Stability of the Bergman kernel on a tower of coverings

Abstract

We obtain several results about stability of the Bergman kernel on a tower of coverings on complex manifolds. An effective estimate for stability of the Bergman kernel is given for a tower of coverings on a compact Riemann surface of genus $\geq 2$. Stability of the Bergman kernel is established for towers of coverings on all hyperbolic Riemann surfaces and on complete Kähler manifolds that satisfy certain potential conditions. As a consequence, stability of the Bergman kernel is established for any tower of coverings of Riemann surfaces.