Limit of Bergman kernels on a tower of coverings of compact K\"ahler manifolds

TL;DR

This paper proves the convergence of Bergman kernels and $L^2$-Hodge numbers on a tower of Galois coverings of compact K"ahler manifolds, with applications to embeddings via canonical sections.

Contribution

It establishes the convergence of Bergman kernels and $L^2$-Hodge numbers on Galois coverings of compact K"ahler manifolds, extending previous results to infinite coverings.

Findings

Bergman kernels converge on Galois tower coverings.

$L^2$-Hodge numbers stabilize in the tower.

Canonical sections induce projective immersions for large coverings.

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 K\"ahler 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 $K_{\widetilde{X}}$.