Hyperbolicity of adjoint linear series on varieties with positive tangent bundle

TL;DR

This paper investigates the hyperbolicity of adjoint linear series on varieties with positive tangent bundles, establishing new bounds for hyperbolicity in various classes of algebraic varieties, including abelian and hyperelliptic varieties.

Contribution

It provides new hyperbolicity results for adjoint linear series on varieties with positive tangent bundles, supporting a conjecture by the authors and establishing sharp bounds for abelian varieties.

Findings

Linear system |K_X + mL| is hyperbolic for m ≥ 3n+1 on certain varieties.

|mL| is hyperbolic for m ≥ n on abelian varieties, with bounds being sharp.

Analogous hyperbolicity results hold for Kummer and hyperelliptic varieties.

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.