Toric partial density functions and stability of toric varieties
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
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.
Constant Scalar Curvature, Tian-Yau-Zelditch, Projective Embeddings, Asymptotic-Expansion, Bergman-Kernel, Kahler-Metrics, Line Bundles, Manifolds, Polytopes, Geometry
IdentifiersURN: urn:nbn:se:kth:diva-131309DOI: 10.1007/s00208-013-0978-2ISI: 000332792900011ScopusID: 2-s2.0-84897655759OAI: oai:DiVA.org:kth-131309DiVA: diva2:655716
QC 20140414. Updated from accepted to published.2013-10-132013-10-132014-04-14Bibliographically approved