Infinite-Dimensionality of Bergman Spaces on\ Complete $d$-Bounded K\"ahler Manifolds

Abstract

Let $M$ be a complete K\"ahler manifold of complex dimension $n\geq 1$ whose K\"ahler form is $d$-bounded in the sense of Gromov. We prove that its Bergman space, namely the Hilbert space of $L^2$ holomorphic $n$-forms, is infinite-dimensional. The proof combines Gromov's construction of auxiliary holomorphic line bundles, parameter-uniform $\db$ estimates, and $L^2$ methods with singular weights.