Quasiprojective varieties admitting Zariski dense entire holomorphic curves

TL;DR

This paper investigates the conditions under which complex quasiprojective varieties admit Zariski dense entire holomorphic curves, revealing structural properties related to their orbifold and quasi-Albanese maps, especially for varieties of log-general type.

Contribution

It demonstrates that the orbifold structure induced by certain birational maps is special when the variety admits a Zariski dense entire curve, extending previous results on the structure of such varieties.

Findings

Orbifold structure is special under birational equivalence for varieties with dense entire curves.

If a variety is of log-general type with high irregularity, entire curves are confined to proper subvarieties.

The quasi-Albanese map being a fiber space is linked to the existence of dense entire curves.

Abstract

Let $X$ be a complex quasiprojective variety. A result of Noguchi-Winkelmann-Yamanoi shows that if $X$ admits a Zariski dense entire curve, then its quasi-Albanese map is a fiber space. We show that the orbifold structure induced by a properly birationally equivalent map on the base is special in this case. As a consequence, if $X$ is of log-general type with $\bar q(X)\geq\dim X$, then any entire curve is contained in a proper subvariety in $X$.