On deformation spaces of toric singularities and on singularities of K-moduli of Fano varieties

Abstract

Firstly, we see that the bases of the miniversal deformations of isolated $\mathbb{Q}$-Gorenstein toric singularities are quite restricted. In particular, we classify the analytic germs of embedding dimension $\le 2$ which are the bases of the miniversal deformations of isolated $\mathbb{Q}$-Gorenstein toric singularities. Secondly, we show that the deformation spaces of isolated Gorenstein toric $3$-fold singularities appear, in a weak sense, as singularities of the K-moduli stack of K-semistable Fano varieties of every dimension $\ge 3$. As a consequence, we prove that the number of local branches of the K-moduli stack of K-semistable Fano varieties and of the K-moduli space of K-polystable Fano varieties is unbounded in each dimension $\ge 3$.

26 pages. v2: title has been changed, exposition has been improved, results are unchanged

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Cited by in corpus (2)

• Reductive quotients of klt singularities
• Foldable fans, cscK surfaces and local K-moduli