English

Campana's orbifold conjecture for numerically equivalent divisors

Complex Variables 2025-06-03 v1 Algebraic Geometry

Abstract

We prove the following version of the Campana's orbifold conjecture: Let XX be a complex non-singular projective variety of dimension nn. Let D1,,Dn+1D_1,\ldots,D_{n+1} be Z\mathbb Z-linearly independent effective divisors in Div(X){\rm Div}(X) and D:=D1++Dn+1D:=D_1+\cdots+D_{n+1} be a normal crossing divisor of XX. Assume furthermore that they are numerically parallel. Let Δ=i=1n+1(1mi1)Di\Delta=\sum_{i=1}^{n+1} (1-m_i^{-1}) D_i and let f:C(X,Δ)f:\mathbb C\to (X,\Delta) be an orbifold entire curve. Then, there exists a positive integer \ell such that, the orbifold (X,Δ) (X,\Delta_{\ell}) is of general type, where Δ=i=1n+1(11)Di\Delta_{\ell}=\sum_{i=1}^{n+1} (1-\frac1{\ell})D_i, and if ff has multiplicity at least \ell along DiD_i, 1in+11\le i\le n+1, then ff must be algebraically degenerate.

Keywords

Cite

@article{arxiv.2506.00873,
  title  = {Campana's orbifold conjecture for numerically equivalent divisors},
  author = {Min Ru and Julie Tzu-Yueh Wang},
  journal= {arXiv preprint arXiv:2506.00873},
  year   = {2025}
}
R2 v1 2026-07-01T02:52:54.210Z