Random almost holomorphic sections of ample line bundles on symplectic manifolds

2000/01/19 by Bernard Shiffman, Steve Zelditch, Shiffman, Bernard +1 · 1 citation
Mathematics · Physics and Astronomy · #53C15 #Advanced Algebra and Geometry #Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Physics (math-ph) #Probability (math.PR) #Symplectic Geometry (math.SG) #math-ph #math.CV #math.MP #math.PR #math.SG #msc:53C15

paper · pdf · doi:10.48550/arxiv.math/0001102

Corrected an attribution and minor typos

openalex publication_date 2000/01/19 · arxiv created 2000/02/02 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The spaces H⁰(M, L^N) 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 H⁰_J(M, L^N) of `almost holomorphic sections' of ample line bundles over an almost complex symplectic manifold (M, J, ω). We consider the unit spheres SH<sup>0</sup><sub>J</sub>(M, L<sup>N</sup>) in the spaces H<sup>0</sup><sub>J</sub>(M, L<sup>N</sup>), 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 s<sub>N</sub> ∈ SH<sup>0</sup><sub>J</sub>(M, L<sup>N</sup>) are `asymptotically holomorphic' in the Donaldson-Auroux sense that ||s_N||_∞/||s_N||₂ = O(√(log N)), ||∂ s_N||_∞/||s_N||₂ = O(√(log N)) and ||∂ s_N||_∞/||s_N||₂ = O(√(N log N)). Our second main result concerns the joint probability distribution of the random variables s_N(z^p), ∇ s_N(z^p), 1≤ p≤ n, for n distinct points z¹,..., zⁿ 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 Π_N onto H⁰_J(M, L^N), which also yields proofs of symplectic analogues of the Kodaira embedding theorem and Tian asymptotic isometry theorem.

Citations

• Nearly Kahlerian Embeddings of Symplectic Manifolds

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• Szego Kernels and Finite Group Actions

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