A note on families of K-semistable log-Fano pairs

Abstract

In this short note, we give an alternative proof of the semipositivity of the Chow-Mumford line bundle for families of K-semistable log-Fano pairs, and of the nefness threeshold for the log-anti-canonical line bundle on families of K-stable log Fano pairs. We also prove a bound on the multiplicity of fibers for families of K-semistable log Fano varieties, which to the best of our knowledge is new.

Slope inequalities for KSB-stable and K-stable families

“…Recall that K-polystability is a stability condition for log Fano pairs equivalent to the existence of a Kähler-Einstein metric; it also allows the construction of projective moduli spaces. It is worth recalling that the study of the Harder-Narashiman filtration, so important for the proof of our slope inequalities, plays also a crucial role in the proof of the projectivity of this moduli space, see [CP21a,CP21b,Pos19,XZ20].…”