Question

Let $X$ be a (smooth) compact complex manifold, and suppose that $H^1(X, \Theta_X) = 0$, where $\Theta_X$ is the tangent sheaf. In other words, suppose that $X$ is rigid.

Suppose moreover that $X$ arises as the complex points of a smooth projective variety over $Q$.

Is it known or expected that the periods of $X$ are algebraic numbers? If $X$ were not rigid, then the periods would be values (at zero) of functions satisfying Picard-Fuchs equations. But the rigidity suggests (to my intuition) that the periods should not be transcendental.

Is anything known? Expected? Written? How about the specific case when $X$ is a rigid Calabi-Yau 3-fold? Has anyone computed such periods? Could one compute them easily?

Answer 1

For rigid, at least in the modular case (known in many events), you can compute the periods of the form, though this supposes you can explicitly write down the weight 4 newform. For instance, Schutt ( http://arxiv.org/pdf/math/0311106 ) gives examples of level 73, and using Magma you can compute the periods as

> M:=NewformDecomposition(NewSubspace(CuspidalSubspace(ModularSymbols(73,4))))[1];
> Periods(M,100);
[ (0.902834199842382836695960181248 + 0.0526923557275574794028757363126*i),
  (0.285105536792331422114513708795 + 0.0175641185758524931342918404798*i) ]

Here $L$-functions are not applicable, as the $L$-functions vanishes at the central point. I extended the above to a few hundred digits and found nothing with PowerRelation.