On hyperbolicity and tautness modulo an analytic subset of Hartogs domains

TL;DR

This paper establishes necessary and sufficient conditions for the hyperbolicity and tautness of Hartogs-type domains over complex spaces, considering the influence of an analytic subset, thereby advancing understanding of complex geometric properties.

Contribution

It provides a comprehensive characterization of hyperbolicity and tautness modulo an analytic subset for Hartogs domains, extending previous results to more general settings.

Findings

Conditions for hyperbolicity modulo an analytic subset

Conditions for tautness modulo an analytic subset

Corollaries for classical Hartogs domains

Abstract

Let $X$ be a complex space and $H$ a positive homogeneous plurisubharmonic function $H$ on $X\times\C^m$. Consider the Hartogs-type domain $\Omega_{H}(X):=\{(z,w)\in X\times \C^m:H(z,w)<1 \}$. Let $S$ be an analytic subset of $X$. We give necessary and sufficient conditions for hyperbolicity and tautness modulo $S\times \C^m$ of $\Omega_{H}(X)$, with the obvious corollaries for the special case of Hartogs domains.