We study the limit as N --> infinity of the correlations between simultaneo
us zeros of random sections of the powers L-N of a positive holomorphic lin
e bundle L over a compact complete 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 prove that Hanna
y's limit pair correlation function for SU(2) polynomials holds for all com
pact Riemann surfaces.