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

Abstract

We define a Gaussian measure on the space $H_J^0(M,L^N)$ 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)

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