A hyperbolicity conjecture for adjoint bundles

2024/12/02 by Joaquí­n Moraga, Moraga, Joaquín, Wern Yeong +1 · 5 citations
Mathematics · #14F17 (Secondary) #14J70 (Primary) 14M25 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2412.01811

openalex publication_date 2024/12/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• Kobayashi hyperbolicity of complete linear systems on abelian varieties
• Hyperbolicity of adjoint linear series on varieties with positive tangent bundle
• Algebraic hyperbolicity of very general hypersurfaces in weighted projective spaces
• Algebraic hyperbolicity of subvarieties of homogeneous varieties
• Algebraic hyperbolicity of adjoint linear systems on spherical varieties

Related

• Hyperbolicity of adjoint linear series on varieties with positive tangent bundle
• Algebraic hyperbolicity of very general surfaces
• Algebraic hyperbolicity of adjoint linear systems on spherical varieties
• Boundedness of hyperbolic varieties
• Algebraic Hyperbolicity for Surfaces in Toric Threefolds