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Hyperbolicity of adjoint linear series on varieties with positive tangent bundle

Atsushi Ito, Joaquín Moraga, Debaditya Raychaudhury, Wern Yeong

Abstract

Let $X$ be a smooth projective variety of dimension $n\geq 3$, and let $L$ be an ample line bundle on $X$. In this article, we study the algebraic hyperbolicity of a very general section of the adjoint linear series $|K_X+mL|$ when the tangent bundle $T_X$ of $X$ has suitable positivity properties. As a consequence, we show that the linear system $|K_X+mL|$ is hyperbolic (or pseudo-hyperbolic) for $m\geq 3n+1$, for various classes of polarized pairs $(X,L)$, thus providing new evidence of a conjecture that was proposed by the second and fourth authors. Moreover, when $X$ is abelian, we show that the linear system $|mL|$ is hyperbolic for $m\geq n$, and the same holds when $m\geq n-1$, if $|L|$ has no base divisors. It turns out that these bounds for abelian varieties are sharp. We also prove analogous statements for Kummer varieties and certain classes of hyperelliptic varieties.

Hyperbolicity of adjoint linear series on varieties with positive tangent bundle

Abstract

Let be a smooth projective variety of dimension , and let be an ample line bundle on . In this article, we study the algebraic hyperbolicity of a very general section of the adjoint linear series when the tangent bundle of has suitable positivity properties. As a consequence, we show that the linear system is hyperbolic (or pseudo-hyperbolic) for , for various classes of polarized pairs , thus providing new evidence of a conjecture that was proposed by the second and fourth authors. Moreover, when is abelian, we show that the linear system is hyperbolic for , and the same holds when , if has no base divisors. It turns out that these bounds for abelian varieties are sharp. We also prove analogous statements for Kummer varieties and certain classes of hyperelliptic varieties.

Paper Structure

This paper contains 20 sections, 39 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on hyperbolicity and positivity
  3. Algebraic hyperbolicity
  4. Positivity of ($\mathbb{Q}$-twisted) sheaves and geometry
  5. Jet-separation and jet-ampleness
  6. Positivity of syzygy bundles
  7. Preliminaries on syzygy bundles
  8. Positivity of syzygy bundles for abelian, hyperelliptic and Kummer varieties
  9. Positivity of syzygy bundles for regular varieties
  10. Positivity of syzygy bundles for smooth fibrations over abelian varieties
  11. Positivity of syzygy bundles for varieties with nef tangent bundles
  12. From positivity of syzygy bundles to hyperbolicity
  13. Almost homogeneous varieties.
  14. Varieties with (pseudo or almost) nef tangent bundle.
  15. The fundamental theorems.
  16. ...and 5 more sections

Key Result

Theorem 1.2

Let $X$ be a smooth regular projective variety of dimension $n\geq 3$ and let ${L}$ be a very ample line bundle on $X$. If the tangent bundle $\mathcal{T}_X$ is nef (resp. pseudo-nef, almost nef), then the linear system $|K_X+mL|$ is hyperbolic (resp. pseudo-hyperbolic, almost hyperbolic) for $m>\ma

Theorems & Definitions (94)

  • Conjecture 1.1: MY24
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Remark 1.6
  • Example 1.7
  • Theorem 1.8
  • Remark 1.9
  • Corollary 1.10
  • ...and 84 more