Minimizing CM degree and specially K-stable varieties

TL;DR

This paper proves that the degree of the CM line bundle is minimized for certain special fibers in a family, introducing a new class called 'specially K-stable' that strengthens the understanding of moduli space separatedness.

Contribution

The paper introduces 'specially K-stable' varieties and proves a minimization property of the CM line bundle degree for these cases, extending previous conjectures.

Findings

Minimization of CM line bundle degree for special fibers.

Introduction of the 'specially K-stable' class.

Demonstration that this class implies K-stability in many cases.

Abstract

We prove that the degree of the CM line bundle for a normal family over a curve with fixed general fibers is strictly minimized if the special fiber is either a smooth projective manifold with a unique cscK metric or ``specially K-stable", which is a new class we introduce in this paper. This phenomenon, as conjectured by Odaka (cf., [Oda20]), is a quantitative strengthening of the separatedness conjecture of moduli spaces of polarized K-stable varieties. The above mentioned special K-stability implies the original K-stability and a lot of cases satisfy it e.g., K-stable log Fano, klt Calabi-Yau (i.e., $K_X\equiv0$), lc varieties with the ample canonical divisor and uniformly adiabatically K-stable klt-trivial fibrations over curves (cf., [Hat22]).