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arXiv · 2412.01811

A hyperbolicity conjecture for adjoint bundles

Joaquín Moraga·Wern Yeong
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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.

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Joaquín Moraga, Wern Yeong. 2024-12-02. A hyperbolicity conjecture for adjoint bundles. https://arxiv.org/abs/2412.01811

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