Asymptotics of almost holomorphic sections of ample line bundles on symplectic manifolds

Bernard Shiffman*, Steve Zelditch

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

73 Scopus citations

Abstract

In their work on symplectic manifolds, Donaldson and Auroux use analogues of holomorphic sections of an ample line bundle L over a symplectic manifold M to create symplectically embedded zero sections and almost holomorphic maps to various spaces. Their analogues were termed 'asymptotically holomorphic' sequences {sN} of sections of LN. We study another analogue Hj0(M, LN) of holomorphic sections, which we call 'almost-holomorphic' sections, following a method introduced earlier by Boutet de Monvel-Guillemin [BoGu] in a general setting of symplectic cones. By definition, sections in Hj0(M, LN) lie in the range of a Szegö projector ΠN. Starting almost from scratch, and only using almost-complex geometry, we construct a simple parametrix for ΠN of precisely the same type as the Boutet de Monvel-Sjöstrand parametrix in the holomorphic case [BoSj]. We then show that ΠN(x, y) has precisely the same scaling asymptotics as does the holomorphic Szegö kernel as analyzed in [BSZ1]. The scaling asymptotics imply more or less immediately a number of analogues of well-known results in the holomorphic case, e.g. a Kodaira embedding theorem and a Tian almost-isometry theorem. We also explain how to modify Donaldson's constructions to prove existence of quantitatively transverse sections in Hj0(M, LN).

Original languageEnglish (US)
Pages (from-to)181-222
Number of pages42
JournalJournal fur die Reine und Angewandte Mathematik
Issue number544
DOIs
StatePublished - 2002

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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