aa r X i v : . [ m a t h . AG ] J un SOME MIRROR PARTNERS WITH COMPLEX MULTIPLICATION
JAN CHRISTIAN ROHDE
For Eckart Viehweg
Abstract.
In this note we provide examples of families of Calabi-Yau 3-manifoldsover Shimura varieties, whose mirror families contain subfamilies over Shimura vari-eties. Therefore these original families and subfamilies on the mirror side contain densesets of complex multiplication fibers. In view of the work of S. Gukov and C. Vafa [6]this is of special interest in theoretical physics.
Introduction
In theoretical physics rational conformal field theories are considered as particularlyinteresting class of conformal field theories. Let ( X , Y ) be a pair of families of Calabi-Yau3-manifolds, which are mirror partners, X be a fiber of X and Y be a fiber of Y . In [6] S.Gukov and C. Vafa explain that X and Y yield a rational conformal field theory, if andonly if both fibers have complex multiplication ( CM ). A family of Calabi-Yau manifoldsover a Shimura variety has a dense set of CM fibers, if the variation of Hodge structures( V HS ) is related to the Shimura datum of the base space in a natural way as in [8].At present several of such families of Calabi-Yau 3-manifolds over Shimura varieties areknown [1], [5], [8], [9], [10]. In general one does not know a Shimura subvariety of the basespace on the mirror side. Here we give new examples of pairs of families of Calabi-Yau3-manifolds over Shimura varieties, which are subfamilies of mirror partners.We start with a family C of degree 3 covers of P with 6 different ramification pointsover an open Shimura subvariety M ⊂ ( P ) . By using the Fermat curve of degree 3and C , one can construct a family of K M as described in [8], Section 8. The Borcea-Voisin construction yields a family W ofCalabi-Yau 3-manifolds, which has a dense set of CM fibers. A. Garbagnati and B. vanGeemen [4] have given a more general method to construct K K C . The latter method allows to construct K M ⊂ ( P ) , where branch pointsof the fibers of C collide. Here we show that the Borcea-Voisin construction yields afamily of Calabi-Yau 3-manifolds over a Shimura subvariety contained in the boundary ofthe base space of W , whose fibers are its own Borcea-Voisin mirrors. Moreover here wefind a Shimura surface on the boundary of the base space of W such that the fibers of afamily of Calabi-Yau 3-manifolds over this surface are Borcea-Voisin mirrors of the fibersof W . We will also see that these families contain dense sets of CM fibers.1. Construction of K surfaces by automorphisms In this Section we recall the construction of K C is the family of genus 2 curves given by V ( y − x ( x − x ) ( x − λx ) x ) → λ ∈ M := P \ { , , ∞} . Moreover for some of these examples [5], [9] the existence of a mirror is not clear.
2) The family C is the family of genus 3 curves given by V ( y − x ( x − x )( x − αx )( x − βx ) x ) → ( α, β ) ∈ M , where M := ( P \ { , , ∞} ) \ { α = β } . (3) The family C is the family of genus 4 curves given by V ( y − x ( x − x )( x − αx )( x − βx )( x − γx ) x ) → ( α, β, γ ) ∈ M , where M := ( P \ { , , ∞} ) \ ( { α = β } ∪ { α = γ } ∪ { β = γ } ) . Remark 1.1.
The families C and C can be obtained by collision of the branch points ofthe fibers of C over the boundary divisor of M ⊂ ( P ) . Let Γ denote the monodromygroup of the V HS of C . Note that for j = 1 , , V HS of C j . For an overview of this topic see also [7]. Due to theDeligne-Mostow theory, the period domain of the family C j is the complex ball B j and M j is a dense open subset of Γ \ B j . In this sense the base spaces M j are modular.Moreover M and M are contained in the complement of M in Γ \ B (follows from [7],Theorem 3 . M j is an open dense subset of a Shimura variety, which is a ballquotient. This can be concluded from the type of V HS of the given families (compare[8], Subsection 6 .
3) and the description of such a
V HS in the proof of [8], Theorem 4 . . j = 1 , , p ∈ M j let f j ( t ) ∈ C [ t ] be a degree 6 polynomial such that ( C j ) p isgiven by the equation v − f j ( t ) = 0. Moreover let ξ = e πi . It is clear that C j has the M j -automorphism fiberwise given by β j : ( v, t ) → ( ξv, t ) . Let F = V ( y z − x − z ) ⊂ P be a genus 1 curve isomorphic to the Fermat curve of degree 3 and α F : F → F be given by ( x : y : z ) → ( ξx : y : z ) . We have chosen this explicite formula due to technical reasons. Moreover let S f j be aminimal model of a surface given by the Weierstrass equation Y = X + f j ( t ) . For the following lemma we will use methods, which occur already in the proof of [4],Proposition 2 . Lemma 1.2.
The surface S f j is a K surface birationally equivalent to F × ( C j ) p / ( α F , β j ) .Proof. The rational map m j : F × ( C j ) p → S f j is given by (( t, v ) , ( x, y )) → ( v x, v y, t ) . The reader checks easily that m j is ( α F , β j )-invariant and of degree 3. Moreover onecomputes Y = ( v y ) = v y = v ( x + 1) = ( v x ) + f j ( t ) = X + f j ( t ) . From [4] we know that the minimal model S f j is a K (cid:3) . Some Automorphisms of our K surfaces The surface S f j has an elliptic fibration given by S f j → P via ( X, Y, t ) → t in the following way (see also [4]):If f j ( t ) = 0, the fiber of t is given by the elliptic curve V ( Y Z = X + uZ ) ⊂ P , where u = f j ( t ) . Now let t ∈ P be a zero of f j ( t ). By using the Tate algorithm, one can compute thesingular fibers. If f j ( t ) has a simple zero in t , the singular fiber ( S f j ) t is of type IV .Thus it consists of three rational curves intersecting transversally in one point.Now assume that f j ( t ) has a double zero in t . Then the fiber ( S f j ) t is of type IV ∗ .Thus it is given by 7 rational curves with the following intersection graph of type ˜ E : D ❝ D ❝ D ❝ D ❝ D ❝ D ❝ D ❝ Let ι f j : S f j → S f j denote the involution given by( X, Y, t ) → ( X, − Y, t )and α f j : S f j → S f j denote the automorphism of degree 3 given by( X, Y, t ) → ( ξX, Y, t ) . The fixed locus of ι f j contains clearly the section s ∞ of the elliptic fibration fiberwisegiven by (0 : 1 : 0) ∈ P for a general t ∈ P and the curve C j,p given by X + f j ( t ) = 0 , which is isomorphic to ( C j ) p (see [8], Remark 2 . . F of type IV over a simple zero of f j ( t ) by blowingup once. This computation shows that ι f j interchanges two irreducible components of F and C j,p intersects F in the intersection point of its irreducible components. Theinvolution ι f j acts non-trivially on the third irreducible component of F .Moreover the fixed locus of the automorphism α f j contains also the section s ∞ and thesections s ± ( t ) = (0 : ± f j ( t ) : 1) ∈ P .Since the fixed loci of ι f j and α f j contain curves, ι f j and α f j are non-symplectic. Recall 2.3.
It is well-known that a non-symplectic involution of a K K ι f j with asingular fiber F ∗ of type IV ∗ . For doing this we also consider the automorphism α f j .First we can a priori state that either ι f j | D = id or that one has without loss ofgenerality ι f j ( D ∩ D ) = D ∩ D . In both cases one obtains without loss of generalitythat ι f j ( D ∪ D ) = D ∪ D . Hence on D ∪ D one finds without loss of generality an solated fixed point of ι f j | F ∗ , which is an intersection point with the section s ∞ or C j,p .This point is also fixed by α f j . Thus the automorphism α f j has to act as a permutation σ ∈ A on the intersection points D ∩ D k for k = 4 , ,
6, which fixes D ∩ D . Hence σ = id. The Hurwitz formula tells us that either α f j | D = id or that the quotient map bythe degree 3 automorphism α f j | D has 2 ramification points. Thus α f j | D = id. Hence for k = 4 , , α f j | D k are given by the intersection points of D k with theother irreducible components of the fiber F ∗ .Since s ∞ is fixed by α f j , we assume without loss of generality that s ∞ hits a singularfiber F ∗ of type IV ∗ in D . Moreover one has also that ι f j ( D ) = D and ι f j acts as anon-trivial involution on D with a fixed point s ∞ ∩ D . Lemma 2.4.
One cannot have ι f j | D = id .Proof. Assume that ι f j | D = id. Hence ι f j | D = id. Since s ∞ hits F ∗ in D , one hasa non-trivial involution on D , whose fixed points are given by the intersection pointswith the fixed curves s ∞ and C j,p . Thus ι F acts on D and D as the identity map.Since the sections s ± are interchanged by ι f j , one concludes s ± ∩ F ∗ / ∈ D , D , D . Since α f j fixes the points D ∪ D and D ∪ D respectively and acts non-trivially on D , thepoint s ± ∩ F ∗ , which is fixed by α f j , cannot be contained in D . By analogue arguments,one concludes s ± ∩ F ∗ / ∈ D , D . Since the section s ∞ , which is also fixed by α f j , hits D , it is not possible that both sections s ± intersect D , on which α f j acts non-trivially.Contradiction! (cid:3) The involution ι f j | F ∗ has an isolated fixed point s ∞ ∩ D . Since the intersection point F ∗ ∩ C j,p is the only additional isolated fixed point of ι f j | F ∗ , one cannot have 3 isolatedfixed points on D ∪ D with respect to ι f j . Thus one concludes: Corollary 2.5.
The curve D is contained in the fixed locus with respect to ι f j and ι f j interchanges the handle consisting of D and D with the handle consisting of D and D . Construction of mirror pairs with complex multiplication
Recall 3.1.
Let S be a K ι , which has a fixedlocus consisting of the curves C , . . . , C N , and E be an elliptic curve with involution ι E fixing 4 points. Moreover let N ′ = N X i =1 g ( C i ) , where g ( C i ) denotes the genus of C i . Then the Calabi-Yau 3-manifold X obtained fromthe Borcea-Voisin construction given by blowing up the singularities of S × E/ ( ι S , ι E )once has the Hodge numbers(1) h , ( X ) = 11 + 5 N − N ′ and h , ( X ) = 11 + 5 N ′ − N (see [11]).In many cases the involution on H ( S, X ), which is given by the action of ι , can beused to construct a second involution ι ′ on the lattice H ( S, X ). The involution ι ′ can berealized as an involution of a family of K S ′ → B over B , whose restrictions toeach fiber of S ′ are non-symplectic involutions. By a relative version of the constructionabove for S ′ , one obtains the Borcea-Voisin mirror family of X (for details see [2], [11]). Remark 3.2.
By using C , one has already constructed families of Calabi-Yau manifoldsover Shimura varieties (see [8], Section 8 and [8], Section 9). For this construction in [8]one has used a family of K M , which occursin [4], Remark 1 . K F × ( C j ) p / ( α F , β j ). Despite this fact it isnot clear that the family X of Calabi-Yau 3-manifolds, which we construct below, is thefamily W in the notation of [8]. Nevertheless both families are contained in the sameBorcea-Voisin family. Thus the fibers have the same Borcea-Voisin mirrors. By 2.2, the involution ι f j on S f j has a fixed locus containing a rational curve s ∞ andthe curve C j,p of genus j + 1. Moreover the elliptic fibration of S f j contains 3 − j singularfibers of type IV ∗ and each of these fibers has one rational curve contained in the fixedlocus of ι f j (see Corollary 2.4). Thus by using (1) and the family of elliptic curves E → M , V ( y z − x ( x − z )( x − λz )) → λ, the Borcea-Voisin construction yields families X j → M j × M of Calabi-Yau 3-manifoldswith the following Hodge numbers: j h , h , Remark 3.4.
By [2], Section 3 and Section 4, one can easily check that X is containedin a family, which is its own Borcea-Voisin mirror family. Moreover the families X and X can be embedded in families, which are Borcea-Voisin mirrors of each other. Remark 3.5.
By the construction above, the families X and X are contained in theboundary of X . Moreover by using Remark 1.1, one can show that the period map of X j is a multivalued map to a dense open subset of B j × B . From these results one canconclude that the base space of X j is an open subset of a Shimura variety with associatedHermitian symmetric domain B j × B .By analogue arguments, one can also see that X is defined over the boundary of X .By [11], 2 .
21, we have a precise description how a fiber of X j provides a (1 , X − j by the mirror map. Due to [6] one can assume that each pair of complexmultiplication fibers of X j and X − j yields a rational conformal field theory. Now we aregoing to show that each X j has a dense set of complex multiplication ( CM ) fibers for j = 1 , ,
3. First recall the definition of CM : Let X be a compact K¨ahler manifold of complex dimension n and S be the R -algebraic group S = Spec( R [ x, y ] /x + y − , where S ( R ) = (cid:26) M = (cid:18) a b − b a (cid:19) ∈ SL ( R ) (cid:27) ∼ = { z ∈ C : | z | = 1 } . The rational Hodge structure on H n ( X, Q ) of weight n corresponds to the representation h X : S → GL( H n ( X, R )) , h X ( z ) v = z p ¯ z q v ( ∀ v ∈ H p,q ( X ) with p + q = n ) . The Hodge group Hg( X ) is the smallest Q -algebraic subgroup G of GL( H n ( X, Q )) suchthat h X ( S ) ⊂ G R . We say that X has CM , if Hg( X ) is a torus.For more details in the case of Calabi-Yau 3-manifolds see [1]. Proposition 3.7.
For j = 1 , , , the family X j has a dense set of CM fibers.Proof. By [8], Subsection 6 .
3, each family C j has a dense set of CM fibers. Note that thefamily of elliptic curves E → M , V ( y z − x ( x − z )( x − λz )) → λ as also a dense set of CM fibers. Since the ramification locus of the involution ι f j on S f j consists of C j,p ∼ = ( C j ) p and some rational curves, it remains to show that S f j has CM ,if ( C j ) p has CM . Using this result one can then conclude as in [8], Subsection 7 . X j ) ( p,q ) has CM , if ( C j ) p and E q have CM .The singularities of the fibers of F × C j / ( α F , β j ) are given by the singular sections. Let m j denote the quotient map by ( α F , β j ). Near the sections of fixed points correspondingto the singular sections of F × C j / ( α F , β j ) the action of ( α F , β j ) is given by ( ξ, ξ ) or( ξ, ¯ ξ ).First consider the case ( ξ, ¯ ξ ). In this case one blows up the corresponding sections on F × C j with exceptional divisor E . The automorphism ( α F , β j ) does not act triviallyon E . Thus we blow up the two fixed sections on each connected component of E withsmooth exceptional divisor E . This divisor is contained in ramification locus of m j . Nowthe quotient by ( α F , β j ) is smooth in a neighbourhood of m j ( E ∪ E ).In the case ( ξ, ξ ) we blow up the section of fixed points and obtain a smooth exceptionaldivisor contained in the ramification locus.Let ^F × C j denote the manifold obtained from the previous blowing up operations on C j × F and F j = ^F × C j / ( α F , β j ) . Thus we obtain a model F j of the quotient F × ( C j ) p / ( α F , β j ) consisting of smooth fibersover M j . The surface F × C j has CM , if F and ( C j ) p have CM . Note that monoidaltransformations of surfaces do not have any effect to the property of CM (compare [8],Corollary 7 . . H (( F j ) p , Q ) is a sub-Hodge structure ofthe Hodge structure on H ( ^ ( F × C j ) p , Q ), one concludes that ( F j ) p has CM , if ( C j ) p has CM . Moreover we can use monoidal transformations to obtain S f j from ( F j ) p . Thus S f j has CM , if ( F j ) p has CM . (cid:3) Acknowledgements
This paper was written at the Graduiertenkolleg “Analysis, Geometry and String The-ory” at Leibniz Universit¨at Hannover. I would like to thank Lars Halle for instructivediscussions about Neron models and for once asking the question “What happens on theboundary of W ?”. This question is partially answered here. I would like to thank KlausHulek and Matthias Sch¨utt for their interest and their comments, which helped to im-prove this text. Moreover I would like to thank Bernd Siebert, who has pointed out thatit is of interest to understand the period maps of the given families, which are outlinedin Remark 1.1 and Remark 3.5. References [1] Borcea, C.: Calabi-Yau threefolds and complex multiplication. Essays on mirror manifolds. In-ternat. Press, Hong Kong (1992) 489-502.[2] Borcea, C.: K II . AMS/IP, Providence, RI (1997) 717-743.[3] Deligne, P., Mostow, G.:Monodromy of hypergeometric functions and non-lattice integral mon-odromy. IHES (1986) 5-89.[4] Garbagnati, A., van Geemen, B.: The Picard-Fuchs equation of a family of Calabi-Yau three-folds without maximal unipotent monodromy. International Mathematics Research Notices (2010)doi:10.1093/imrn/rnp238.[5] Garbagnati, A.: New families of Calabi-Yau 3-folds without maximal unipotent monodromy(2010) arXiv:1005.0094.[6] Gukov, S., Vafa, C.: Rational conformal field theories and complex multiplication. Comm. Math.Phys. (2004) 181-210.
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GRK1463/Institut f. Algebraische Geometrie, Leibniz Universit¨at Hannover, Welfen-garten 1, 30167 Hannover, Germany
E-mail address : [email protected]@math.uni-hannover.de