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 $K_{ \widetilde{ X}}$.
Categories: math.CV math.DG
