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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 cre- ate 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 H0 J(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 H0 J(M, LN) lie in the range of a Szego projectorN. Starting almost from scratch, and only using almost- complex geometry, we construct a simple parametrix forN of precisely the same type as the Boutet de Monvel-Sjostrand parametrix in the holomorphic case (BoSj). We then show that N(x, y) has precisely the same scaling asymptotics as does the holomorphic Szego 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 the- orem and a Tian almost-isometry theorem. We also explain how to modify Donaldson's constructions to prove existence of quantitatively transverse sections in H0 J(M, L N).

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