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Theoretical analysis demonstrates openness of K-semistability in families of log Fano pairs, establishing stability threshold constructibility and canonical properties of test configurations.
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.
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Blum et al. (2022) studied this question.
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