Second Main Theorem on the Moduli Spaces of Polarized Varieties

TL;DR

This paper establishes a second main theorem for moduli spaces of polarized varieties using advanced geometric techniques, leading to new insights into the distribution of entire curves in these spaces.

Contribution

It introduces a novel second main theorem for polarized moduli spaces leveraging Viehweg-Zuo and McQuillan's inequalities, extending classical results.

Findings

Proves a second main theorem for moduli spaces of polarized varieties.

Generalizes Nadel's classical result on entire curves.

Provides new tools for studying the distribution of entire curves.

Abstract

Let $(X,D)$ be a smooth log pair over $\mathbb{C}$ such that the complement $U := X \setminus D$ carries a maximally varied family of polarized manifolds. We prove a version of second main theorem on $(X,D)$ by using the Viehweg-Zuo construction of the family and McQuillan's tautological inequality. As an application, we generalize a classical result of Nadel about the distribution of entire curves in the (compactified) base space of polarized families.