Condition for infinite dimensional complex manifold to be Kähler by pullback form

Question

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!

Answer 1

There are three definitions of infinite-dimensional complex manifolds: strong (existence of holomorphic charts), weak (abundance of holomorphic functions) and formal (vanishing of Nijenhuis tensor). The implications "strong" $\Rightarrow$ "weak" $\Rightarrow$ "formal" are strict.The Kahler property is closedness of Hermitian form, and it is no different from finite-dimensional case. All these properties are inherited for submanifolds or immersed manifolds.

Of course, the original question as stated ("if N→fM is a holomorphic map with maximal rank at each point, then N is Kähler with the pullback form induced by M") makes no sense either finite-dimensional or infinite-dimensional: if $f$ has non-Kahler fibers, $N$ would be non-Kahler regardless, and there are examples of holomorphic fibrations with Kahler fibers and base and non-Kahler total space. I guess David meant that dimension of $N$ is less or equal to that of $M$.

Some reference to infinite-dimensional Kahler manifolds:

J. L. Brylinski, {\em Loop Spaces, Characteristic Classes and Geometric Quantization,} Progr. Math., vol. 107, Birkh\"auser Boston, Boston, MA, 1993.

LeBrun, Claude, {\em A K\"ahler structure on the space of string worldsheets}, Classical Quantum Gravity 10 (1993), no. 9, L141--L148.

Lempert, L\'aszl\'o, {\em Loop spaces as complex manifolds,} J. Differential Geom. 38 (1993), no. 3, 519--543.

Lempert, L\'aszl\'o, {\em The Dolbeault complex in infinite dimensions. I}, J. Amer. Math. Soc. 11 (1998), no. 3, 485--520.

M. V. Movshev, {\em The structure of a symplectic manifold on the space of loops of 7-manifold}, arXiv:math/9911100, 10 pages.

Verbitsky, M., {\em A formally Kaehler structure on a knot space of a $G_2$-manifold}, arXiv:1003.3174, 24 pages.