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

## TL;DR

This paper investigates the properties of $L^p$ Bergman spaces and $L^2$ cohomology on complex submanifolds in ${\mathbb C}^N$, establishing finiteness and infiniteness theorems and deriving curvature growth rigidity results.

## Contribution

It provides new finiteness theorems for $L^p$ Bergman spaces and $L^2$ cohomology on complex submanifolds, along with infiniteness results and curvature growth rigidity applications.

## Key findings

- Finiteness theorems for $L^p$ Bergman spaces on complex submanifolds.
- Infiniteness theorems testing the limits of finiteness results.
- Rigidity results relating curvature growth to geometric properties.

## 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.