On toric geometry and K-stability of Fano varieties

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 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 3 -fold, while in the other they are non-reduced near the closed point associated to the toric Fano 3 3 -fold. Second, we study K-stability of the general members of two deformation families of smooth Fano 3 3 -folds by building degenerations to K-polystable toric Fano 3 3 -folds.

On K‐stability of some del Pezzo surfaces of Fano index 2

“…More precisely one can prove that the affine quotient $$H^0(\mathcal {T}^{\mathrm{qG}, 1}_{Y}) / T_N$$ has dimension $$2a-3$$. Since every facet of the polytope $$P^\circ$$ has no interior lattice points, by [26, Proposition 2.6] the automorphism group of $$Y$$ is $$T_N \rtimes \operatorname{Aut}(P)$$, where $$\operatorname{Aut}(P) \subseteq \mathrm{GL}(N)$$ is the finite group consisting of the lattice automorphisms which keep the polytope $$P$$ invariant. Since the difference between $$T_N$$ and $$\operatorname{Aut}(Y)$$ is just a finite group, we deduce that the affine quotient the affine quotient $$H^0(\mathcal {T}^{\mathrm{qG}, 1}_{Y}) / \operatorname{Aut}(Y)$$ has dimension $$2a-3$$.…”

“…)∕𝑇 𝑁 has dimension 2𝑎 − 3. Since every facet of the polytope 𝑃 • has no interior lattice points, by [26,Proposition 2.6] the automorphism group of 𝑌 is 𝑇 𝑁 ⋊ Aut(𝑃), where Aut(𝑃) ⊆ GL(𝑁) is the finite group consisting of the lattice automorphisms which keep the polytope 𝑃 invariant. Since the difference between 𝑇 𝑁 and Aut(𝑌) is just a finite group, we deduce that the affine quotient the affine quotient 𝐻 0 (𝒯 qG,1 𝑌 )∕ Aut(𝑌) has dimension 2𝑎 − 3.…”

“…It is possible to give a proof of K‐polystability for a general hypersurface in $$\mathbb {P}(1,1,a,a)$$ of degree $$2a$$, when $$a$$ is odd, by analysing the deformation theory of the toric surface appearing in Proposition 3 similar to [26] and without using Theorem 1.…”