Random almost holomorphic sections of ample line bundles on symplectic manifolds
| dc.creator | Shiffman, Bernard | |
| dc.creator | Zelditch, Steve | |
| dc.date | 2000-01-19 | |
| dc.date | 2000-02-02 | |
| dc.date.accessioned | 2026-07-07T04:33:21Z | |
| dc.date.available | 2026-07-07T04:33:21Z | |
| dc.description | The spaces $H^0(M, L^N)$ of holomorphic sections of the powers of an ample line bundle $L$ over a compact Kähler manifold $(M,ω)$ 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 $(M, J, ω)$. 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 Szegö projector $Π_N$ onto $H^0_J(M, L^N)$, which also yields proofs of symplectic analogues of the Kodaira embedding theorem and Tian asymptotic isometry theorem. | |
| dc.description | Corrected an attribution and minor typos | |
| dc.identifier | https://arxiv.org/abs/math/0001102 | |
| dc.identifier | http://arxiv.org/abs/math/0001102 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/58544 | |
| dc.subject | Symplectic Geometry | |
| dc.subject | Mathematical Physics | |
| dc.subject | Complex Variables | |
| dc.subject | Probability | |
| dc.subject | 53C15 | |
| dc.title | Random almost holomorphic sections of ample line bundles on symplectic manifolds | |
| dc.type | text |