which enjoy appropriate convexity conditions, using the method of analytic discs for local extensional steps and Morse-theoretical tools for the global topological control of monodromy.

In its original form, the theorem states that in an arbitrary bounded domain $\Omega \Subset {\mathbb{C}}^n$ ($n \geqslant 2$), every compact set $K
\subset \Omega$ with $\Omega \backslash K$ connected is an illusory singularity for holomorphic functions, namely $\mathcal{ O}( \Omega
\backslash K) = \mathcal{ O}( \Omega ) \big\vert_{ \Omega \backslash
K}$ (for history, motivations and background, we refer [*e.g.*]{} to [@hm2002; @ra1986; @ra2002]). By now, the shortest proof, due to Ehrenpreis, follows easily from the simple proposition that $\overline{ \partial }$-cohomology with compact support vanishes in bidegre $(0, 1)$ ([*see*]{} [@ho1966]). Along these lines and after results due to Kohn-Rossi, the Hartogs theorem was generalized to $(n -
1)$-complete complex manifolds by Andreotti-Hill [@ah1972], [*i.e.*]{} manifolds exhausted by a $\mathcal{ C}^\infty$ function whose Levi-form has at least $2$ positive eigenvalues at every point. We also refer to [@ks2001] for an approach via the holomorphic Plateau boundary problem.

To endeavor the theory in general singular complex spaces $\big( X,
\mathcal{ O}_X \big)$, it is at present advisable to look for methods avoiding global $\overline{ \partial 