A hyperbolicity conjecture for adjoint bundles
Joaquín Moraga, Wern Yeong
TL;DR
Addressing whether adjoint linear systems $|K_X+(3n+1)L|$ are algebraically hyperbolic for smooth projective varieties, the paper develops a toric framework and leverages Lazarsfeld kernel bundles and vanishing theorems to propagate hyperbolicity via dimension induction. It proves the conjecture for smooth toric varieties, and establishes pseudo hyperbolicity for Gorenstein toric varieties, with a sharp hyperbolicity result $|K_X+9L|$ for Gorenstein toric $3$-folds; the approach centers on positivity of $N+2nL$, restriction maps to torus-invariant divisors, and descent of hyperbolicity to lower-dimensional strata. The results yield broad new evidence for the conjecture and provide numerous explicit examples (e.g., products of projective spaces and Grassmannians) while highlighting natural questions about adjoint systems in toric geometry and Fujita-type basepoint-freeness phenomena. Overall, the work integrates toric geometry, positivity techniques, and deformation-theoretic tools to advance understanding of when adjoint systems exhibit hyperbolicity with potential implications for Kobayashi hyperbolicity and related arithmetic-geometric properties.
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.