Holomorphic curves in Base Spaces of Families of Polarized Manifolds

TL;DR

This paper proves that entire curves in the base space of certain polarized manifold families are contained within fibers of the classifying map, confirming the Relative Isotriviality Conjecture.

Contribution

It establishes the Relative Isotriviality Conjecture for families of polarized manifolds with semi-ample canonical sheaves.

Findings

Entire curves are contained in fibers of the classifying map.

Confirms the Relative Isotriviality Conjecture.

Advances understanding of holomorphic curves in moduli spaces.

Abstract

For a smooth family $V \to U$ of polarized manifolds with semi-ample canonical sheaves, we show the following result: any entire curve must be contained in the fibers of the classifying map from the base space $U$ to the moduli space. This settles the Relative Isotriviality Conjecture, \cite[Conjecture 1.5]{DLSZ}.