On deformations of toric Fano varieties

Abstract

In this note we collect some results on the deformation theory of toric Fano varieties.

Some observations on the dimension of Fano K-moduli

“…Since X l is toric it does not admit equisingular deformations (i.e. non-trivial deformations to a non-isomorphic projective variety with the same singularities), e.g., [13,Lemma 4.4]. Hence all Q-Gorenstein deformations must come from local Q-Gorenstein deformations of the singularities.…”

A 1-dimensional component of K-moduli of del Pezzo surfaces

“…By [1, Lemma 6] there are no local-to-global obstructions for Q-Gorenstein deformations of X, so the Q-Gorenstein smoothings of the two 1 4 (1, 1) points of X, which we denote p 1 and p 2 , can be realised globally and simultaneously. More precisely, since H i (X, T X ) = 0 for i ≥ 1 by [27], the product of the restriction morphisms to the germs (p i ∈ X)…”

On toric geometry and K-stability of Fano varieties

“…As X is a toric Fano variety, H 1 (X, T 0 X ) = 0 and H 2 (X, T 0 X ) = 0 (see [56, Proof of Theorem 5.1] and [47,Lemma 4.4]). From the spectral sequence…”

“…As in the smooth case, this implies that moduli of del Pezzo surfaces with cyclic quotient singularities are smooth [45]. In dimension 3, there are examples of Fano varieties with obstructed deformations and isolated (canonical) singularities [30,47,48]. Note however that Fano 3-folds with terminal singularities have unobstructed deformations [43,54].…”