Weight sensitivity in K-stability of Fano varieties

TL;DR

This paper investigates the relationship between weighted K-stability and the Futaki invariant in Fano varieties, revealing surprising differences in certain threefolds and providing explicit examples and generalizations.

Contribution

It establishes conditions under which weighted K-polystability is equivalent to Futaki invariant vanishing for specific Fano threefolds and extends the analysis to higher dimensions.

Findings

Weighted K-polystability equals Futaki invariant vanishing in most spherical Fano threefolds.

Counterexamples are found for the Fano threefold 2-29 and certain torus actions.

The results are generalized to higher-dimensional quadrics and their blowups.

Abstract

We prove that, for a spherical Fano threefold not in the Mori-Mukai family 2-29, and a weight function associated with the action of the connected center of a Levi subgroup of its automorphism group, weighted K-polystability is equivalent to vanishing of the weighted Futaki invariant. This is surprising since unlike the case of toric Fano manifold, there exist non-product, special, equivariant test configurations. For the K\"ahler-Einstein Fano threefold 2-29, and for well-chosen torus action on the three dimensional quadric, we show that this property is false and exhibit explicit examples of weighted optimal degenerations. We then generalize this to higher-dimensional quadrics and blowups of quadrics along a codimension 2 subquadric.