Algebroid maps and hyperbolicity of symmetric powers

TL;DR

This paper investigates the hyperbolicity of symmetric powers of complex projective varieties using Nevanlinna theory, providing bounds and applications to algebraic points and subvarieties.

Contribution

It introduces a new approach to bounding hyperbolicity of symmetric powers via algebroid maps and applies these results to curves and abelian varieties.

Findings

Lower bounds for hyperbolicity indices of symmetric powers.

Explicit bounds for algebraic points of bounded degree.

Applications to curves in surfaces and subvarieties of abelian varieties.

Abstract

Given a complex projective algebraic variety $X$ we define $ h(X)$ as the largest $n$ such that the $n$-th symmetric power of $X$ is (Brody) hyperbolic. Using Nevanlinna theory for algebroid maps, we give non-trivial lower bounds for $ h(X)$. From an arithmetic point of view, the problem is closely related to the finiteness of algebraic points of bounded degree in varieties over number fields. We provide explicit applications of our results in the case of curves embedded in surfaces and in the case of subvarieties of abelian varieties.