A hyperbolicity conjecture for adjoint bundles

JoaquÃ­n Moraga; Wern Yeong

paper

A hyperbolicity conjecture for adjoint bundles

arXiv:2412.01811

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 $∣ K_{X} + (3 n + 1) L ∣$ 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 $∣ K_{X} + (3 n + 1) L ∣$ is pseudo hyperbolic. For a Gorenstein toric threefold $X$ , we show that $∣ K_{X} + 9 L ∣$ is hyperbolic.

15 pages, comments welcome. v2: Corrected a gap in proof of main theorems from previous draft (statements unchanged), added results about pseudo hyperbolicity on Gorenstein toric varieties