Examples of non-Kahler compact symplectic manifolds. - MathOverflow

Examples of non-Kahler compact symplectic manifolds.

2012-09-21T18:49:33Z

I am trying to gather a list of all known symplectic manifolds which don't have Kahler structure. If you know any please add to the list and give references for it.

Please avoid giving repetitive examples.

Thanks.

Answer by Agol for Examples of non-Kahler compact symplectic manifolds.

2012-09-21T23:10:56Z

The original example of a torus bundle over a torus which is symplectic but admits no Kahler structure is due to Thurston.

Answer by eigenbunny for Examples of non-Kahler compact symplectic manifolds.

2012-09-21T23:48:31Z

Honestly, this is like asking for a list of all animals that are not dolphins.

Instructions for making your own:

Step 1 - use the symplectic mapping torus for a symplectomorphism of your favorite Kaehler manifold (acting suitably on homology) to produce something whose Betti numbers violate being Kaehler (one odd degree Betti number is odd)

Step 2 - take the manifold from Step 1 and embed it symplectically in projective space, then blow up. That gets rid of the fundamental group, but raises the dimension a lot.

Step 3 - take Donaldson hypersurfaces to get the dimension down again.

Answer by Mike Usher for Examples of non-Kahler compact symplectic manifolds.

2012-09-22T01:15:54Z

There are lots of simply-connected four-dimensional examples in Gompf's symplectic sum paper paper which are shown to be non-Kahler by virtue of violating the Noether inequality (see Theorem 6.2). There are also higher-dimensional examples in the last section of Gompf's paper, including infinitely many simply-connected non-Kahler ones in any even dimension at least 8. Some of these are obtained directly by symplectic sum, and others by taking four-dimensional examples and embedding them in CP^n and blowing up, as suggested by Tim and eigenbunny--in these cases Gompf uses the Hard Lefschetz theorem to prove that the result isn't Kahler.

The standard symplectic surgery operations in four-dimensions (symplectic sum, Luttinger surgery, rational blowdown...) should generally be expected to produce non-Kahler manifolds more often than not, though it's not always feasible to see that the result isn't Kahler--the standard ways of doing so are by the parity of $b_1$ or by the Noether inequality. In particular this paper of Fintushel-Park-Stern gives another large collection of simply connected examples violating the Noether inequality.

One can also show that a symplectic manifold is not Kahler by showing that it is not formal (in the sense of rational homotopy theory). Often nonformal examples also don't satisfy hard Lefschetz (so could instead just be shown to be non-Kahler by that criterion), but there is a nonformal example of Cavalcanti which does satisfy hard Lefschetz.

Answer by Francois Ziegler for Examples of non-Kahler compact symplectic manifolds.

2012-09-22T02:45:28Z

I would recommend the Tralle-Oprea book, Symplectic manifolds with no Kähler structure.

Answer by Bart Van Steirteghem for Examples of non-Kahler compact symplectic manifolds.

2012-09-22T03:37:34Z

Edited: Corrected after Eugene Lerman's comment of Sep 22, 2012, which also implies this is not an answer to the question (edits are in italic).

In [Invent. Math. 131 (1998), no. 2, 311–319] Chris Woodward gives examples of multiplicity-free compact Hamiltonian manifolds that are not compatibly Kähler. This proved that Delzant's result that all compact multiplicity-free torus actions are compatibly Kähler [Bul. Soc. math. France 116, 315–339 (1988)] does not extend to the nonabelian case. Woodward uses a result from Susan Tolman's paper "Examples of non Kähler Hamiltonian torus actions" [Invent. Math. 131 (1998), no. 2, 299–310].

So, back-to-back papers with examples of symplectic manifolds with a lot of symmetry that are not compatibly Kähler.