Question

Are there any symplectic but not complex Calabi- Yau manifolds in real dimensions 4 and 6?

Answer 1

First of all, the notion Symplectic Calabi-Yau is quite new. A few persons who use it (including myself) usually mean by this symplectic manifolds with $c_1=0$, (this is just to make sure that we speak about the same thing)

In real dimension $4$ we know for the moment only two types of symplectic Calabi-Yau manifolds - $K3$ surfaces and $T^2$ bundles over $T^2$. These manifolds have as well the structure of a complex manifold with a non-vanishing holomorphic volume form. It is conjectured that there are no other symplectic Calabi Yau manifolds in dimension $4$.

In real dimension six there are quite a lot of symplectic CY manifolds coming from the twistor construction (you can check here: http://arxiv.org/abs/0802.3648), and some of them do have a complex structure, but this is not known for all of them.

At the same time, probably you know that in dimension $2n\ge 6$ the following question is open:

Question. Is it true that every manifold $M^{2n}$ that has an almost complex structure $J$ has as well a holomorphic structure homothopic to J?

This is an old question and apparently no one has an idea of how to answer it. Now, the answer to your question in dimension $6$ depends on what you mean by a complex Calabi-Yau. This notation is not used in math literature. If by such a manifold you mean a complex manfiold with $c_1=0$, then you would not be able (for the moment) to get any example in dimension $6$ where the answer to your question is no (because the above Question is open). On the other hand, if by complex Calabi-Yau you mean a complex manifold with a non-vanishing holomorphic volume form, then the answer to your question is yes, an example is given in http://arxiv.org/abs/0905.3237. There is a symplectic Calabi Yau 6-manifold in this paper, that has $b_3=0$, hence it can not have a holomorphic volume form of top degree for any complex structure. One can construct further such examples.