Let $\phi$ be a stricly plurisubharmonic function on a domain in ${\Bbb C}^n$, and $S=\phi^{-1}(c)$ its level set. Consider $S$ as a Riemannian manifold equipped with a metric induced by $dd^c\phi$. I am interested in curvature restrictions on the Riemannian structure of $S$. In all examples I could check, its Ricci curvature is positive. Is it always true? What kind of restrictions we get? Any ideas or reference would be appreciated.

I am thinking by analogy with a strictly convex function on a flat space: its Hessian is a metric, and its level set has (I think) positive curvature, though I don't have a formal proof of this either. I would appreciate a reference, or a refutation, if this is false.