Analytic structure of $q$-pseudoconcave subsets of continuous graphs

TL;DR

This paper demonstrates that certain $n$-pseudoconcave subsets of continuous graphs in complex space can be decomposed into disjoint unions of complex manifolds, revealing their analytic structure.

Contribution

It establishes that $n$-pseudoconcave subsets of continuous graphs are unions of complex manifolds, extending previous understanding of their geometric and analytic properties.

Findings

$Z$ can be realized as a disjoint union of $n$-dimensional complex manifolds.

The result applies to subsets of graphs of continuous functions over specific domains.

The conclusion holds for subsets of graphs of continuous functions with certain dimensional conditions.

Abstract

Let $Z\subset\mathbb{C}^N$ be an $n$-pseudoconcave subset, for $1\leq n<N$, which is locally the graph of a continuous function over a closed subset of $\mathbb{C}^n\times\mathbb{R}$. We show that $Z$ can be realised as the disjoint union of $n$-dimensional complex manifolds. In particular, the same conclusion can be made for any $n$-pseudoconcave subset $Z$ of the graph $\Gamma(g)$ of a continuous function $g:D\subset\mathbb{C}^n\times\mathbb{R}\to\mathbb{R}\times\mathbb{C}^p$, for $n\geq 1$ and $p\geq 0$.