Limit of Bergman kernels on a tower of coverings of compact Kähler manifolds

Abstract

We prove the convergence of the Bergman kernels and the \(L^{2}\)-Hodge numbers on a tower of Galois coverings \(\{X_j\}\) of a compact Kähler manifold X converging to an infinite Galois (not necessarily universal) covering \({\widetilde{X}}\). We also show that, as an application, sections of canonical line bundle \(K_{X_j}\) for sufficiently large j give rise to an immersion into some projective space, if so do sections of \(K_{{\widetilde{X}}}\).