Mathematische Annalen

, Volume 358, Issue 3–4, pp 879–923 | Cite as

Toric partial density functions and stability of toric varieties

Article

Abstract

Let \((L, h)\rightarrow (X, \omega )\) denote a polarized toric Kähler manifold. Fix a toric submanifold \(Y\) and denote by \(\hat{\rho }_{tk}:X\rightarrow \mathbb {R}\) the partial density function corresponding to the partial Bergman kernel projecting smooth sections of \(L^k\) onto holomorphic sections of \(L^k\) that vanish to order at least \(tk\) along \(Y\), for fixed \(t>0\) such that \(tk\in \mathbb {N}\). We prove the existence of a distributional expansion of \(\hat{\rho }_{tk}\) as \(k\rightarrow \infty \), including the identification of the coefficient of \(k^{n-1}\) as a distribution on \(X\). This expansion is used to give a direct proof that if \(\omega \) has constant scalar curvature, then \((X, L)\) must be slope semi-stable with respect to \(Y\) (cf. Ross and Thomas in J Differ Geom 72(3): 429–466, 2006). Similar results are also obtained for more general partial density functions. These results have analogous applications to the study of toric K-stability of toric varieties.

Mathematics Subject Classification (2000)

32Q15 32A25 14M25 53D20 53C21 58Jxx 

Notes

Acknowledgments

We thank Julius Ross, Richard Thomas and Steve Zelditch for useful conversations. The second author was supported by a Leverhulme Research Fellowship while this work was being completed.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Computer Vision and Active Perception Laboratory/Centre for Autonomous Systems, School of Computer Science and CommunicationKTH Royal Institute of TechnologyStockholmSweden
  2. 2.Department of MathematicsUniversity College LondonGreater LondonUK

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