Openness of K-semistability for Fano varieties

Abstract

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.

37 pages. Comments are very welcome. v3: to appear in Duke Math. J.. v2: added Theorem 4.5 showing openness of K-stability and constructibility of K-polystability

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