Optimal test-configurations for toric varieties

Details Optimal test-configurations for toric varieties Optimal test-configurations for toric varieties

2007-09-17

arxiv.org/abs/0709.2687v1

J. Differential Geom. 80 (2008), 501--523

Mathematics

Differential Geometry

24 pages

Scientific paper

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.

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