Random almost holomorphic sections of ample line bundles on symplectic manifolds

Abstract

The spaces $H^0(M,L^N)$ of holomorphic sections of the powers of an ample line bundle $L$ over a compact Kähler manifold $(M,\omega )$ have been generalized by Boutet de Monvel and Guillemin to spaces $H_J^0(M,L^N)$ of `almost holomorphic sections' of ample line bundles over an almost complex symplectic manifold $(M,J,\omega )$. We consider the unit spheres $S{H}_J^0(M,L^N)$ in the spaces $H_J^0(M,L^N)$, 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 $s_N\in S{H}_J^0(M,L^N)$ 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 Szegö projector ${\Pi }_N$ onto $H_J^0(M,L^N)$, which also yields proofs of symplectic analogues of the Kodaira embedding theorem and Tian asymptotic isometry theorem.

Corrected an attribution and minor typos