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arXiv · 2605.20762

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

Nguyen Linh Chi·Si Duc Quang
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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.

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Nguyen Linh Chi, Si Duc Quang. 2026-05-20. A Brownian-Motion Approach to the Second Main Theorem for Meromorphic Mappings and Hypersurfaces with Truncated Counting Functions. https://arxiv.org/abs/2605.20762

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