Examples of $T^2$-bundle over $T^{2n}$ whose first Chern class does not vanish.

Question

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.

Answer 1

If $X$ is a principal $T^2$-bundle over a compact complex manifold $M$ then $X$ is a complex manifold and $c_1(X)$ equals the pullback of $c_1(M)$. This is Corollary 5.7 here. In your case (if your bundle is principal) you have $c_1(M)=0$ and so $c_1(X)=0$ too. Here $c_1(X)$ is the first Chern class of the complex structure on $X$.

If $X$ is not a principal bundle, then I don't think there is any general result. For example it is true that every symplectic $T^2$-bundle over $T^2$ has torsion symplectic first Chern class, but the proof involves a lot of case-by-case examinations. See for example Chapter 7 of this survey of T.-J. Li.