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Algebraic hyperbolicity of adjoint linear systems on spherical varieties

Minseong Kwon, Haesong Seo

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 $|ω_{X} \otimes A^{\otimes m}|$ 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.

Algebraic hyperbolicity of adjoint linear systems on spherical varieties

Abstract

Moraga and Yeong conjectured that for a smooth complex projective variety of dimension , an ample line bundle on and an integer , very general elements of the adjoint linear system 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.

Paper Structure

This paper contains 8 sections, 15 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Spherical geometry
  3. Preliminaries
  4. Characteristic p method: Frobenius splitting
  5. Positivity of twisted kernel bundles
  6. Hyperbolicity of adjoint bundles on spherical varieties
  7. Setup
  8. Proof of Main Theorem \ref{['main thm: hyperbolic if orbit closures are smooth']}

Key Result

Theorem 1.2

Let $G$ be a connected reductive group, and $X$ a smooth projective spherical $G$-variety of dimension $n \ge 2$. If all $G$-orbit closures in $X$ are smooth, then Conjecture conj: adjoint bundles holds for $X$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (39)

  • Conjecture 1.1: MY2024
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Theorem 2.2: Perrin-sanya, Timashev
  • Definition 2.3
  • Remark 2.4
  • ...and 29 more