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

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_j⁰(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_j⁰(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_j⁰(M, L^N).