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?

  • $\begingroup$ 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? $\endgroup$ – JBorger Jun 14 '10 at 4:25
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    $\begingroup$ 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. $\endgroup$ – JS Milne Jun 14 '10 at 11:27
  • $\begingroup$ 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. $\endgroup$ – Marty 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))))[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.

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  • $\begingroup$ 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? $\endgroup$ – Marty Jun 14 '10 at 2:08
  • $\begingroup$ 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. $\endgroup$ – Junkie Jun 14 '10 at 5:13
  • $\begingroup$ 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. $\endgroup$ – Marty Jun 14 '10 at 16:42
  • $\begingroup$ Needs a www in front. mpim-bonn.mpg.de/preprints/send?bid=27 Helena Verrill, The L-series of certain rigid Calabi-Yau threefolds $\endgroup$ – Junkie Jun 15 '10 at 1:34

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