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 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.

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