On toric geometry and K-stability of Fano varieties

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.

Toric Degenerations, Tropical Curve, and Gromov–Witten Invariants of Fano Manifolds

In this paper, we give a tropical method for computing Gromov–Witten type invariants of Fano manifolds of special type. This method applies to those Fano manifolds that admit toric degenerations to toric Fano varieties with singularities allowing small resolutions. Examples include (generalized) flag manifolds of type A and some moduli space of rank two bundles on a genus two curve.