Deux exemples sur la dimension moyenne d'un espace de courbes de Brody

TL;DR

This paper investigates the mean dimension of the space of 1-Brody curves in two complex surfaces, demonstrating zero mean dimension for Hopf surfaces and positive mean dimension for the projective plane minus a line.

Contribution

It provides new results on the mean dimension of Brody curves in specific complex surfaces, using growth bounds and deformation techniques.

Findings

Mean dimension is zero for Hopf surfaces.

Mean dimension is positive for the projective plane minus a line.

Deformations of elliptic curves contribute to positive mean dimension.

Abstract

We study the mean dimension of the space of 1-Brody curves lying in two complex surfaces: first for Hopf surfaces, then for the projective plane minus a line. We show in the first case that the mean dimension is zero via a bound on the growth of meromorphic curves involving the logarithmic derivative lemma. In the second case, we show its positivity by lifting from the line to its complement a space of Brody curves of positive mean dimension containing deformations of an elliptic curve.