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

Petracci, Andrea

doi:10.1090/tran/8636

On deformation spaces of toric singularities and on singularities of K-moduli of Fano varieties
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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 $\leq 2$ 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 $\geq 3$. 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 $\geq 3$.

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