Symplectic but Not Complex Manifolds.

[Bacle2]

Hi, All:
AFAIK, every complex manifold can be given a symplectic structure, by using

w:=dz/\dz^ , where dz^ is the conjugate of dz, i.e., this form is closed,

and symplectic. Still, I think the opposite is not true, i.e., not every symplectic

manifold can be given a complex structure. Does anyone know of examples/results?

I heard something about an equivalence between Lefschetz fibrations (or pencils)

and existence of symplectic structures, but I cannot think of examples.

Any Ideas?

 

[quasar987]

Hi, All:
AFAIK, every complex manifold can be given a symplectic structure, by using

w:=dz/\dz^ , where dz^ is the conjugate of dz, i.e., this form is closed,

and symplectic.

But this form is defined only locally. And trying to patch it globally, you may lose closedness. In her (free, available on her website) book, Ana Cana da Silva mentions the Hopf surface S^1 x S^3 as a complex manifold that does not admit a symplectic structure (obviously since H^2(S^1 x S^3) = 0 x H^2(S^3)=0.) See p.122.

Still, I think the opposite is not true, i.e., not every symplectic

manifold can be given a complex structure. Does anyone know of examples/results?

For this Silva gives a reference for an example of Ferdandez-Gotay-Gray that is a circle bundle over a circle bundle over the 2-torus. (p.121)

 

[zhentil]

Bacle, I think you mean to say that every Kahler manifold can be given a symplectic structure. That the converse is not true is much more delicate. Gompf discovered non-Kahler symplectic manifolds in the 80's, using some surgery construction to produce symplectic manifolds that do not satisfy the Hodge decomposition on cohomology.

 

[Bacle2]

Thanks, both for your comments, refs., I will look into them.