Algebraic hyperbolicity of adjoint linear systems on spherical varieties

TL;DR

This paper proves Moraga and Yeong's conjecture on algebraic hyperbolicity of adjoint linear systems for spherical varieties, including horospherical and toroidal cases, with results on the generic hyperbolicity of these systems.

Contribution

It establishes the conjecture for spherical varieties with smooth orbit closures and extends results to horospherical and toroidal varieties, also addressing the generic case.

Findings

Proved the conjecture for spherical varieties with smooth orbit closures.

Confirmed the conjecture for horospherical and toroidal spherical varieties.

Showed the conjecture holds generically for all spherical varieties.

Abstract

Moraga and Yeong conjectured that for a smooth complex projective variety $X$ of dimension $n$, an ample line bundle $A$ on $X$ and an integer $m \ge 3 n + 1$, very general elements of the adjoint linear system $|\omega_{X} \otimes A^{\otimes m}|$ are algebraically hyperbolic. We prove the conjecture for spherical varieties with smooth orbit closures. As a corollary, we conclude that the conjecture holds for horospherical varieties, and for toroidal spherical varieties. Furthermore, for any spherical variety, we show that the conjecture holds modulo the complement of an open dense orbit.