Asymptotics of almost holomorphic sections on symplectic manifolds

2002/12/12 by B. Shiffman, S. Zelditch · 1 citation
Mathematics · #math.SG #math.PR #msc:53C15

paper · pdf

published as J. Reine Angew. Math. 544 (2002), 181--222 · A substantially revised and published version of our posted papter math.SG/0001102. Much more detail on the Szego projector

arxiv created 2002/12/12 · arxiv updated 2009/11/30

Abstract

We study the asymptotics of almost holomorphic sections s ∈ H⁰_J(M, ω) of an ample line bundle L → M over an almost complex symplectic manifold in the sense of Boutet de Monvel-Guillemin. Such sections are defined as the kernel of a complex which is analogous to the ∂ complex for a positive line bundle over a complex manifold. Our main result is the scaling limit asymptotics of the Szego projectors Π_N of powers L^N. The Kodaira embedding theorem and Tian almost isometry theorem are almost immediate consequences of the scaling limit. We also relate such almost holomorphic sections to the asymptotically holomorphic sections in the sense of Donaldson and Auroux.

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• Fluctuations of two-dimensional determinantal processes associated with Berezin--Toeplitz operators

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