Torus equivariant K-stability

TL;DR

This paper investigates the conjecture that torus-equivariant test configurations suffice for checking K-polystability of polarized varieties, providing partial results and implications for constant scalar curvature manifolds.

Contribution

It offers partial proofs supporting the conjecture and demonstrates how this approach could simplify proving K-polystability.

Findings

Partial results towards the torus-equivariant K-stability conjecture

Implication that this approach could yield a new proof of K-polystability for constant scalar curvature manifolds

Potential simplification in testing K-polystability using torus-equivariant configurations

Abstract

It is conjectured that to test the K-polystability of a polarised variety it is enough to consider test-configurations which are equivariant with respect to a torus in the automorphism group. We prove partial results towards this conjecture. We also show that it would give a new proof of the K-polystability of constant scalar curvature polarised manifolds.