Random Almost Holomorphic Sections of Ample Line Bundles on Symplectic Manifolds

The spaces H(M, L ) of holomorphic sections of the powers of an ample line bundle L over a compact Kähler manifold (M, ω) have been generalized by Boutet de Monvel and Guillemin to spaces H J (M, L ) of ‘almost holomorphic sections’ of ample line bundles over an almost complex symplectic manifold (M, J, ω). We consider the unit spheres SH J(M, L N ) in the spaces H 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 → ∞. Our first main result is that almost all sequences of sections sN ∈ SH J(M, L ) are ‘asymptotically holomorphic’ in the Donaldson-Auroux sense that ||sN ||∞/||sN ||2 = O( √ logN), ||∂̄sN ||∞/||sN ||2 = O( √ logN) and ||∂sN ||∞/||sN ||2 = O( √ N logN). Our second main result concerns the joint probability distribution of the random variables sN (z ), ∇sN (z), 1 ≤ p ≤ n, for n distinct points z, . . . , z in a neighborhood of a point P0 ∈ M . We show that this joint distribution has a universal scaling limit about P0 as N → ∞. 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 ΠN onto H 0 J (M, L ), which also yields proofs of symplectic analogues of the Kodaira embedding theorem and Tian asymptotic isometry theorem.