Seminar: Filippo Valnegri (G.15.25)
Let $V\subset\mathbb{C}^n$ be an open subset and let $Z\subset V$ be a closed one. We say that $Z$ is a \emph{$q$-pseudoconcave subset} of V (for $1\leq q\leq n$) if $V\setminus Z$ is $(n-q-1)$-pseudoconvex in $V$, in the sense of Słodkowski. This is equivalent to ask that for each $z\in Z$ there exists a neighbourhood $U\subset\mathbb{C}^n$ of $z$ such that, for every compact $K\subset U$ and for every $(q-1)$-plurisubharmonic function $\phi:U\to\mathbb{R}$, the following condition is satisfied
\[
\max_{Z\cap K} \phi=\max_{Z\cap bK} \phi
\]
where $bK$ denotes the topological boundary of $K$.
In 2001, Chirka proved that if $D\subset\mathbb{C}^n\times\mathbb{R}$ is a domain and $g:D\to\mathbb{R}$ is a continuous function, then the graph $\Gamma(g)$ of $g$ is an $n$-pseudoconcave subset of $D\times i\mathbb{R}\subset\mathbb{C}^{n+1}$ if and only if $\Gamma(g)$ is the disjoint union of complex hypersurfaces.
In 2022, Pawlaschyk and Shcherbina extended this result to graphs of higher codimension, i.e., to graphs given by continuous functions of the form $g:D\to\mathbb{R}\times\mathbb{C}^p$ (for $p\geq 0$).
In this talk, we will discuss another extension of Chirka's result. Namely, let $D\subset\mathbb{C}^n\times\mathbb{R}$ be a domain and let $g:D\to\mathbb{R}$ be a continuous function. If $Z_0\subset\Gamma(g)$ is a closed $n$-pseudoconcave subset, then there exists a unique foliation on $Z_0$ given by $n$-dimensional complex manifolds. The same conclusion can be made also for graphs of higher codimension, as in Pawlaschyk and Shcherbina's case.
As an application, we obtain a foliation on closed $n$-pseudoconcave subsets of $(2n+1)$-dimensional real $\mathscr{C}^1$-submanifolds of $\mathbb{C}^N$ (for $N>n$).
