A note on envelopes of holomorphy

TL;DR

This paper investigates the relationship between different notions of envelopes of holomorphy for Riemann domains over complex submanifolds, clarifying their connections in complex analysis.

Contribution

It establishes the relations between the $\

Findings

Clarifies the connection between envelopes of holomorphy in different contexts

Provides a framework for comparing envelopes over Riemann domains and complex manifolds

Abstract

Let $p:X\longrightarrow M$ be a Riemann domain over a connected $n$-dimensional complex submanifold $M$ of $\mathbb C^N$ and let $\mathcal F\subset\mathcal O(X)$ be such that $p\in\mathcal F^N$. Our aim is to discuss relations between the $\mathcal F$-envelope of holomorphy of $(X,p)$ in the sense of Riemann domains over $M$ and the $\mathcal F$-envelope of holomorphy of $X$ in the sense of complex manifolds.