Campana's orbifold conjecture for 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 $D_1,\dots ,D_{n+1}$ be $\mathbb{Z}$-linearly independent effective divisors in $\mathrm{Div}(X)$ and $D:=D_1+\cdots +D_{n+1}$ be a normal crossing divisor of $X$. Assume furthermore that they are numerically parallel. Let $\hat{I}”={\sum }_{i=1}^{n+1}(1-m_i^{-1})D_i$ and let $f:\mathbb{C}\to (X,\hat{I}”)$ be an orbifold entire curve. Then, there exists a positive integer $\ell$ such that, the orbifold $(X,\hat{I}{”}_{\ell })$ is of general type, where $\hat{I}{”}_{\ell }={\sum }_{i=1}^{n+1}(1-\frac{1}{\ell })D_i$, and if $f$ has multiplicity at least $\ell$ along $D_i$, $1\le i\le n+1$, then $f$ must be algebraically degenerate.