Infinite-Dimensionality of Bergman Spaces on\ Complete $d$-Bounded Kähler Manifolds

Let $M$ be a complete Kähler manifold of complex dimension $n\ge 1$ whose Kähler 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 $\text{db}$ estimates, and $L^2$ methods with singular weights.