Minimal models and boundedness of stable varieties

TL;DR

This paper proves that a class of stable smoothable n-dimensional varieties is bounded under the assumption of the minimal model program in dimension n+1, and establishes the existence of a projective moduli space for these varieties.

Contribution

It demonstrates boundedness and moduli space existence for stable smoothable varieties assuming the minimal model program in one higher dimension.

Findings

Boundedness of stable smoothable n-folds proved under MMP assumption

Existence of projective coarse moduli space established

Connection between MMP in dimension n+1 and moduli space construction

Abstract

We consider a class of stable smoothable n-dimensional varieties, the analogs of stable curves. Assuming the minimal model program in dimension n+1, we prove that this class is bounded. From Kollar's method of constructing projective moduli spaces we get as a corollary that minimal model program in dimension n+1 implies the existence of a projective coarse moduli space for stable smoothable n-folds.