Abstract

The Green—Griffiths—Lang conjecture is regarded as a central problem in arithmetic geometry, as it relates the notion of complex hyperbolicity of a variety to its geometric properties and the distribution of its rational points. In particular, the case of m-th symmetric powers of varieties is related to the distribution of algebraic points of degree at most m. By using the theory of jet differentials, in 2022 Cadorel, Campana, and Rousseau gave geometric conditions to obtain pseudo-hyperbolicity of symmetric powers of varieties, which was applied, for instance, to obtain hyperbolicity for symmetric powers of generic hypersurfaces of high degree. In this talk, we will show a new technique coming from Nevanlinna theory that gives geometric criteria to obtain hyperbolicity of symmetric powers of varieties. Using this method, we provide concrete examples of varieties with hyperbolic symmetric powers. This is joint work with Hector Pasten.