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

Bo-Yong Chen; Yuanpu Xiong

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

Complex Variables 2020-09-21 v2

Authors: Bo-Yong Chen , Yuanpu Xiong

Abstract

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.

Keywords

complex submanifold kahler metric l^p bergman space hermitian line bundle l^2 cohomology finiteness theorem rigidity curvature growth

Cite

@article{arxiv.1907.11873,
  title  = {Curvature and $L^p$ Bergman spaces on complex submanifolds in ${\mathbb C}^N$},
  author = {Bo-Yong Chen and Yuanpu Xiong},
  journal= {arXiv preprint arXiv:1907.11873},
  year   = {2020}
}

Comments

Corollary 1.4, Theorem 1.5 and Theorem 1.6 are new

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

Abstract

Keywords

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