Universality and scaling of zeros on symplectic manifolds

This article is concerned with random holomorphic polynomials and their generalizations to algebraic and symplectic geometry. A natural algebro-geometric generalization studied in our prior work involves random holomorphic sections $H^0(M,L^N)$ of the powers of any positive line bundle $L \to M$ over any complex manifold. Our main interest is in the statistics of zeros of $k$ independent sections (generalized polynomials) of degree $N$ as $N\to \mathrm{\infty }$. We fix a point $P$ and focus on the ball of radius $1\mathrm{/}\sqrt{N}$ about $P$. Under a microscope magnifying the ball by the factor $\sqrt{N}$, the statistics of the configurations of simultaneous zeros of random $k$-tuples of sections tends to a universal limit independent of $P,M,L$. We review this result and generalize it further to the case of pre-quantum line bundles over almost-complex symplectic manifolds $(M,J,\omega )$. Following [SZ2], we replace $H^0(M,L^N)$ in the complex case with the `asymptotically holomorphic’ sections defined by Boutet de Monvel-Guillemin and (from another point of view) by Donaldson and Auroux. Using a generalization to an $m$-dimensional setting of the Kac-Rice formula for zero correlations together with the results of [SZ2], we prove that the scaling limits of the correlation functions for zeros of random $k$-tuples of asymptotically holomorphic sections belong to the same universality class as in the complex case.