Convergence of Bergman measures of high powers of a line bundle

Preprint, 2008

Let L be a holomorphic line bundle on a compact complex manifold X of dimension n, and let exp(-\phi) be a continuous metric on L. Fixing a measure dμ on X gives a sequence of Hilbert spaces consisting of holomorphic sections of tensor powers of L. We prove that the corresponding sequence of scaled Bergman measures converges, in the high tensor power limit, to the equilibrium measure of the pair (K,\phi), where K is the support of dμ, as long as dμ is stably Bernstein-Markov with respect to (K,\phi). Here the Bergman measure denotes dμ times the restriction to the diagonal of the pointwise norm of the corresponding orthogonal projection operator. In particular, an extension to higher dimensions is obtained of results concerning random matrices and classical orthogonal polynomials.