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?