Degenerate Second Main Theorems for Holomorphic Curves in Different Geometric Settings

TL;DR

This paper proves advanced second main theorems for holomorphic curves in various geometric contexts, providing explicit truncation levels and improved defect bounds, with applications to Schmidt's subspace theorem.

Contribution

It introduces new second main theorems with explicit truncation levels for holomorphic curves in multiple settings, enhancing previous results and applying to Diophantine approximation.

Findings

Explicit truncation levels independent of the number of hypersurfaces

Total defect bounds that improve upon previous results

Extension of results to Kähler manifolds and other settings

Abstract

We establish second main theorems for holomorphic curves into a projective subvary $V \subset \mathbb{P}^n(\mathbb{C})$ 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 $\mathbb{C}$, annuli, complex discs with finite growth index, and K\"ahler 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.