# Are periods of rigid Calabi-Yau threefolds over $Q$ algebraic?

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?

• Can you be a bit more precise about your context, or at least explain why you think this should be true? I ask because the projective line gives a trivial counterexample---the periods for $H^2$ are the multiples of $2\pi i$. Do you want to look at the middle cohomology or something? Also do expect whatever you expect to be true to hold at the level of Hodge structures? Jun 14 '10 at 4:25
• This looks wrong to me. Compute the Mumford-Tate group G of the relevant cohomology group (as a rational Hodge structure); then conjecturally, the transcendence degree of the field generated by the periods is the dimension of G (Grothendieck). The point is that algebraic classes force algebraic relations between the periods; hence conjecturally also Hodge classes do (this is known for abelian varieties by Deligne); and Grothendieck conjectured that these are all the relations. So if you want lots of relations, you need to find lots of Hodge classes. Jun 14 '10 at 11:27
• At the very least, I should have adjusted the periods by powers of $2 \pi i$, according to the appropriate weights. My interest is in whether there might be a relationship between a motivic Galois group and a differential Galois group. Most transcendence of periods results (that I know of) rely on putting the period into a nontrivial family. I wonder whether the method has a theoretical limit, or whether all (conjecturally) transcendental (after scaling by $\pi^n$) periods can be viewed as values of nonalgebraic G-functions at zero. Jun 14 '10 at 15:08

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))));
> 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.

• Is it easy to see a relationship between the periods of the modular form (arising from a chunk of the motive the the Calabi-Yau) and the concrete periods of the Calabi-Yau? Jun 14 '10 at 2:08
• Hmm, shows how much theory I am missing. I don't know if there is any relation. Certainly computing periods of a C-Y looks harder. Verrill has a row called "periods" in her table on page 2 of mpim-bonn.mpg.de/preprints/send?bid=27 , and on page 3 she says what it means in terms of period lattices. Her examples are rigid, though I still don't see an exact relation. Jun 14 '10 at 5:13
• I guess that the 2-dim (modular) Galois representation arises from a motivic chunk (direct summand) of the $H^2$ or $H^4$ of the C-Y 3-fold. Thus, the period matrix of the motive assoc. to your modular form should arise as a submatrix of the period matrix of the C-Y. I don't recall how such periods are related to the usual periods of a modular form that you've computed, but I can look it up in Deligne. But thanks for the very good example/computation. Do you have Verrill's 1st name or a title? The link seems broken. Jun 14 '10 at 16:42
• Needs a www in front. mpim-bonn.mpg.de/preprints/send?bid=27 Helena Verrill, The L-series of certain rigid Calabi-Yau threefolds Jun 15 '10 at 1:34