Universality and Scaling of Correlations Between Zeros on Complex Manifolds

Pavel Bleher, Bernard Shiffman, and Steve Zelditch

We study the limit as $N\to\infty$ of the correlations between simultaneous zeros of random sections of the powers $L^N$ of a positive holomorphic line bundle $L$ over a compact complex manifold $M$, when distances are rescaled so that the average density of zeros is independent of $N$. We show that the limit correlation is independent of the line bundle and depends only on the dimension of $M$ and the codimension of the zero sets. We also provide some explicit formulas for pair correlations. In particular, we provide an alternate derivation of Hannay's limit pair correlation function for SU$(2)$ polynomials, and we show that this correlation function holds for all compact Riemann surfaces.