A Brownian-Motion Approach to the Second Main Theorem for Meromorphic Mappings and Hypersurfaces with Truncated Counting Functions

TL;DR

This paper introduces a novel approach using Brownian motion and stochastic calculus to establish a second main theorem for holomorphic curves into projective varieties, with applications to uniqueness theorems.

Contribution

It develops a new second main theorem for meromorphic mappings into projective varieties using stochastic methods, accommodating arbitrary hypersurface families and truncated counting functions.

Findings

Established a second main theorem with truncated counting functions for holomorphic curves.

Proved a uniqueness theorem for holomorphic curves sharing hypersurfaces.

Applied stochastic calculus to complex value distribution theory.

Abstract

By using Brownian motion and stochastic calculus, we establish a second main theorem for holomorphic curves into a projective subvariety $V\subset\mathbb P^n(\mathbb C)$ with an arbitrary family $\mathcal Q$ of $q$ hypersurfaces $Q_1,\ldots,Q_q$ concerning its distributive constant $\Delta_{\mathcal Q,V}$. In our result, the counting functions are truncated to level $H_V(d)-1$, where $d=lcd(\deg Q_1,\ldots,\deg Q_d)$ and $H_V(d)$ is the Hilbert function of $V$. As an application of the second main theorem, we give a uniqueness theorem for holomorphic curves from $\mathbb C$ into $V$ sharing an arbitrary family of hypersurfaces regardless of multiplicity.