0

Hi. I have a question.

When X is a symplectic manifold which is diffeomorphic to T2-bundle over T2n, then does the first Chern class c1(X) vanishes in H2(X;R)? (i.e. a symplectic Calabi Yau)

Or, is there any example of T2-bundle over T2n 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 T2-bundle structure over T4. 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.

1 Answer 1

2

If X is a principal T2-bundle over a compact complex manifold M then X is a complex manifold and c1(X) equals the pullback of c1(M). This is Corollary 5.7 here. In your case (if your bundle is principal) you have c1(M)=0 and so c1(X)=0 too. Here c1(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 T2-bundle over T2 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.

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .