The arithmetic and the geometry of Kobayashi hyperbolicity

Applications of computational invariant theory to Kobayashi hyperbolicity and to Green-Griffiths algebraic degeneracy

A major unsolved problem (according to Demailly 1997) towards the Kobayashi hyperbolicity conjecture in optimal degree is to understand jet differentials of germs of holomorphic discs that are invariant under any reparametrization of the source. The underlying group action is not reductive, but we provide a complete algorithm to generate all invariants, in arbitrary dimension n and for jets of arbitrary order k. Two main new situations are studied in great details. For jets of order 4 in dimension 4, we establish that the algebra of Demailly-Semple invariants is generated by 2835 polynomials, …

Effectivity in the Hyperbolicity-related problems

The aim of this work is to deal with effective questions related to the Kobayashi and Debarre conjectures, and based on the work of Damian Brotbek and Lionel Darondeau. We first show that if a line bundle $L$ generates $k$-jets, the $k$-th Wronskian ideal sheaf associated with global sections of $L$ reaches its largest possible value. Then we obtain an effective estimate for a weaker version of a theorem of Nakamaye on certain universal Grassmannians. We finally derive from this explicit effective bounds for the Kobayashi and Debarre conjectures.

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.

Kobayashi non-hyperbolicity of Calabi-Yau manifolds via mirror symmetry

A compact complex manifold is Kobayashi non-hyperbolic if there exists an entire curve on it. Using mirror symmetry we establish that there are (possibly singular) elliptic or rational curves on any Calabi-Yau manifold $X$, whose mirror dual $\check X$ exists and is not "Hodge degenerate", therefore proving that $X$ is Kobayashi non-hyperbolic. We are not aware of any higher dimensional simply connected Calabi-Yau manifolds that satisfy the "Hodge degenerate" condition.

一般曲面的Kobayashi双曲性：基于Poincaré问题

Kobayashi Hyperbolicity of General Surfaces via the Poincaré Problem

通过Poincaré问题与2-jet微分叶状结构，证明$\mathbb{P}^3$中次数至少18的一般曲面无有理或椭圆曲线，进而建立其Kobayashi双曲性，并推广到平面曲线补集。

Kobayashi hyperbolicity in degree > n^{2n}

For a generic hypersurface $\mathbb{X}^{n-1} \subset \mathbb{P}^n(\mathbb{C})$ of degree \[ d \,\geqslant\, n^{2n} \] (1) $\mathbb{P}^n \big\backslash \mathbb{X}^{n-1}$ is Kobayashi-hyperbolically imbedded in $\mathbb{P}^n$; (2) $\mathbb{X}^{n-1}$ is Kobayashi($\Leftrightarrow$ Brody)-hyperbolic. (1) improves Brotbek-Deng 1804.01719: $d \geqslant (n+2)^{n+3}\, (n+1)^{n+3} = n^{2n}\,n^6\, \big(e^3+{\rm O}(\frac{1}{n}) \big)$. (2) supersedes Demailly 1801.04765: $d \geqslant \frac{1}{3}\, \big( e^1(n-1) \big)^{2n} = n^{2n}\, e^{2n}\, \big( \frac{1}{3\, e^2} + {\rm O} (\frac{1}{n}) \big)$. The me…

Hyperbolicity of coarse moduli spaces and isotriviality for certain families

In this paper, we prove the Kobayashi hyperbolicity of the coarse moduli spaces of canonically polarized or polarized Calabi-Yau manifolds in the sense of complex $V$-spaces (a generalization of complex $V$-manifolds in the sense of Satake). As an application, we prove the following hyperbolic version of Campana's isotriviality conjecture: for the smooth family of canonically polarized or polarized Calabi-Yau manifolds, when the Kobayashi pseudo-distance of the base vanishes identically, the family must be isotrivial, that is, any two fibers are isomorphic. We also prove that for the smooth pr…

Kobayashi hyperbolicity, negativity of the curvature and positivity of the canonical bundle

This survey article mainly addresses to graduate students and young researchers in complex geometry willing to enter the beautiful word of connections between curvature and Kobayashi hyperbolicity. It is a detailed account of a recent breakthrough by Wu and Yau, shortly after generalized by Tosatti and Yang (and others), which sits on the crossroad between complex differential geometry and Kobayashi hyperbolicity. More specifically, an old conjecture by Kobayashi, stated at the very beginning of the theory, predicts that a compact hyperbolic manifold should have ample canonical bundle. Now, on…

Characterization of cycle domains via Kobayashi hyperbolicity

A real form $G$ of a complex semisimple Lie group $G^C$ has only finitely many orbits in any given $G^C$-flag manifold $Z=G^C/Q$. The complex geometry of these orbits is of interest, e.g., for the associated representation theory. The open orbits $D$ generally possess only the constant holomorphic functions, and the relevant associated geometric objects are certain positive-dimensional compact complex submanifolds of $D$ which, with very few well-understood exceptions, are parameterized by the Wolf cycle domains $Ω_W(D)$ in $G^C/K^C$, where $K$ is a maximal compact subgroup of $G$. Thus, for t…

About the hyperbolicity of complete intersections

This note is an extended version of a thirty minutes talk given at the "XIX Congresso dell'Unione Matematica Italiana", held in Bologna from September 12th to September 17th, 2011. This was essentially a survey talk about connections between Kobayashi hyperbolicity properties and positivity properties of the canonical bundle of projective algebraic varieties.

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 general deformations

We present two methods of constructing low degree Kobayashi hyperbolic hypersurfaces in the projective space: the projection method and the deformation method. The talk is based on joint works of the speaker with B. Shiffman and C. Ciliberto.

通过数值不变量研究复流形的定量双曲性

Quantitative hyperbolicity for complex manifolds via numerical invariants

研究通过引入双曲指标这一数值不变量来研究复流形的双曲性，结合喷流微分和密度流方法，证明了\(\mathbb{P}^{n + 1}\)中一般超曲面\(X_d\)双曲指标随\(d\)至少线性增长趋于无穷，并讨论了小林猜想的解析方法。

A survey on hyperbolicity of projective hypersurfaces

These are lecture notes of a course held at IMPA, Rio de Janiero, in september 2010: the purpose was to present recent results on Kobayashi hyperbolicity in complex geometry. Our ultimate goal is to describe the results obtained on questions related to the geometry of entire curves traced in generic complex projective hypersurfaces of high degree. For the convenience of the reader, this survey tries to be as self contained as possible.

Hyperbolicity of bases of log Calabi-Yau families

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.

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。

Complex-analytic intermediate hyperbolicity, and finiteness properties

Motivated by the finiteness of the set of automorphisms Aut(X) of a projective manifold X, and by Kobayashi-Ochiai's conjecture that a projective manifold dim(X)-analytically hyperbolic (also known as strongly measure hyperbolic) should be of general type, we investigate the finiteness properties of Aut(X) for a complex manifold satisfying a (pseudo-) intermediate hyperbolicity property. We first show that a complex manifold X which is (dim(X) -- 1)-analytically hyperbolic has indeed finite automorphisms group. We then obtain a similar statement for a pseudo-(dim(X) -- 1)-analytically hyperbol…

双曲性性质的标记超kähler流形模空间

Hyperbolicity properties of moduli spaces of marked hyperk{ä}hler manifolds

研究标记超kähler流形模空间的双曲性性质，证明Kobayashi伪距离消失，并推广Greb和Schwald的定理到任意维数和非紧曲线情况。

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。

非约化几何不变量理论与双曲性

Non-reductive geometric invariant theory and hyperbolicity

本文利用非约化几何不变量理论商的相交理论及 Riedl 和 Yang 的工作，证明了一般超曲面的 Green--Griffiths--Lang 和 Kobayashi 双曲性猜想。

Algebraically hyperbolic manifolds have finite automorphism groups

A projective manifold $M$ is algebraically hyperbolic if there exists a positive constant $A$ such that the degree of any curve of genus $g$ on $M$ is bounded from above by $A(g-1)$. A classical result is that Kobayashi hyperbolicity implies algebraic hyperbolicity. It is known that Kobayashi hyperbolic manifolds have finite automorphism groups. Here we prove that, more generally, algebraically hyperbolic projective manifolds have finite automorphism groups.

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.

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.

Varietes faiblement speciales a courbes entieres degenerees

We show that certain non-special but weakly special threefolds $X$ constructed by Bogomolov-Tschinkel enjoy strong complex hyperbolicity properties: their entire curves are algebraically degenerate and lie either on a fixed divisor or on the fiber of the unique elliptic fibration on $X$. These properties are consistent with the conjectural link between hyperbolicity and arithmetics of projective manifolds. The arithmetic counterpart of this hyperbolicity property (the non-potential density of $X$, object of two conflicting conjectures) remains however open.

On the hyperbolicity of general hypersurfaces

In 1970, Kobayashi conjectured that general hypersurfaces of sufficiently large degree in $P^n$ are hyperbolic. In this paper we prove that a general sufficiently ample hypersurface in a smooth projective variety is hyperbolic. To prove this statement, we construct hypersurfaces satisfying a property which is Zariski open and which implies hyperbolicity. These hypersurfaces are chosen such that the geometry of their higher order jet spaces can be related to the geometry of a universal family of complete intersections. To do so, we introduce a Wronskian construction which associates a (twisted)…

Demailly-Semple jets of orders 4 and 5 in dimension 2

In view of Kobayashi's hyperbolicity conjecture, Demailly-Semple jets of orders 4 and 5 in dimension 2 are studied, some expectations about their algebraic tameness being (dis)proved, after systematic, substantial, formal, manual calculations.

Hyperbolic surfaces in ${\bf P}^3$: examples

This is a recent conference report on the Kobayashi Problem on hyperbolicity of generic projective hypersurfaces. As an appendix, a (non-updated) author's survey article of 1992 on the same subject, published in an edition with a limited distribution, is added.

Hyperbolicity of Generic Surfaces of High Degree in Projective 3-Space

The main goal of this work is to prove that a very generic surface of degree at least 21 in complex projective 3-dimensional space is hyperbolic in the sense of Kobayashi. This means that every entire holomorphic map $f:{\Bbb C} \to X$ to the surface is constant. In 1970, Kobayashi conjectured more generally that a (very) generic hypersurface of sufficiently high degree in projective space is hyperbolic (here, the terminology "very generic" refers to complements of countable unions of proper algebraic subsets). Our technique follows the stream of ideas initiated by Green and Griffiths in 1979,…

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.

非Kähler超曲率变形及退化平衡流形的修改

Deformations of Non-Kähler Hyperbolicity Notions and Modifications of Degenerate Balanced Manifolds

本文提出两种新的非Kähler流形超曲率概念，并研究其在复变形下的行为及与其他超曲率概念的关系。

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.

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.

A strong hyperbolicity property of locally symmetric varieties

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.

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.

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。

On the Hyperbolicity of the Complements of Curves in Algebraic Surfaces: The Three Component Case

The paper is a contribution to the conjecture of Kobayashi that the complement of a generic curve in the projective plane is hyperbolic, provided the degree is at least five. Previously the authors treated the cases of two quadrics and a line and three quadrics. The main results are Let C be the union of three curves in P_2 whose degrees are at least two, one of which is at least three. Then for generic such configurations the complement of C is hyperbolic and hyperbolically embedded. The same statement holds for complements of curves in generic hypersurfaces X of degree at least five and curv…

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…

On the degeneracy of integral points and entire curves in the complement of nef effective divisors

As a consequence of our recently established generalized Schmidt's subspace theorem for closed subschemes in general position, we prove a degeneracy theorem for integral points on the complement of a union of nef effective divisors. A novel aspect of our result is the attainment of a strong degeneracy conclusion (arithmetic quasi-hyperbolicity) under weak positivity assumptions on the divisors. The proof hinges on applying our recent theorem with a well-situated ample divisor realizing a certain lexicographical minimax. We also explore the connections with earlier work by other authors and mak…

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…

双曲嵌入在托里几何中的应用：有效性与变形稳定性

Hyperbolic Embeddings in Toric Geometry: Effectivity and Deformation Stability

本文研究了托里几何中双曲嵌入的变形行为，通过有效方法分析例外集并证明其变形稳定性。