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Toric partial density functions and stability of toric varieties
KTH, School of Computer Science and Communication (CSC), Computer Vision and Active Perception, CVAP. KTH, School of Computer Science and Communication (CSC), Centres, Centre for Autonomous Systems, CAS.ORCID iD: 0000-0003-1114-6040
2014 (English)In: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, Vol. 358, no 3-4, 879-923 p.Article in journal (Refereed) Published
Abstract [en]

Let (L, h) -> (X, omega) denote a polarized toric Kahler manifold. Fix a toric submanifold Y and denote by (rho) over cap (tk) : X -> 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 is an element of N. We prove the existence of a distributional expansion of (rho) over cap (tk) as k -> infinity, 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.

Place, publisher, year, edition, pages
2014. Vol. 358, no 3-4, 879-923 p.
Keyword [en]
Constant Scalar Curvature, Tian-Yau-Zelditch, Projective Embeddings, Asymptotic-Expansion, Bergman-Kernel, Kahler-Metrics, Line Bundles, Manifolds, Polytopes, Geometry
National Category
Mathematics Geometry
URN: urn:nbn:se:kth:diva-131309DOI: 10.1007/s00208-013-0978-2ISI: 000332792900011ScopusID: 2-s2.0-84897655759OAI: diva2:655716

QC 20140414. Updated from accepted to published.

Available from: 2013-10-13 Created: 2013-10-13 Last updated: 2014-04-14Bibliographically approved

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Pokorny, Florian T.
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Computer Vision and Active Perception, CVAPCentre for Autonomous Systems, CAS
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