A hyperbolicity conjecture for adjoint bundles

TL;DR

This paper investigates a conjecture that certain linear systems on smooth projective varieties are hyperbolic, confirming it for smooth toric varieties and providing partial results for Gorenstein toric varieties.

Contribution

It proves the hyperbolicity conjecture for smooth projective toric varieties and establishes pseudo hyperbolicity for Gorenstein toric varieties, advancing understanding of hyperbolicity in algebraic geometry.

Findings

Confirmed the conjecture for smooth projective toric varieties.

Showed that for Gorenstein toric varieties, the linear system is pseudo hyperbolic.

Established hyperbolicity for Gorenstein toric threefolds with a specific linear system.

Abstract

Let $X$ be a $n$-dimensional smooth projective variety and $L$ be an ample Cartier divisor on $X$. We conjecture that a very general element of the linear system $|K_X+(3n+1)L|$ is a hyperbolic algebraic variety. This conjecture holds for some classical varieties: surfaces, products of projective spaces, and Grassmannians. In this article, we investigate the conjecture for $X$ a toric variety. We confirm the conjecture in the case of smooth projective toric varieties. When $X$ is a Gorenstein toric variety, we show that $|K_X+(3n+1)L|$ is pseudo hyperbolic. For a Gorenstein toric threefold $X$, we show that $|K_X+9L|$ is hyperbolic.