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.

 

[blue_raver22]

I would first like to clarify the definitions of a symplectic manifold and a complex manifold. A symplectic manifold is a smooth manifold equipped with a non-degenerate, closed 2-form called a symplectic form. A complex manifold, on the other hand, is a smooth manifold equipped with a complex structure, which is a smooth choice of complex coordinates on the manifold.

Based on these definitions, it is indeed true that every complex manifold can be given a symplectic structure, as complex coordinates can be used to define a symplectic form. However, the opposite is not necessarily true. Not every symplectic manifold can be given a complex structure.

One example of a symplectic but not complex manifold is the 4-dimensional symplectic manifold known as the K3 surface. This manifold has a symplectic form, but it does not have a complex structure. This was proven by mathematician Kunihiko Kodaira in 1957.

There are also other examples of symplectic manifolds that do not admit a complex structure, such as certain toric varieties and symplectic 6-manifolds with minimal models. In general, it is a difficult problem to determine which symplectic manifolds can be given a complex structure.

As for the equivalence between Lefschetz fibrations and symplectic structures, this is known as the SYZ conjecture and has been studied extensively in recent years. However, it remains a conjecture and has not been fully proven.

In conclusion, while every complex manifold can be given a symplectic structure, the opposite is not true. There are examples of symplectic manifolds that do not admit a complex structure, and this remains an active area of research in mathematics.