Plurisubharmonic exhaustion functions without critical points at infinity

Question

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 $K\subset X$?

Thanks in advance.

Answer 1

Expanded version of my earlier comment, which was directed at the unedited version of the question.

Stein manifolds are those complex manifolds $X$ which have a strictly psh exhaustion function, i.e., a proper, bounded below $C^\infty$ function $\psi$ such that the closed $(1,1)$-form $\omega:=-dd^c \psi$ is positive with respect to the complex structure $J$. (My convention is that $d^c f = J \circ df$ where $J$ is the complex structure acting on cotangent vectors.)

Those for which one can take $\psi$ to have compact critical set are called "finite-type". They include smooth affine algebraic varieties $X\subset \mathbb{C}^N$, which one can see by compactifying to a projective variety $\bar{X}=X\cup D$ and considering $\log \|s\|^2$ where $s$ is a section of a hermitian holorphic line bundle cutting out the divisor $D$.

Any open Riemann surface is Stein, but those of infinite genus (i.e. with infinite rank $H_1$) do not have finite type. If there were a psh exhaustion with compact critical set one could perturb it near the critical set so as to make it a Morse function. The downward gradient flow exists and converges to critical points, and so Morse theory bounds the rank of $H_\ast$ from above by the number of critical points.