Separatedness of moduli of K-stable varieties

TL;DR

This paper proves the uniqueness of K-stable limits in families of polarized varieties, ensuring moduli spaces are Hausdorff, and provides characterizations and applications related to the CM line bundle across various characteristics.

Contribution

It establishes the uniqueness of K-stable limits in families, ensuring moduli spaces are Hausdorff, and offers new characterizations using the CM line bundle applicable in all characteristics.

Findings

K-stable limits are unique in flat families

Moduli spaces of K-stable varieties are Hausdorff

Characterization of K-stability via the CM line bundle

Abstract

Given a one parameter flat family of polarized algebraic varieties, we show that any K-stable limit is unique. In particular, moduli spaces of K-stable polarized varieties are automatically Hausdorff when they exist. We also give a characterization of K-stable limits in terms of the CM line bundle, and some applications to moduli. Our methods work for arbitrary projective schemes in any characteristic.