Stability of the Bergman kernel on a tower of coverings (Q503067)

Let \(\widetilde M\) be a Riemannian manifold and let \(\Gamma\) be a free and properly discontinuous group of isometries of \(\widetilde M\). A tower of subgroups of \(\Gamma\) is a nested sequence of subgroups \(\Gamma=\Gamma_1\supset\Gamma_2\supset\dots\supset\Gamma_j\supset\dots\supset\bigcap\Gamma_j=\{\mathrm{id}\}\) such that \(\Gamma_j\) is a normal subgroup of \(\Gamma\) of finite index \([\Gamma:\Gamma_j]\) for each \(j\). The family of smooth manifolds \(M_j=\widetilde M/\Gamma_j\), equipped with the push-downs of the Riemannian metric on \(\widetilde M\), is called a tower of coverings on the Riemannian manifold \(M=\widetilde M/\Gamma\). \(\widetilde M\) is referred as the top manifold of the tower of coverings. In the paper under review the authors prove the following results. {\parindent=6mm \begin{itemize}\item[1.] Let \(M_j=\mathbb D/\Gamma_j\) be a tower of coverings on a compact Riemann surface of genus \(g\geq2\). Let \(\tau_j\) be the injectivity radius of \(M_j\) and let \(|\cdot |_{\mathrm{hyp}}\) be the pointwise length with respect to the hyperbolic metric. Then the Bergman kernel \(K_{M_j}\) of \(M_j\) satisfies \[ |4\pi|K_{M_j}|_{\mathrm{hyp}}-1|\leq\frac{12\cdot3^{2/3}}{\pi}(g-1)^{1/3}e^{-\tau_j/3}, \] when \(\tau_j\geq\log3\). Moreover, a similar estimate also holds for the Bergman metric. \item[2.] Let \(M\) and \(\widetilde M\) be complete Kähler manifolds with associated Kähler forms \(\omega\) and \(\widetilde\omega\), respectively. Let \(M_j=\widetilde M/\Gamma_j\) be a tower of coverings on \(M\). Then the tower is Bergman stable provided two certain potential conditions are satisfied. \end{itemize}} As a consequence of the last result the authors obtain the following.{\parindent=6mm \begin{itemize}\item[3.] Any tower of coverings of Riemann surfaces with a simply connected top manifold is Bergman stable. \end{itemize}}