Arithmetic and intermediate Jacobians of some rigid Calabi-Yau threefolds
We construct Calabi-Yau threefolds defined over via quotients of abelian threefolds, and re-verify the rigid Calabi-Yau threefolds in this construction are modular by computing their L-series, without [Dieulefait] or [GouveaYui]. We compute the intermediate Jacobians of the rigid Calabi-Yau threefolds as complex tori, then compute a -model for the 1-torus given a -structure on the rigid Calabi-Yau threefolds, and find infinitely many examples and counterexamples for a conjecture of Yui about the relation between the -series of the rigid Calabi-Yau threefolds and the -series of their intermediate Jacobians.
Associated to any smooth projective variety, the intermediate Jacobian varieties generalize Jacobian varieties of curves. For an -dimensional complex variety we define the (Griffiths) intermediate Jacobians of to be the varieties
where the quotient involves only the torsion free part of and is the -th level in the Hodge filtration, i.e.,
Thus, for a curve , the only intermediate Jacobian is , which is the Jacobian variety of the curve. We will be interested mostly in Calabi-Yau varieties, so for the one-dimensional case we have elliptic curves which are known to be isomorphic to their (intermediate) Jacobians, and these are all the possibilities. For two dimensional examples one has K3 surfaces which have trivial first and third cohomology, hence no-non-trivial intermediate Jacobians, and so our focus will be with Calabi-Yau threefolds. Here the only non-trivial intermediate Jacobian is
Not much is known about these varieties, but they are very useful tools. As complex varieties one has e.g., [ClemensGriffiths] who used the intermediate Jacobian of a cubic threefold to show there exist unirational varieties that are not rational, and [VoisinIJ] where intermediate Jacobians are used to show the Griffiths group of a general member in a family of non-rigid Calabi-Yau threefolds is infinite dimensional. An interest in the physics literature comes from the use of Calabi-Yau varieties in string theory, e.g., in [BKNPP], [DDP] and [Morrison]. As is Kähler we have that the intermediate Jacobian is a complex torus of dimension . Thus, when is rigid, is a 1-torus, possibly with a canonical structure of an elliptic curve.
The work to date studies the geometry of , i.e., the complex structure. However, as intermediate Jacobians generalize the classical Jacobian variety of a curve, as well as the Picard varieties and Albanese varieties of any -dimensional varieties, it is natural to ask if one can study arithmetic on all intermediate Jacobians by finding a canonical -structure when is defined over . All of our examples of Calabi-Yau threefolds will be defined over , but one can similarly try to associate a canonical -structure for any number field if is defined over instead. We will construct some rigid Calabi-Yau threefolds in which we are able to compute their intermediate Jacobians as complex varieties, and then refine this computation to give a natural -structure as well, given a choice of model for the Calabi-Yau threefolds defined over .
Shafarevich conjectures that every variety of CM-type (meaning its Hodge group is abelian) has the -series of a Grossencharacter, a Hecke -series, and Borcea shows that a rigid Calabi-Yau threefold with CM-type has a CM elliptic curve as its intermediate Jacobian, which is well known to have a Hecke -series. Thus, if the conjecture is true, it is a natural question to ask if there is any relation between the associated Grossencharacters of a rigid Calabi-Yau threefold and its intermediate Jacobian. The motivation for the current work is to study a precise conjecture of Yui to this effect.
Conjecture 1 (Yui, [YuiUpdate]).
Let be a rigid Calabi-Yau threefold of CM-type defined over a number field . Then the intermediate Jacobian is an elliptic curve with CM by an imaginary quadratic field , and has a model defined over the number field .
If is a Hecke character associated to and
Consequently, is modular.
We will show that many of our rigid Calabi-Yau threefolds of CM-type satisfy this conjecture, but not all. In particular, we will show that if one of our examples, , satisfies the conjecture, then all quadratic twists of satisfy the conjecture, while all non-quadratic twists of do not. After this we generalize our construction to Calabi-Yau -folds and show that for infinitely many satisfying a congruence with the order of the CM automorphisms we have the natural generalization of the conjecture is true. Similarly, for infinitely many the conjecture will not be true because of the CM twists on the varieties.
We start by constructing our varieties and studying their geometry over . We use a generalized Borcea construction of Calabi-Yau threefolds using (finite) quotients of products of elliptic curves, and determine which quotients give rigid Calabi-Yau threefolds. We then choose a -structure for the Calabi-Yau threefolds, via the underlying elliptic curves, and compute their respective -series. Once this is done we compute the intermediate Jacobians as complex tori, and then over , via the choice of -structure given on the threefolds. We are then able to compare the -series of the intermediate Jacobians and their respective threefolds and check the conjecture.
Our construction also gives rise to non-rigid Calabi-Yau threefolds, which we leave to future work [Molnar3], as the intermediate Jacobians are no longer elliptic curves, and the arithmetic of 2-tori and 4-tori is more complicated. Moreover, the question of modularity (automorphy) is nowhere near as resolved, as [Dieulefait] and [GouveaYui] no longer apply.
The author was partially supported by Ontario Graduate Scholarships, as well as partial support and hospitality at the Fields Institute for the thematic program on Calabi-Yau varieties in 2013, the University of Copenhagen in 2014, and the Leibniz Universität Hannover in 2015. We are very grateful for these opportunities. We are also grateful for the patience and many insightful discussions we had throughout writing this work, in particular with Ian Kiming, Hector Pasten, Andrija Peruničić, Simon Rose, Matthias Schütt, and our supervisor Noriko Yui, without whom we may never have been introduced to the interesting questions we address here.
2. Construction of threefolds
We will generalize a construction of Calabi-Yau threefolds due to Borcea [Borcea], using elliptic curves with complex multiplication. To start, we first consider our threefolds over , determine their Hodge numbers.
Over , there is only one elliptic curve with an automorphism of order 3 and one elliptic curve with an automorphism of order 4, up to isomorphism. Denote these by and , with their respective CM automorphisms and , and note that is an automorphism of order 6 on .
On the triple product we have an action of the group for and . As preserves the holomorphic threeform of we have and , so a crepant resolution of is a Calabi-Yau threefold. As is a global quotient orbifold of dimension 3, such a crepant resolution exists by [BridgelandKingReid]. The same is true for many subgroups of but the geometry varies widely with the choice of subgroup.
Consider the following groups of automorphisms acting on .
Crepant resolutions of the respective quotients are Calabi-Yau threefolds with Hodge numbers
The example using was studied in [CynkHulek], while all of the Hodge numbers using and can be found in [FilippiniGarbagnati] as well as references therein. The pair can be found in [Garbagnati] and [Kreuzer], and a large set of pairs from a toric construction including can be found in [Kreuzer]. Lastly, as mentioned above, the pair (51,3) is the original Borcea construction [Borcea]. These exhaust all the Calabi-Yau threefolds one can obtain from this construction, up to isomorphism, noting that any subgroup of that does not act trivially on one coordinate is isomorphic to one of the above subgroups.
While the rigid examples cannot have Calabi-Yau mirror partners, all of the non-rigid examples have (topological) mirrors that have been constructed in the literature. All mirror pairs except can be found in the toric construction of [Kreuzer], while the last mirror can be found in [BatyrevKreuzer] which constructs Calabi-Yau varieties and their mirrors via conifold transitions. It would be interesting to see if there is a relationship between the intermediate Jacobians in a mirror pair.
As all the examples are similar, we only look at the cyclic example
which contains all the geometry necessary for the resolution of each example.
We must investigate the fixed points under the action of each element of this group, so we break things up into steps.
By continuously extending the appropriate automorphisms we have a birational diagram