On toric geometry and K-stability of Fano varieties

TL;DR

This paper explores the deformation theory of toric Fano varieties and its implications for K-stability, revealing complex behaviors of moduli spaces and establishing methods to assess stability through degenerations.

Contribution

It demonstrates how deformation theory can be applied to analyze K-stability of Fano varieties, including examples with obstructed deformations and reducible or non-reduced moduli spaces.

Findings

Existence of K-polystable toric Fano 3-folds with obstructed deformations

Moduli spaces can be reducible or non-reduced near certain toric Fano 3-folds

Degeneration techniques to study K-stability of smooth Fano 3-folds

Abstract

We present some applications of the deformation theory of toric Fano varieties to K-(semi/poly)stability of Fano varieties. First, we present two examples of K-polystable toric Fano 3-fold with obstructed deformations. In one case, the K-moduli spaces and stacks are reducible near the closed point associated to the toric Fano 3-fold, while in the other they are non-reduced near the closed point associated to the toric Fano 3-fold. Second, we study K-stability of the general members of two deformation families of smooth Fano 3-folds by building degenerations to K-polystable toric Fano 3-folds.