together with holomorphic isomorphisms $\varphi_j : \ U_j \to {\sf A}_j$ onto some closed complex analytic sets ${\sf A }_j$ contained in balls $\widetilde{ \sf B}_j \subset
{\mathbb{C}}^{ N_j}$, some $N_j \geqslant 2$. By definition ([@fi1976; @gu1990]), a $\mathcal{ C}^\infty$ function $f : X \to {\mathbb{C}}$ is locally represented as $f \vert_{ U_j} = \widetilde{ \sf f}_j \circ \varphi_j$ for some collection of $\mathcal{ C }^\infty$ “ambient” functions $\widetilde{ \sf f}_j : \widetilde{ \sf B}_j \to {\mathbb{C}}$, $j\in J$. A real-valued continuous function $\rho$ on $X$ is an [*exhaustion function*]{} if sublevel sets $\{ z \in X : \, \rho(z) < c\}$ are relatively compact in $X$ for every $c\in {\mathbb{R}}$. A $\mathcal{
C}^\infty$ function $\rho : X \to {\mathbb{R}}$ is called [*strongly $q$-convex*]{} if the $\mathcal{ C}^\infty$ ambient $\widetilde{
\rho}_j : \widetilde{ \sf B}_j \to {\mathbb{R}}$ can be chosen to be strongly $q$-convex, [*i.e.*]{} their Levi-forms $i\, \partial \overline{
\partial} ( \widetilde{ \rho }_j)$ have at least $N_j - q + 1$ positive eigenvalues at every point, for all $j \in
J$. Finally[^2], $X$ is called [*$q$-complete*]{} if it possesses a $\mathcal{ C }^\infty$ strongly $q$-convex exhaustion function. Note that the $1$-complete spaces are precisely the Stein spaces.

We 