Hyperbolicity and GCD for n+1 divisors with non-empty intersection

TL;DR

This paper investigates hyperbolicity of certain complex varieties with boundary divisors intersecting non-trivially, extending existing methods to higher dimensions and establishing new degeneracy and GCD estimates.

Contribution

It extends the Levin-Huang-Xiao method to higher dimensions, providing new hyperbolicity criteria and GCD estimates for varieties with intersecting boundary divisors.

Findings

All entire curves are algebraically degenerate under certain conditions.

Established a second main theorem for regular sequences of closed subschemes.

Derived a GCD-type estimate in the geometric setting.

Abstract

We study hyperbolicity for quasi-projective varieties where the boundary divisor consists of n+1 numerically parallel effective divisors on a complex projective variety of dimension n, allowing non-empty intersection. Under explicit local conditions on beta constants or intersection multiplicities, we prove that all entire curves are algebraically degenerate. Our approach extends the method of Levin-Huang-Xiao to higher dimensions, establishing a second main theorem for regular sequences of closed subschemes. This also yields a GCD-type estimate in the same geometric setting.