On the dimension of bundle-valued Bergman spaces on compact Riemann surfaces (Q6631924)

Let \((M,g)\) be a compact Riemann surface equipped with a Hermitian metric, and \((E,h)\) be a Hermitian holomorphic vector bundle over \(M.\) For a nonempty open subset \(D \subset M\) let \(A^2(D,g; E,h)\) denote the Bergman space of \(E\)-valued holomorphic sections \(s\in \mathcal O (D;E)\) such that \(\|s\|^2_h := \int_D h(s,s) \omega_g <\infty\), where \(\omega_g\) is the volume form associated with \(g\). \N\NThe authors indicate that, as a vector space, \(A^2(D,g; E,h)=A^2(D; E)\) is independent of the choice of \(g\) and \(h,\) and they prove that the following are equivalent: (a) \(M\setminus D\) is nonpolar, (b) \(\mathcal O (D;E)\) is strictly contained in \(A^2(D;E),\) (c) \(\dim A^2(D; E)= \infty,\) (d) there exists a bounded subharmonic function \(\psi \in \mathcal C^\infty (D)\) such that \(i \partial \bar \partial \psi \ge \omega\) on \(D\) for some volume form \(\omega\) on \(M\). \N\NThe proof reduces to showing that (a) implies (d), and (d) implies (c), where the authors use Green's function to obtain the desired strictly subharmonic function and a version of Hörmander's \(L^2\)-method for solving the Cauchy-Riemann equations for bundle-valued forms due to Demailly.