Optimal test-configurations for toric varieties

TL;DR

This paper investigates optimal destabilising functions on K-unstable toric varieties, linking them to geometric decompositions and flow minimization, confirming conjectures relating stability invariants and flow behavior.

Contribution

It introduces the concept of an optimal destabilising convex function on toric varieties and connects it to geometric decompositions and flow minimization, advancing stability theory.

Findings

Existence of an optimal destabilising convex function on K-unstable toric varieties.

Piecewise linear destabilising functions induce a Harder-Narasimhan type decomposition.

Calabi flow minimizes the Calabi functional when it exists for all time, aligning with Futaki invariants.

Abstract

On a K-unstable toric variety we show the existence of an optimal destabilising convex function. We show that if this is piecewise linear then it gives rise to a decomposition into semistable pieces analogous to the Harder-Narasimhan filtration of an unstable vector bundle. We also show that if the Calabi flow exists for all time on a toric variety then it minimises the Calabi functional. In this case the infimum of the Calabi functional is given by the supremum of the normalised Futaki invariants over all destabilising test-configurations, as predicted by a conjecture of Donaldson.