In this paper, we prove the openness of K-semistability in families of log Fano pairs by showing that the stability threshold is a constructible function on the fibers. We also prove that any special test configuration arises from a log canonical place of a bounded complement and establish properties of any minimizer of the stability threshold.
"Openness of K-semistability for Fano varieties." Duke Math. J. 171 (13) 2753 - 2797, 15 September 2022. https://doi.org/10.1215/00127094-2022-0054