Optimal test-configurations for toric varieties

Gabor Szekelyhidi Imperial College London

Differential Geometry mathscidoc:1609.10088

Journal of Differential Geometry, 80, (3), 501-523, 2008
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 minimizes the Calabi functional. In this case the infimum of the Calabi functional is given by the supremum of the normalized Futaki invariants over all destabilising test-configurations, as predicted by a conjecture of Donaldson.
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@inproceedings{gabor2008optimal,
  title={OPTIMAL TEST-CONFIGURATIONS FOR TORIC VARIETIES},
  author={Gabor Szekelyhidi},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160909141953229820745},
  booktitle={Journal of Differential Geometry},
  volume={80},
  number={3},
  pages={501-523},
  year={2008},
}
Gabor Szekelyhidi. OPTIMAL TEST-CONFIGURATIONS FOR TORIC VARIETIES. 2008. Vol. 80. In Journal of Differential Geometry. pp.501-523. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160909141953229820745.
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