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



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


