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

TL;DR

This paper provides new proofs of semipositivity and nefness thresholds for families of K-semistable log-Fano pairs and introduces a novel bound on fiber multiplicities, advancing understanding in algebraic geometry.

Contribution

It offers an alternative proof of key positivity results and establishes a new bound on fiber multiplicities for K-semistable log Fano families.

Findings

Semipositivity of the Chow-Mumford line bundle proved via alternative method

Nefness threshold for log-anti-canonical line bundle established

New bound on fiber multiplicity for K-semistable log Fano families

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.