Campana's orbifold conjecture for numerically equivalent divisors

TL;DR

This paper proves a version of Campana's orbifold conjecture, showing that under certain conditions, orbifold entire curves are algebraically degenerate, advancing understanding of hyperbolicity in complex geometry.

Contribution

It establishes that for a class of orbifolds with numerically parallel divisors, the orbifold is of general type after a finite modification, implying algebraic degeneracy of entire curves.

Findings

Existence of a positive integer ll making the orbifold of general type.

Orbifold entire curves with sufficient multiplicity are algebraically degenerate.

Advancement in understanding hyperbolicity for orbifolds with numerically equivalent divisors.

Abstract

We prove the following version of the Campana's orbifold conjecture: Let $X$ be a complex non-singular projective variety of dimension $n$. Let $D_1,\ldots,D_{n+1}$ be $\mathbb Z$-linearly independent effective divisors in ${\rm Div}(X)$ and $D:=D_1+\cdots+D_{n+1}$ be a normal crossing divisor of $X$. Assume furthermore that they are numerically parallel. Let $\Delta=\sum_{i=1}^{n+1} (1-m_i^{-1}) D_i$ and let $f:\mathbb C\to (X,\Delta) $ be an orbifold entire curve. Then, there exists a positive integer $\ell$ such that, the orbifold $ (X,\Delta_{\ell}) $ is of general type, where $\Delta_{\ell}=\sum_{i=1}^{n+1} (1-\frac1{\ell})D_i$, and if $f$ has multiplicity at least $\ell$ along $D_i$, $1\le i\le n+1$, then $f$ must be algebraically degenerate.