Equidistribution of zeros of holomorphic sections in the non compact setting

TL;DR

This paper extends the understanding of zero distribution of random holomorphic sections from compact to non-compact complex manifolds, under certain curvature bounds, with applications to number theory and orthogonal polynomials.

Contribution

It establishes a non-compact analogue of the equidistribution of zeros of random sections, including convergence rates and diverse applications.

Findings

Zeros of random sections become equidistributed in non-compact settings.

Provides convergence speed estimates for zero-divisors.

Applications include cusp forms, arithmetic quotients, and orthogonal polynomials.

Abstract

We consider N-tensor powers of a positive Hermitian line bundle L over a non-compact complex manifold X. In the compact case, B. Shiffman and S. Zelditch proved that the zeros of random sections become asymptotically uniformly distributed with respect to the natural measure coming from the curvature of L, as N tends to infinity. Under certain boundedness assumptions on the curvature of the canonical line bundle of X and on the Chern form of L we prove a non-compact version of this result. We give various applications, including the limiting distribution of zeros of cusp forms with respect to the principal congruence subgroups of SL2(Z) and to the hyperbolic measure, the higher dimensional case of arithmetic quotients and the case of orthogonal polynomials with weights at infinity. We also give estimates for the speed of convergence of the currents of integration on the zero-divisors.