For a finite dimensional Kähler manifold $M$, there is the condition that if $N\xrightarrow{f}M$ is a holomorphic map with maximal rank at each point, then $N$ is Kähler with the pullback form induced by $M$ (p.78). Let's say we're in the very nice case where our Kähler manifold $M$ is already a complex Hilbert manifold.

Is it possible to have a similarly simple condition as in the finite dimensional case?

I think that $N$ is always Kähler via the pullback form if we already know that $f$ is the inclusion and $N$ is a closed complex submanifold of $M$. At least I don't see where it should go wrong. Is this correct?

While I'm asking, is there a brief (not neccessarily with proofs) introduction to the infinite dimensional case, in particular for Kähler manifolds?

Thank you!