Asymptotics of almost holomorphic sections of ample line bundles on symplectic manifolds: An addendum

By Bernard Shiffman, Steve Zelditch. 2002

139 citations recorded by the source corpus.

We define a Gaussian measure on the space H 0 J (M, L N ) of almost holomorphic sections of powers of an ample line bundle L over a symplectic manifold (M, ω), and calculate the joint probability densities of sections taking prescribed values and covariant derivatives at a finite number of points. We prove that they have a universal scaling limit as N → ∞. This result will be used in another paper to extend our previous work on universality of scaling limits of correlations between zeros to the almost-holomorphic setting.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fda067ff3a4a95bb4f0ed2666f2c904243542cbd

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In Applied Mathematics

1. G. Marinescu
2. Xiaonan Ma
3. Bernard Shiffman
4. Steve Zelditch
5. Kefeng Liu
6. Xiaonan Ma
7. X. Dai
8. Zhiqin Lu

In Geometry and Topology

1. G. Marinescu
2. Xiaonan Ma
3. Bernard Shiffman
4. Steve Zelditch
5. Kefeng Liu
6. Xiaonan Ma
7. X. Dai
8. Zhiqin Lu

In Mathematical Physics

1. G. Marinescu
2. Xiaonan Ma
3. Bernard Shiffman
4. Kefeng Liu
5. X. Dai
6. Xiaonan Ma
7. Steve Zelditch
8. Zhiqin Lu

Citations in the corpus

• Number Variance of Random Zeros on Complex Manifolds (2006, methodology): “Sk(z)Sk(w) = Nm πm eiN(θ −ϕ)+Nz·w¯ N 2 ( |z 2+ w ) (16) of level Nfor the Bargmann-Fock space of functions on Cm (see [BSZ2]). We shall apply the following (near and far) off-diagonal asymptotics from [SZ2]: NUMBER VARIANCE OF RANDOM ZEROS ON COMPLEX MANIFOLDS 11 Theorem 2.4. Let (L,h) → (M,ω) be as ”
• Random polynomials of high degree and Levy concentration of measure (2003, methodology): “Szeg¨o kernels. It is proved in [Ca, Ze2] in the holomorphic case and [ ShZe2] in the almostholomorphic case that there exists a complete asymptotic expansion: �N(z,0; z,0) = a0N m + a1(z)N m−1 + a2(z)N m−2 + . . . (13) for certain smooth coefficientsaj(z) with a0 = π −m . Hence, the maps �N are we”
• Generalized Bergman kernels on symplectic manifolds (2004, methodology): “We have thus a unified treatment of the convergence of the induced Fubini–Study metric [10,12,43,48], the holomorphic Morse inequalities [2,8,22] and the characterization of Moishezon spaces [8,22,29,45] from the point of view of Bergman kernels.”
• On the asymptotic expansion of Bergman kernel (2004, methodology): “he holomorphic case, the Szeg¨o kernels are exactly (modulo smooth operators) the Szeg¨o kernel associated to the holomorphic sections by Boutet de Monvel-Sj¨ostrand [13]. In the very important paper [30], Shiffman and Zelditch also gave a simple way to construct first the Szeg¨o kernels, then the ope”