  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.
