}$ techniques, as well as global integral kernels, because such tools are not yet available. The geometric Hartogs theory was attacked long ago by Rothstein, who introduced the notion of $q$-convexity. On the other hand, within the modern sheaf-theoretic setting, the so-called [*Andreotti-Grauert theory*]{} allows to perform extension (of holomorphic functions, of differentials forms, of coherent sheaves, [*etc.*]{}) from shell-like regions of the form $\big\{ z\in X: \, a < \rho ( z) <b \big\}$ into their inside $\big\{ z\in X : \rho ( z) < b \big\}$, where $\rho$ is a fixed $(n-1)$-convex exhaustion function for $X$. Geometrically speaking, one performs holomorphic extension by means of the [*Grauert bump method*]{} through the level sets of $\rho$ in the direction of decreasing values, jumping finitely many times across the critical points of $\rho$.

However, a satisfying, complete generalization of the Hartogs theorem should apply to general excised bounded domains $\Omega \backslash K$ lying in an $(n-1)$-complete complex space $\big( X, \mathcal{ O}_X
\big)$, not only to shells $\{ a < \rho < b\}$ relative to the $(n-1)$-convex exhaustion function. But then, after perturbing and smoothing out $\partial \Omega$, one must unavoidably take account of the critical points of $\rho \big\vert_{ 