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

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

2002-12-12

arxiv.org/abs/math/0212181v1

Proc. Amer. Math. Soc. 131 (2003), no. 1, 291--302

Mathematics

Symplectic Geometry

Addendum to math.SG/0212180. Supplements and completes results of math-ph/0002039

Scientific paper

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 $(M, \omega)$, 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 $N \to \infty$. This result completes our proof (with P. Bleher) that correlations between zeros of sections in the almost-holomorphic setting have the same universal scaling limit as in the complex case (see Universality and scaling of zeros on symplectic manifolds, Random matrix models and their applications, 31--69, Math. Sci. Res. Inst. Publ., 40)

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