Quasi-Projective Moduli for Polarized klt Good Minimal Models

TL;DR

This paper proves the quasi-projectivity of moduli spaces for polarized klt good minimal models by establishing weak positivity of direct images and applying Viehweg's ampleness criterion.

Contribution

It introduces a new approach to show the quasi-projectivity of moduli spaces for polarized klt good minimal models using weak positivity and Gabber's Extension Theorem.

Findings

Weak positivity of direct images for stable families of klt good minimal models.

Quasi-projectivity of the moduli space normalization for polarized klt good minimal models.

Application of Viehweg's ampleness criterion in this context.

Abstract

We prove the weak positivity of direct images for locally stable families of klt good minimal models over reduced quasi-projective bases using Gabber's Extension Theorem. As an application, we apply Viehweg's ampleness criterion to show that the normalization of the moduli space of polarized klt good minimal models of arbitrary Kodaira dimension, constructed in [Jia23], is quasi-projective.