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

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 $p\geq 0$.