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!


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.