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.

中文翻译：

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$$ \ pmb {{\ mathbb {C}} ^ N} $$ CN中复杂子流形上的曲率和$$ L ^ p $$ L p Bergman空间

令M为\（{\ mathbb {C}} ^ N \）中的闭合复子流形，具有由欧几里得度量导出的完整Kähler度量。得到了给定的Hermitian线束L在M上的全同形部分的\（L ^ p \） Bergman空间上的几个有限性定理，以及相关的\（L ^ 2 \）同调群。为了检验有限性定理的准确性，还给出了一些无限性定理。作为应用，我们获得一些有关曲率增长的刚度结果。