Minimizing CM Degree and Specially K-stable Varieties

Abstract

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., [ 46]), 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, for example, K-stable log Fano, klt Calabi-Yau (i.e., $K_{X}\equiv 0$), lc varieties with the ample canonical divisor and uniformly adiabatically K-stable klt-trivial fibrations over curves (cf., [ 27]).