Degenerate Second Main Theorems for Holomorphic Curves in Different Geometric 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 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.

The part concerning SMTs for holomorphic curves from annuli and hypersurfaces using Nochka weights has been removed. The total defect bound are re-estimated more optimal than the previous one. The title of the paper has been changed, and Nguyen Van An has joined this work as a coauthor