Hi. I have a question.

When $X$ is a symplectic manifold which is diffeomorphic to $T^2$-bundle over $T^{2n}$, then does the first Chern class $c_1(X)$ vanishes in $H^2(X;\mathbb{R})$? (i.e. a symplectic Calabi Yau)

Or, is there any example of $T^2$-bundle over $T^{2n}$ which is not symplectic Calabi-Yau?

In fact, I proved that some 6-dimensional compact symplectic manifold $M$ is symplectic Calabi-Yau, and $M$ admits a $T^2$-bundle structure over $T^4$. But the proof is not pretty (the computation is really awful). So, I am worry about that my question might be true..

Thank you in advance.