Umehara algebra and complex submanifolds of indefinite complex space forms

TL;DR

This paper investigates the Umehara algebra to understand the non-existence of common complex submanifolds in indefinite complex space forms, providing new theoretical results and applications to complex geometry.

Contribution

It establishes new results on Umehara algebra and demonstrates the non-existence of certain holomorphic isometric immersions between indefinite complex space forms.

Findings

Holomorphic polynomial isometric immersions between different indefinite complex space forms are impossible.

No common complex submanifolds exist for indefinite complex projective or hyperbolic spaces.

Results apply to various complex manifolds with intrinsic Bergman metrics, extending previous work.

Abstract

The Umehara algebra is studied with motivation on the problem of the non-existence of common complex submanifolds. In this paper, we prove some new results in Umehara algebra and obtain some applications. In particular, if a complex manifolds admits a holomorphic polynomial isometric immersion to one indefinite complex space form, then it cannot admits a holomorphic isometric immersion to another indefinite complex space form of different type. Other consequences include the non-existence of the common complex submanifolds for indefinite complex projective space or hyperbolic space and a complex manifold with a distinguished metric, such as homogeneous domains, the Hartogs triangle, the minimal ball, the symmetrized polydisc, etc, equipped with their intrinsic Bergman metrics, which generalizes more or less all existing results.