Poincare-Lelong approach to universality and scaling of correlations between zeros

This note is concerned with the scaling limit as N --> infinity of n-point correlations between zeros of random holomorphic polynomials of degree N in m variables. More generally we study correlations between zeros of holomor phic sections of powers L-N of any positive holomorphic line bundle L over a compact Kahler manifold. Distances are rescaled so that the average densi ty of zeros is independent of N. Our main result is that the scaling limits of the correlation functions and, more generally, of the "correlation form s" are universal, i.e. independent of the bundle L, manifold M or point on M.