情况下，代数双曲的对数簇是一个必要条件的度量。
Given a log canonical pair $(X, Δ)$, we show that $K_X+Δ$ is nef assuming there is no non-constant map from the affine line with values in the open strata of the stratification induced by the non-klt locus of $(X, Δ)$. This implies a generalization of the Cone Theorem where each $K_X+Δ$-negative extremal ray is spanned by a rational curve that is the closure of a copy of the affine line contained in one of the open strata of $\mathrm{Nklt}(X, Δ)$. Moreover, we give a criterion of Nakai type to determine when under the above condition $K_X+Δ$ is ample and we prove some partial results in the ca…
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