Asymptotics of almost holomorphic sections on symplectic manifolds

Details Asymptotics of almost holomorphic sections on symplectic manifolds Asymptotics of almost holomorphic sections on symplectic manifolds

2002-12-12

arxiv.org/abs/math/0212180v1

J. Reine Angew. Math. 544 (2002), 181--222

Mathematics

Symplectic Geometry

A substantially revised and published version of our posted papter math.SG/0001102. Much more detail on the Szego projector

Scientific paper

We study the asymptotics of almost holomorphic sections $s \in H^0_J(M, \omega)$ of an ample line bundle $L \to M$ 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 $\bar{\partial}$ complex for a positive line bundle over a complex manifold. Our main result is the scaling limit asymptotics of the Szego projectors $\Pi_N$ 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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