Random almost holomorphic sections of ample line bundles on symplectic manifolds

Details Random almost holomorphic sections of ample line bundles on symplectic manifolds Random almost holomorphic sections of ample line bundles on symplectic manifolds

2000-01-19

arxiv.org/abs/math/0001102v2

Mathematics

Symplectic Geometry

Corrected an attribution and minor typos

Scientific paper

The spaces $H^0(M, L^N)$ of holomorphic sections of the powers of an ample line bundle $L$ over a compact K\"ahler manifold $(M,\omega)$ have been generalized by Boutet de Monvel and Guillemin to spaces $H^0_J(M, L^N)$ of `almost holomorphic sections' of ample line bundles over an almost complex symplectic manifold $(M, J, \omega)$. We consider the unit spheres $SH^0_J(M, L^N)$ in the spaces $H^0_J(M, L^N)$, which we equip with natural inner products. Our purpose is to show that, in a probabilistic sense, almost holomorphic sections behave like holomorphic sections as $N \to \infty$. Our first main result is that almost all sequences of sections $s_N \in SH^0_J(M, L^N)$ are `asymptotically holomorphic' in the Donaldson-Auroux sense that $||s_N||_{\infty}/||s_N||_{2} = O(\sqrt{\log N})$, $||\bar{\partial} s_N||_{\infty}/||s_N||_{2} = O(\sqrt{\log N})$ and $||\partial s_N||_{\infty}/||s_N||_{2} = O(\sqrt{N \log N})$. Our second main result concerns the joint probability distribution of the random variables $s_N(z^p),\ \nabla s_N(z^p)$, $1\le p\le n$, for $n$ distinct points $z^1,..., z^n$ in a neighborhood of a point $P_0\in M$. We show that this joint distribution has a universal scaling limit about $P_0$ as $N \to \infty$. In particular, the limit is precisely the same as in the complex holomorphic case. Our methods involve near-diagonal scaling asymptotics of the Szeg\"o projector $\Pi_N$ onto $H^0_J(M, L^N)$, which also yields proofs of symplectic analogues of the Kodaira embedding theorem and Tian asymptotic isometry theorem.

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