Weak analytic hyperbolicity of generic hypersurfaces of high degree in the complex projective space of dimension 4

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。

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 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…

Pseudo Kobayashi hyperbolicity of base spaces of families of minimal projective manifolds with maximal variation

In this paper we prove that every quasi-projective base space $V$ of smooth family of minimal projective manifolds with maximal variation is pseudo Kobayashi hyperbolic, i.e. $V$ is Kobayashi hyperbolic modulo a proper subvariety $Z\subsetneq V$. In particular, $V$ is algebraically degenerate, that is, every nonconstant entire curve $f:\mathbb{C}\to V$ has image $f(\mathbb{C})$ which lies in that proper subvariety $Z\subsetneq V$. As a direct consequence, we prove the Brody hyperbolicity of moduli spaces of minimal projective manifolds, which answers a question by Viehweg-Zuo in 2003.

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…

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。

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…

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.

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。

利用 Green-Griffiths 射流微分研究多项式次数一般超曲面的双曲性

Hyperbolicity of generic hypersurfaces of polynomial degree via Green-Griffiths jet differentials

本文改进 Bérczi-Kirwan 的结果，基于 Siu 斜向量场、全纯 Morse 不等式及经典 Green-Griffiths 射流空间，证明一般射影超曲面在次数满足多项式下界时满足 Kobayashi 与 Green-Griffiths-Lang 猜想。

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。

Hyperbolicity of cyclic covers and complements

We prove that a cyclic cover of a smooth complex projective variety is Brody hyperbolic if its branch divisor is a generic small deformation of a large enough multiple of a Brody hyperbolic base-point-free ample divisor. We also show the hyperbolicity of complements of those branch divisors. As an application, we find new examples of Brody hyperbolic hypersurfaces in $\mathbb{P}^{n+1}$ that are cyclic covers of $\mathbb{P}^n$.

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.

On the logarithmic Kobayashi conjecture

We study the hyperbolicity of the log variety $(\mathbb{P}^n, X)$, where $X$ is a very general hypersurface of degree $d\geq 2n+1$ (which is the bound predicted by the Kobayashi conjecture). Using a positivity result for the sheaf of (twisted) logarithmic vector fields, which may be of independent interest, we show that any log-subvariety of $(\mathbb{P}^n, X)$ is of log-general type, give a new proof of the algebraic hyperbolicity of $(\mathbb{P}^n, X)$, and exclude the existence of maximal rank families of entire curves in the complement of the universal degree $d$ hypersurface. Moreover, we…

On the Brody hyperbolicity of moduli spaces for canonically polarized manifolds

Consider the moduli functor of canonically polarized complex manifolds with Hilbert polynomial h, and let M_h be the corresponding coarse quasi-projective moduli scheme. We show that M_h is Brody hyperbolic in the following sense: Assume that for some quasi-projective variety U there exists a morphism U --> M_h, quasi-finite over its image, which is induced by a family f: V --> U, belonging to the moduli problem. Then all holomorphic maps from the complex plane to U are constant.

On Algebraic Hyperbolicity of Log Varieties

Fix a very general hypersurface D in P^n of degree at least 2n + 1 and we show that the complement P^n - D does not contain any algebraic torus C^*.

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。

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。

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。

Examples of hyperbolic hypersurfaces of low degree in projective spaces

We construct families of hyperbolic hypersurfaces of degree $2n$ in the projective space $\mathbb{P}^n(\mathbb{C})$ for $3 \leq n \leq 6$.

Nondeformability of entire curves in projective hypersurfaces of high degree

In this article, we prove that there does not exist a family of entire curves in the universal family of hypersurfaces of degree $d\geq 2n$ in the complex projective space ${\mathbb P}^n$. This can be seen as a weak version of the Kobayashi conjecture asserting that a general projective hypersurface of high degree is hyperbolic in the sense of Kobayashi.

Hyperbolicity and Specialness of Symmetric Powers

Inspired by the computation of the Kodaira dimension of symmetric powers Xm of a complex projective variety X of dimension n $\ge$ 2 by Arapura and Archava, we study their analytic and algebraic hyperbolic properties. First we show that Xm is special if and only if X is special (except when the core of X is a curve). Then we construct dense entire curves in (suf-ficiently hig) symmetric powers of K3 surfaces and product of curves. We also give a criterion based on the positivity of jet differentials bundles that implies pseudo-hyperbolicity of symmetric powers. As an application, we obtain the…

Low pole order frames on vertical jets of the universal hypersurface

Of the two techniques introduced by Bloch, Green-Griffiths and developed by Siu, Demailly to establish Kobayashi hyperbolicity of generic high degree complex algebraic hypersurfaces X in P^(n+1), the second one, initiated by Clemens, Ein, Voisin and developed by Siu, Paun, Rousseau consists in constructing meromorphic frames on the space of the so-called vertical k-jets J_vert^k (X_univ) of the universal hypersurface X_univ parametrizing all X in P^(n+1) of degree d. In 2004, Siu announced that there exists a constant c_n such that the twisting of the tangent bundle to J_vert^n (X_univ) by O (…

Logarithmic vector fields and hyperbolicity

In this article we prove that the complement of a very generic curve of degree at least equal to 14 in the complex projective plane is hyperbolic in the sense of Kobayashi. Thus, using a new method, we improve the former known bound obtained by El Goul. We also improve the case of a very generic curve with two components.

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 $π_1$.

Vanishing homology of projective hypersurfaces with 1-dimensional singularities

We introduce and study the vanishing homology of singular projective hypersurfaces. We prove its concentration in two levels in case of 1-dimensional singular locus $Σ$, and moreover determine the ranks of the nontrivial homology groups. These two groups depend on the monodromy at special points of $Σ$ and on the effect of the monodromy of the local system over its complement.

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。

Differential Equations on Complex Projective Hypersurfaces of Low Dimension

Let $n=2,3,4,5$ and let $X$ be a smooth complex projective hypersurface of $\mathbb P^{n+1}$. In this paper we find an effective lower bound for the degree of $X$, such that every holomorphic entire curve in $X$ must satisfy an algebraic differential equation of order $k=n=\dim X$, and also similar bounds for order $k>n$. Moreover, for every integer $n\ge 2$, we show that there are no such algebraic differential equations of order $k<n$ for a smooth hypersurface in $\mathbb P^{n+1}$.

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.

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,…

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。

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。

Existence of global invariant jet differentials on projective hypersurfaces of high degree

Let $X\subset\mathbb P^{n+1}$ be a smooth complex projective hypersurface. In this paper we show that, if the degree of $X$ is large enough, then there exist global sections of the bundle of invariant jet differentials of order $n$ on $X$, vanishing on an ample divisor. We also prove a logarithmic version, effective in low dimension, for the log-pair $(\mathbb P^n,D)$, where $D$ is a smooth irreducible divisor of high degree. Moreover, these result are sharp, \emph{i.e.} one cannot have such jet differentials of order less than $n$.

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。

Hodge Theory and Higgs Bundle on Moduli Spaces of Smooth Varieties with Semi-Ample Canonical Line Bundles, Negativity and Hyperbolicity

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.

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…

射影曲面大自同构群的双曲性

Hyperbolicity for large automorphism groups of projective surfaces

本文研究紧复曲面上非初等自同构群作用的双曲性，证明含抛物元素时 Zariski 扩散不变测度具有非零 Lyapunov 指数，并给出全曲面一致扩张判据及其动力学推论。

The varieties of tangent lines to hypersurfaces in projective spaces

For a hypersurface in a projective space, we consider the set of pairs of a point and a line in the projective space such that the line intersects the hypersurface at the point with a fixed multiplicity. We prove that this set of pairs forms a smooth variety for a general hypersurface.

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…

Hyperbolicity Related Problems for Complete Intersection Varieties

In this paper we examine different problems regarding complete intersection varieties of high degree in a complex projective space. First we show how one can deduce hyperbolicity for generic complete intersection of high multidegree and high codimension from the known results on hypersurfaces. Then we prove a existence theorem for jet differentials that generalizes a theorem of S. Diverio. Finally, motivated by a conjecture of O. Debarre, we focus on the positivity of the cotangent bundle of complete intersections, and prove some results towards this conjecture; among other things, we prove th…

射影空间中超曲面上完全交的变分Hodge猜想

Variational Hodge conjecture for complete intersections on hypersurfaces in projective space

该研究针对射影空间中超曲面上完全交闭链的变分Hodge猜想，提出了一种全新的简化证明方法，为相关代数几何问题的研究提供了新的论证思路。

Low-degree Rational curves on hypersurfaces in projective spaces and their degenerations

We study rational curves on general Fano hypersurfaces in projective space, mostly by degenerating the hypersurface along with its ambient projective space to reducible varieties. We prove results on existence of low-degree rational curves with balanced normal bundle, and reprove some results on irreducibility of spaces of rational curves of low degree.

Classical Computation of Number of Lines in Projective Hypersurfaces: Origin of Mirror Transformation

In this paper, we discuss classical derivation of the residue integral representation of the d=1 rational Gromov-Witten invariants of projective hypersurfaces that followed from localization technique.

Some criteria to check if a projective hypersurfaces is smooth or singular

In this paper we present some properties for projective hypersurfaces, smooth and singular, to be criteria for identification. To make the decision with these criteria, we have included procedures written in Singular language.

Viehweg's hyperbolicity conjecture is true over compact bases

We prove Viehweg's hyperbolicity conjecture over compact bases and over bases with non-uniruled compactification. The most general case of the conjecture states that the the base space of a maximal variation family of smooth projective manifolds with semi-ample canonical sheaf is of log-general type.

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。

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。

On smooth surfaces in projective four-space lying on quartic hypersurfaces with isolated singularities

We prove that a smooth surface, non of general type, in projective four-space, which lies on a quartic hypersurface with isolated singularities has degree at most 27 (in fact we prove a slightly more general result).

Number of points of a nonsingular hypersurface in an odd-dimensional projective space

The numbers of $\mathbb{F}_q$-points of nonsingular hypersurfaces of a fixed degree in an odd-dimensional projective space are investigated, and an upper bound for them is given. Also we give the complete list of nonsingular hypersurfaces each of which realizes the upper bound. This is a natural generalization of our previous study of surfaces in projective $3$-space.

射影超曲面的霍奇猜想

Hodge conjecture for projective hypersurface

本文通过证明霍奇类为解析对数De Rham类，并证明此类为代数闭链类，从而证明了光滑射影超曲面的霍奇猜想。

Invariants and rigidity of projective hypersurfaces

This paper continues our researches \cite{DS1, DS2, DS3} by computing some invariants based on Hilbert-Poincaré series associated to Milnor algebras. Our computations are for some of the classical surfaces and 3-folds with different configurations of isolated singularities. As a by-product of a recent result of E. Sernesi, we give examples of classical hypersurfaces which are (or are not) projectively rigid. We also include a Singular program to compute the invariants and to decide if a singular projective hypersurface is nodal and projectively rigid.

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…

Gauss-Manin systems of wild regular functions: Givental-Hori-Vafa models of smooth hypersurfaces in weighted projective spaces as an example

We study Gauss-Manin systems of non tame Laurent polynomial functions. We focuse on Hori-Vafa models, which are the expected mirror partners of the small quantum cohomology of smooth hypersurfaces in weighted projective spaces.

Scrolls and hyperbolicity

Using degeneration to scrolls, we give an easy proof of non-existence of curves of low genera on general surfaces in P3 of degree d >=5. We show, along the same lines, boundedness of families of curves of small enough genera on general surfaces in P3. We also show that there exist Kobayashi hyperbolic surfaces in P3 of degree d = 7 (a result so far unknown), and give a new construction of such surfaces of degree d = 6. Finally we provide some new lower bounds for geometric genera of surfaces lying on general hypersurfaces of degree 3d > 15 in P4.