Non-integrated defect relation for meromorphic maps from a K\"{a}hler manifold intersecting hypersurfaces in subgeneral of $\mathbb P^n(\mathbb C)$

TL;DR

This paper establishes a new defect relation for meromorphic maps from Kähler manifolds intersecting hypersurfaces in subgeneral position, generalizing previous results and applying to the distribution of Gauss maps of minimal surfaces.

Contribution

It introduces a truncated non-integrated defect relation for meromorphic maps intersecting hypersurfaces in subgeneral position, extending known theorems in complex geometry.

Findings

Derived a new inequality for defect sums involving hypersurfaces in subgeneral position.

Generalized previous defect relation results to broader settings.

Applied the theoretical results to analyze the distribution of Gauss maps of minimal surfaces.

Abstract

In this article, we establish a truncated non-integrated defect relation for meromorphic mappings from an $m$-dimensional complete K\"{a}hler manifold into $\mathbb P^n(\mathbb C)$ intersecting $q$ hypersurfaces $Q_1,...,Q_q$ in $k$-subgeneral position of degree $d_i$, i.e., the intersection of any $k+1$ hypersurfaces is emptyset. We will prove that $$ \sum_{i=1}^q\delta_f^{[u-1]}(Q_i)\le (k-n+1)(n+1)+\epsilon+\frac{\rho u(u-1)}{d}, $$ where $u$ is explicitly estimated and $d$ is the least common multiple of $d_i'$s. Our result generalizes and improves previous results. In the last part of this paper we will apply this result to study the distribution of the Gauss map of minimal surfaces.