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A complex manifold $X$ is said to be weakly pseudoconvex if there exists on $X$ a smooth plurisubharmonic exhaustion function $\psi$.

For example, Stein manifolds are weakly pseudoconvex (in this case $\psi$ can be chosen to be even strictly plurisubharmonic), compact complex manifolds are weakly pseudoconvex (take $\psi\equiv 0$), etc...

Now, let $X$ be a non-compact weakly pseudoconvex manifold. The question is the following:

Does there exist a smooth plurisubharmonic exhaustion function $\varphi$ on $X$ such that $d\varphi$ is never zero outside some compact set $K\subset X$?

This is indeed true for $X$ a Stein manifold or, equivalently by the solution of the Levi problem, for strongly pseudoconvex manifolds: these are in fact embeddable as close submanifolds of some affine complex space and $\varphi=|z|^2$ will do the job.

Thanks in advance.

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# Plurisubharmonic exhaustion functions without critical points at infinity

A complex manifold $X$ is said to be weakly pseudoconvex if there exists on $X$ a smooth plurisubharmonic exhaustion function $\psi$.

For example, Stein manifolds are weakly pseudoconvex (in this case $\psi$ can be chosen to be even strictly plurisubharmonic), compact complex manifolds are weakly pseudoconvex (take $\psi\equiv 0$), etc...

Now, let $X$ be a non-compact weakly pseudoconvex manifold. The question is the following:

Does there exist a smooth plurisubharmonic exhaustion function $\varphi$ on $X$ such that $d\varphi$ is never zero outside some compact set $K\subset X$?

This is indeed true for $X$ a Stein manifold or, equivalently by the solution of the Levi problem, for strongly pseudoconvex manifolds: these are in fact embeddable as close submanifolds of some affine complex space and $\varphi=|z|^2$ will do the job.

Thanks in advance.