首页/论文/2007.07572

arXiv 2007.075722020-07-15math.AGmath.CV

Hyperbolicity and Specialness of Symmetric Powers

Benoit Cadorel, Frédéric Campana, Erwan Rousseau

英文摘要：

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 Kobayashi hyperbolicity of symmetric powers of generic projective hypersur-faces of sufficiently high degree. On the algebraic side, we give a criterion implying that subvarieties of codimension $\le$ n -- 2 of symmetric powers are of general type. This applies in particular to varieties with ample cotangent bundles. Finally, based on a metric approach we study symmetric powers of ball quotients.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

On h-principle and specialness for complex projective manifolds

We show that a complex projective manifold X which satisfies the Gromov's h-principle is `special', and raise some questions about the reverse implication, the extension to the quasi-K\" ahler case, and the relationships of these properties to the `Oka' property. The guiding principle is that the existence of many Stein manifolds which have degenerate Kobayashi pseudometric gives strong obstructions to the complex hyperbolicity of X satisfying the h-principle.

2012-11-04math.AGmath.CV

Hyperbolicity of singular spaces

We study the hyperbolicity of singular quotients of bounded symmetric domains. We give effective criteria for such quotients to satisfy Green-Griffiths-Lang's conjectures in both analytic and algebraic settings. As an application, we show that Hilbert modular varieties, except for a few possible exceptions, satisfy all expected conjectures.

2018-09-28math.AGmath.CV

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.

2015-02-28math.AGmath.CV

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.

2016-06-13math.AG

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…

2021-10-12math.AG

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

CNRS, Institut Élie Cartan de Lorraine, Université de Lorraine（法国国家科学研究中心，洛林伊莱·卡当研究所，洛林大学）

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。

2023-04-20math.AGmath.CV

更多相关论文收起更多论文

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。

2024-06-02math.AGmath.NT

对称双曲多项式

Symmetric Hyperbolic Polynomials

本文研究对称双曲多项式，完整刻画三次情形、给出四次大类，并将钩形多项式与保双曲性线性映射关联，证明相关双曲锥的谱像性。

2023-08-18math.AGmath.OC

Carathéodory双曲性、体积估计与函数域上的水平结构

Carathéodory hyperbolicity, volume estimates and level structures over function fields

本研究将水平结构不存在性结论推广至强Carathéodory双曲复流形一致化的拟射影流形，建立了相关曲线映射体积估计，拓展了已有结果并给出了非光滑复Finsler度量流形的Schwarz引理版本。

2025-01-16math.AG

$A_n$ 奇点的局部欧拉示性数及其在双曲性中的应用

Local Euler characteristics of $A_n$-singularities and their application to hyperbolicity

Simon Fraser University（西蒙菲莎大学）；University of Oregon（俄勒冈大学）

利用环面几何工具，研究了 $A_n$ 型孤立曲面奇点的对称微分层的局部欧拉示性数，给出了余切丛 $m$ 次对称幂的显式公式（周期 $n+1$ 的拟多项式），并将其应用于构造 $\mathbb{P}^3$ 中低次代数拟双曲曲面的新例子。

2025-02-04math.AGmath.NT

Subvarieties of quotients of bounded symmetric domains

We present a new criterion for the complex hyperbolicity of a non-compact quotient X of a bounded symmetric domain. For each p $\ge$ 1, this criterion gives a precise condition under which the subvarieties V $\subset$ X with dim V $\ge$ p are of general type, and X is p-measure hyperbolic. Then, we give several applications related to ball quotients, or to the Siegel moduli space of principally polarized abelian varieties. For example, we determine effective levels l for which the moduli spaces of genus g curves with l-level structures are of general type.

2018-09-28math.AGmath.DG

Hyperbolic Secant Varieties of M-Curves

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…

2020-02-02math.AG

曲线高割线簇的Ulrich层与行列式表示

Ulrich sheaves and determinantal representations for higher secant varieties of curves

Eberhard Karls Universität Tübingen（蒂宾根大学）；Technische Universität Dresden（德累斯顿工业大学）；KAIST（韩国科学技术院）

研究光滑射影曲线高割线簇的行列式表示，利用高阶Szegő核等工具，在嵌入充裕时给出对称表示，对实曲线割线簇给出正定条件，对有理正规曲线割线簇给出明确表示，还能在特定条件下将曲线凸包表示为谱面体。

2026-07-27math.AG