Degenerate Second Main Theorems for Holomorphic Curves in Different Geometric Settings

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.