Let's say we're in the very nice case where our Kähler manifold $M$ is already a complex Hilbert manifold. Is there a nice condition, similar to the maximal rank condition in the finite case, for a possibly infinite dimensional submanifold to again be Kähler?

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!