Holomorphic curves in moduli spaces of polarized Abelian varieties

TL;DR

This paper develops a Nevanlinna theory framework for holomorphic curves into moduli spaces of polarized Abelian varieties, using stochastic calculus to derive key inequalities and applying them to value distribution problems.

Contribution

It introduces a tautological inequality and a logarithmic derivative lemma for logarithmic pairs, leading to a Second Main Theorem applicable to moduli spaces of Abelian varieties.

Findings

Established a Second Main Theorem for holomorphic curves in moduli spaces

Derived a tautological inequality and a logarithmic derivative lemma

Applied the theory to boundary-divisor intersections in moduli spaces

Abstract

We study the value distribution of holomorphic curves from a general open Riemann surface into a smooth logarithmic pair $(X, D).$ By stochastic calculus, we first obtain a version of tautological inequality (proposed by McQuillan) and a logarithmic derivative lemma. Then, one uses them to establish a Second Main Theorem of Nevanlinna theory for pair $(X, D)$ under certain conditions. Finally, we apply the Second Main Theorem to study the holomorphic curves from a general open Riemann surface into certain special moduli spaces of polarized Abelian varieties intersecting boundary divisors.