  We show that every strictly pseudoconvex domain $\Omega$ with smooth boundary
in a complex manifold $\mathcal{M}$ admits a global defining function, i.e., a
smooth plurisubharmonic function $\varphi \colon U \to \mathbb R$ defined on an
open neighbourhood $U \subset \mathcal{M}$ of $\overline{\Omega}$ such that
$\Omega = \{\varphi < 0\}$, $d\varphi \neq 0$ on $b\Omega$ and $\varphi$ is
strictly plurisubharmonic near $b\Omega$. We then introduce the notion of the
core $\mathfrak{c}(\Omega)$ of an arbitrary domain $\Omega \subset \mathcal{M}$
as the set of all points where every smooth and bounded from above
plurisubharmonic function on $\Omega$ fails to be strictly plurisubharmonic. If
$\Omega$ is not relatively compact in $\mathcal{M}$, then in general
$\mathfrak{c}(\Omega)$ is nonempty, even in the case when $\mathcal{M}$ is
Stein. It is shown that every strictly pseudoconvex domain $\Omega \subset
\mathcal{M}$ with smooth boundary admits a global defining function that is
strictly plurisubharmonic precisely in the complement of
$\mathfrak{c}(\Omega)$. We then investigate properties of the core. Among other
results we prove 1-pseudoconcavity of the core, we show that in general the
core does not possess an analytic structure, and we investigate Liouville type
properties of the core.
