Curvature and $L^p$ Bergman spaces on complex submanifolds in ${\mathbb{C}}^N$

Let $M$ be a closed complex submanifold in ${\mathbb{C}}^N$ with the complete K"ahler metric induced by the Euclidean metric. Several finiteness theorems on the $L^p$ Bergman space of holomorphic sections of a given Hermitian line bundle $L$ over $M$ and the associated $L^2$ cohomology groups are obtained. Some infiniteness theorems are also given in order to test the accuracy of finiteness theorems. As applications we obtain some rigidity results concerning growth of curvatures.