Campana's orbifold conjecture for numerically equivalent divisors

2025/06/01 by Min Ru, Julie Tzu‐Yueh Wang, Ru, Min +1 · 1 citation
Mathematics · #32H30 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2506.00873

openalex publication_date 2025/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 D₁,…,D_n+1 be \mathbb Z-linearly independent effective divisors in \rm Div(X) and D:=D₁+⋯+D_n+1 be a normal crossing divisor of X. Assume furthermore that they are numerically parallel. Let Δ=∑_i=1ⁿ⁺¹ (1-m_i^-1) D_i and let f:\mathbb C→ (X,Δ) be an orbifold entire curve. Then, there exists a positive integer ℓ such that, the orbifold (X,Δ_ℓ) is of general type, where Δ_ℓ=∑_i=1ⁿ⁺¹ (1-\frac1ℓ)D_i, and if f has multiplicity at least ℓ along D_i, 1≤ i≤ n+1, then f must be algebraically degenerate.

Citations

• Vojta's abc conjecture for entire curves in toric varieties highly ramified over the boundary
• Defect relation of n+1 components through the GCD method]
• A complex case of Vojta's general abc conjecture and cases of Campana's orbifold conjecture
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• Vojta's abc Conjecture for algebraic tori and applications over function fields
• A truncated second main theorem for algebraic tori with moving targets and applications
• Degeneracy of Holomorphic Curves into Algebraic Varieties
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Cited by

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

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• Greatest Common Divisor results on semiabelian varieties and a Conjecture of Silverman