Openness of uniform K-stability in families of $\mathbb{Q}$-Fano varieties

TL;DR

This paper proves that uniform K-stability is a Zariski open property in families of Q-Fano varieties, using the behavior of the stability threshold (delta-invariant) which is shown to be lower semicontinuous.

Contribution

It establishes the openness of uniform K-stability in Q-Gorenstein families of Q-Fano varieties via analysis of the stability threshold's behavior.

Findings

Uniform K-stability is Zariski open in families.

The stability threshold is lower semicontinuous in families.

The delta-invariant characterizes K-stability of log pairs.

Abstract

We show that uniform K-stability is a Zariski open condition in Q-Gorenstein families of Q-Fano varieties. To prove this result, we consider the behavior of the stability threshold in families. The stability threshold (also known as the delta-invariant) is a recently introduced invariant that is known to detect the K-semistability and uniform K-stability of a Q-Fano variety. We show that the stability threshold is lower semicontinuous in families and provide an interpretation of the invariant in terms of the K-stability of log pairs.