OPENNESS OF UNIFORM K-STABILITY IN FAMILIES OF Q-FANO VARIETIES

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 provides an interpretation of the invariant in terms of the K-stability of log pairs.

Automated summary

We prove that uniform K-stability is a Zariski open condition in Q-Gorenstein families of Q-Fano varieties and investigate the behavior of the stability threshold in families.