Convergence of Bergman measures for high powers of a line bundle

2008/05/19 by Berman, Robert, Nystrom, David Witt · 1 citation
#32A25 #32U20 #42C05 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.0805.2846

Abstract

Let L be a holomorphic line bundle on a compact complex manifold X of dimension n, and let e^-ϕ 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,ϕ), where K is the support of dμ, as long as dμ is stably Bernstein-Markov with respect to (K,ϕ). 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.

Cited by

• Quantization for Semipositive Adjoint Line Bundles

Related

• Bergman kernels and equilibrium measures for line bundles over projective manifolds
• Bergman kernels and equilibrium measures for ample line bundles
• Bergman kernels and equilibrium measures for polarized pseudoconcave domains
• Upper bounds for Bergman kernels associated to positive line bundles with smooth Hermitian metrics
• Bergman kernel estimates and Toeplitz operators on holomorphic line bundles