Do there exist closed symplectic manifolds with Euler characteristic zero? - MathOverflow

Do there exist closed symplectic manifolds with Euler characteristic zero?

2011-01-03T19:11:48Z

By symplectic manifold I mean a pair $(M^{2n},\omega)$ consisting of a smooth, connected, even dimensional manifold and a non-degenerate $2$-form. I am interested in compact, boundarlyess examples where $\chi(M)=0$. If none such exist, can anyone provide a simple proof (understandable to a Topologist who knows a little geometry)?

In case the answer to the question in the title is a quick "yes", I have several follow up questions:

What about if we restrict to the case $n=2$ (in which case $M$ would have to be non-simply-connected)?
What about if we restrict to closed symplectically aspherical manifolds? [Recall that $(M^{2n},\omega)$ is called symplectically aspherical if the symplectic class $[\omega]\in H^2(M;\mathbb{R})\cong \mathrm{Hom}(H_2(M),\mathbb{R})$ vanishes on the image of the Hurewicz homomorphism $h\colon \pi_2(M)\to H_2(M)$ .]

Thanks.

Answer by Ben Webster for Do there exist closed symplectic manifolds with Euler characteristic zero?

2011-01-03T19:18:29Z

Yes. $T^2 \times T^2$ with the sum of the volume forms on each factor.

Answer by Noz for Do there exist closed symplectic manifolds with Euler characteristic zero?

2011-01-03T22:15:36Z

You can find plenty of example of symplectic $4$-manifolds with $0$ Euler characteristic by taking $Y\times S^1$ where $Y$ is a fibered $3$-manifolds. See: http://www.zentralblatt-math.org/zmath/en/search/?q=an:0324.53031&format=complete for a proof (I couldn't find an on-line version of the paper, however the construction is outlined there: http://arxiv.org/abs/1001.0132).

For the vanishing of $\omega$ on $\pi_2$ just take the fiber to be something not a sphere (the fibers are symplectic so it couldn't work with the sphere as a fiber).

Those manifolds are non-trivial fibrations over the torus if you take the monodromy to be non trivial.

Answer by Mohammad F.Tehrani for Do there exist closed symplectic manifolds with Euler characteristic zero?

2011-01-09T19:14:20Z

Also there are C.Y 3-folds with this property constructed via toric geometry (I think due to Batyrev)