Submanifolds of some Hartogs domain and the complex Euclidean space

TL;DR

This paper proves that certain Hartogs domains over symmetric spaces are not compatible with complex Euclidean spaces as shared Kähler submanifolds, extending previous results to cases with non-Nash algebraic Bergman kernels.

Contribution

It demonstrates that Hartogs domains over irreducible bounded symmetric domains with Bergman metrics are not relatives of complex Euclidean spaces, even when the Bergman kernel is not Nash algebraic.

Findings

Hartogs domains over symmetric spaces are not relatives of Euclidean space

The result holds without the Nash algebraic condition on the Bergman kernel

Extends previous compatibility results to more general Bergman kernels

Abstract

Two Kahler manifolds are called relatives if they admit a common Kahler submanifold with the same induced metrics. In this paper, we show that a Hartogs domain over an irreducible bounded symmetric domain equipped with the Bergman metric is not a relative to the complex Euclidean space. This generalizes the results in [5, 4] and the novelty here is that the Bergman kernel of the Hartogs domain is not necessarily Nash algebraic.