Degenerate Second Main Theorems for Holomorphic Curves in Different Geometric Settings

2024/06/04 by Si Duc Quang, Quang, Si Duc, Nguyen Van An +2
Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.2406.02371

openalex publication_date 2024/06/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We establish second main theorems for holomorphic curves into a projective subvary V ⊂ ℙⁿ(ℂ) of dimension k, intersecting hypersurfaces in N-subgeneral position with respect to V (N > k). Our results provide explicit truncation levels for the counting functions that are independent of the number of hypersurfaces. The theorems are obtained in several settings, including holomorphic curves on ℂ, annuli, complex discs with finite growth index, and Kähler manifolds. We obtain a total defect bound that improves upon the previously known results. As an application, we establish a corresponding form of Schmidt's subspace theorem for families of homogeneous polynomials in subgeneral position.

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