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

Abstract

Let $M$ be a closed complex submanifold in ${\mathbb{C}}^N$ with the complete Kähler 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.

Corollary 1.4, Theorem 1.5 and Theorem 1.6 are new