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

TL;DR

This paper investigates the deformation spaces of toric singularities and their relation to the singularities in K-moduli of Fano varieties, revealing classification results and unbounded complexity in higher dimensions.

Contribution

It classifies the bases of miniversal deformations for certain toric singularities and links these to the singularities of K-moduli stacks, showing unbounded local complexity in dimensions three and higher.

Findings

Classified bases of miniversal deformations for low-dimensional toric singularities.

Established that Gorenstein toric 3-fold singularities appear as singularities in K-moduli stacks.

Proved the number of local branches in K-moduli is unbounded in dimensions ≥3.

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 $\geq 3$.