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arXiv · 1808.09070

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

Harold Blum·Yuchen Liu
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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.

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Harold Blum, Yuchen Liu. 2018-08-28. Openness of uniform K-stability in families of $\mathbb{Q}$-Fano varieties. https://arxiv.org/abs/1808.09070

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