## plurisubharmonic sublevel sets

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Let $X$ be a complex manifold, let $\Omega \subseteq {\bf C} \times X$ be defined by $\Omega = \{ (z,p) \in {\bf C} \times X : a(p) < Im z < - b(p) \}$ where $a$ and $b$ are plurisubharmonic functions on $X$ with $a + b < 0.$ Assume that $\Omega$ is a Stein manifold, is it true that $X$ is a Stein manifold ? If with the same assumtpion we suppose that $X$ is locally eucledean countable basis "complex manifold" but not in general Hausdorff, can we conclude that $X$ must be Hausdorff ? This question is related to the possibility to embedding a manifold with the holomorphic action of a real Lie group into some space where there is an action of the complexified group.

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