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.
Categories: math.CV math.AG