Global $L^2$ Approximation in the Bergman Spaces with Values in Holomorphic Vector Bundles

In this paper, we get global $L^2$ approximation theorems in Bergman spaces with values in singular holomorphic vector bundles over Stein manifolds generalizing results of Li and Zhou. The main results combine strong openness property of multiplier submodule sheaf and twisted $L^2$ estimates of $\bar{\mathrm{\partial }}$-equations.