## 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 {s_{N}} of sections of L^{N}. We study another analogue H_{j}^{0}(M, L^{N}) 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 H_{j}^{0}(M, L^{N}) lie in the range of a Szegö projector Π_{N}. Starting almost from scratch, and only using almost-complex geometry, we construct a simple parametrix for Π_{N} of precisely the same type as the Boutet de Monvel-Sjöstrand parametrix in the holomorphic case [BoSj]. We then show that Π_{N}(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 H_{j}^{0}(M, L^{N}).

Original language | English (US) |
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Pages (from-to) | 181-222 |

Number of pages | 42 |

Journal | Journal fur die Reine und Angewandte Mathematik |

Issue number | 544 |

DOIs | |

State | Published - 2002 |

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics