Examples of Calabi-Yau threefolds parametrised by Shimura varieties

Examples of Calabi–Yau threefolds parametrised by Shimura varieties 2751.6. The Picard–Fuchs equationIn this section we will assume for simplicity that q=dimH 2,1 (X) = 1. With the notationof section 1.5, the diffeomorphism φ : X → B×X induces an isomorphism ofsheaves R 3 π ∗ Z ∼ = → H 3 (X,Z) B on B, where the last sheaf is just the locally constant sheafdefined by the abelian group H 3 (X,Z). Using the marking H 3 (X,Z)/torsion ∼ = V Z andthe Hodge decomposition of H 3 (X t ,C) for each deformation X t of X, we obtain a (trivial)vector bundle V C × B over B with holomorphic subbundlesF 3 ⊂ F 2 ⊂ F 1 ⊂F 0 ∼ = VB := V C × B, F 3 t = H 3,0 (X t ),where we identify V C ×{t} with H 3 (X t ,C).The period map P describes the variation of these subbundles inside the trivialbundle V C ×B. Another way to describe this variation is to take a non-vanishing sectionω of the rank one bundleF 3 , so ω(t) is a basis of H 3,0 (X t ) for all t ∈ B. The trivial bundleV B comes with the Gauss–Manin connection ∇ which maps the horizontal sectionss v := t ↦→(v,t) to zero, where v∈ V C :∇ = ∇ ∂/∂t : V C × B −→ V C × B.Applying the connection i times to the section ω, we get a section ∇ i ω. As dimV C =2+2q=4, there must be a linear relation, with coefficients p i (t) which will be holomorphicin t:Dω = 0, D :=4∑i=0p i (t)∇ i .This linear relation is known as the Picard–Fuchs equation.Instead of considering this rank four bundle with its section ω, one can alsochoose a basis γ 1 ,...,γ 4 of H 3 (X,Z)/torsion and define four holomorphic functionsϕ i (t) := ∫ γ iω(t) on B, where γ i is identified with a cycle in H 3 (X t ,Z)/torsion using thediffeomorphism φ. These four functions are a basis of the solutions of the degree fourdifferential operator ∑ 4 i=0 p i(t)(d/dt) i which is also called the Picard–Fuchs equationfor the familyX→B.1.7. An exampleThe Dwork pencil of quintic threefolds in P 4 is defined by the equationX t : X 5 1 +...+ X 5 5 − 5tX 1X 2···X 5 = 0.For general t ∈C, the variety X t is a CY threefold with h 1,1 (X t )=1and q=h 2,1 (X t )=101. However, there is a finite subgroup G ∼ = (Z/5Z) 3 acting on P 4 which inducesautomorphisms on each X t and the third cohomology group splits under this action:H 3 (X t ,Q) = T t ⊕ S t ,withT t := H 3 (X t ,Q) G ∼ = Q 4 , H 3,0 (X t )⊂T t ⊗ Q C.

276 A. Garbagnati and B. van GeemenThus the G-invariant part of the cohomology gives a four-dimensional variation ofpolarized Hodge structures and thus it gives a degree four Picard–Fuchs equation.In the context of Mirror Symmetry, it was observed that the (singular) quotientvariety X t /G has a resolution of singularities M t which is a CY threefold, moreover itsHodge numbers are:h 1,1 (M t ) = 101, h 2,1 (M t ) = 1, M t := ˜X t /G.Note that h p,q (M t )=h 3−p,q (X t ), which is one of the requirements for the (101-dimensional)family of quintic CY threefolds and the one parameter family of M t ’s to beMirror CY families.The quotient map induces an isomorphism T t∼ = H 3 (M t ,Q). In particular, thedegree four Picard–Fuchs equation obtained from the variation of the T t is the Picard–Fuchs equation of the one parameter family of CY threefolds M t . A spectacular resultfrom Mirror Symmetry is that a certain solution of this Picard–Fuchs equation defines apower series in one variable whose coefficients a d allow one to compute the Gromov–Witten invariants of a quintic threefold, that is, roughly, the number of rational curvesof degree d on a quintic threefold.In the paper [16], Greene, Plesser and Roan verify that there is an action of thegroup H ∼ =Z/41Z onP 4 such that each member of the pencil of quintic threefoldsY t : X 1 X 4 2 + X 2 X 4 3 +...+ X 5 X 4 1 − 5tX 1 X 2···X 5 = 0is invariant under H. This leads, as above, to a splittingwhereH 3 (Y t ,Q) = T ′t ⊕ S ′ t ,T ′t := H 3 (Y t ,Q) H ∼ = Q 4 , H 3,0 (Y t )⊂T ′t ⊗ Q C.Moreover, the degree four Picard–Fuchs equation defined by the variation of the Hodgestructures T t′ is the same as the Picard–Fuchs equation obtained from the variation ofthe T t∼ = H 3 (M t ,Q). In [11] more such examples are given. A possible explanationwould be that the CY threefold M t is birationally isomorphic to a desingularization ofY t /H.This is indeed the case. Using results of Shioda, in the recent paper [3] it isshown that there is a commutative diagram, where the arrows are rational maps whichare quotients by certain finite groups on suitable Zariski open subsets:˜X dI,tւ ցX t Y t ,ց ւM twhere˜X dI,t : X d 1 +...+ X d 5 − 5t(X 1X 2···X 5 ) d/5 = 0,

278 A. Garbagnati and B. van Geemen2.1. ExampleThe reference is [4, §3]. Let E i , i=1,2,3 be elliptic curves and let ι i : E i → E i be theinversion z↦→−z for the group law on E i . LetG 4 := 〈ι 1 × ι 2 × 1 E3 , ι 1 × 1 E2 × ι 3 〉 ⊂ Aut(E 1 × E 2 × E 3 ).Then the (singular) variety (E 1 × E 2 × E 3 )/G 4 has a resolution of singularities whichis a CY threefold X with h 2,1 = 3 (and h 1,1 = 51). Thus the deformation space of Xis three dimensional. Obviously, it contains the CY varieties obtained by deformingthe three elliptic curves. Thus the period points of deformations of X are in B =H 3 1 ,where H 1 is the upper half plane which parametrises elliptic curves. Thus these CY’sare parametrised by a Shimura variety.2.2. Examples of Borcea–Voisin typeLet S be a K3 surface admitting an involution α S such that H 2,0 (S) is in the eigenspaceof the eigenvalue−1 for the action of α ∗ S on H2 (S,C). We will assume moreover thatthe fixed locus of the involution α S is made up of k rational curves. The dimensionof the family of K3 surfaces admitting an involution acting non trivially on H 2,0 andfixing k rational curves is 10−k and such a family is parametrised by a Shimura varietyassociated to SO(2,10−k).Let E be an elliptic curve and let ι be the involution z↦→−z on E. The quotientthreefold (S×E)/(α S × ι) admits a desingularization which is a CY threefold X (thisconstruction is called Borcea–Voisin construction). In [29, 5] the Hodge numbers of Xare computed:h 1,1 (X) = 15+5k, h 2,1 (X) = 11−k.Hence the dimension of the family of the Calabi–Yau threefolds determined by X isthe sum of the dimension of the family of the K3 surfaces with involution and the dimensionof the family of elliptic curves. Thus these CY threefolds are parametrised bythe product of the Shimura varieties parametrising these two families, see [24, Section11.3].Example 2.1 is a particular case of this construction, indeed the desingularizationof the quotient (E 1 × E 2 )/(ι 1 × ι 2 ) is a K3 surface S (in fact, it is a Kummersurface). The automorphism α S induced on S by 1 E × ι acts non trivially on H 2,0 (S)and fixes 8 rational curves. Hence(S×E 3 )/(α S ×ι) is birational to(E 1 ×E 2 ×E 3 )/G 4 .For the Shimura varieties, one should remember that the real Lie groups SO(2,2) 0 andSL(2,R)×SL(2,R) are isogeneous and thus the Shimura variety associated to SO(2,2)is indeed a quotient ofH 1 ×H 1 .2.3. The easiest caseAnother particular case of the Borcea–Voisin construction is obtained by choosing Sto be the unique K3 surface with an automorphism α S which fixes k = 10 rational

Examples of Calabi–Yau threefolds parametrised by Shimura varieties 283which is a 3:1 cyclic cover ofP 1 with covering automorphismβ f : C f → C f ,(t,v) ↦−→ (t,ξv).Substituting f = v 3 in the Weierstrass equation of S f , one finds the birational isomorphism:C f × E −→ S f ≈(C f × E)/(β f × α E ),((t,v),(x,y)) ↦−→ (X,Y,t) = (v 2 x,v 3 y,t).The automorphism α f on S f is induced by α E . This leads to an isomorphism of Hodgestructures:T f∼ =(H 1 (C f ,Q)⊗H 1 (E,Q) ) β f ×α E,which implies another isomorphism of Hodge structures:H 3 (X f ,Q) ∼ = ( H 1 (C f ,Q)⊗H 1 (E,Q)⊗H 1 (E,Q) ) H,where H ∼ =(Z/3Z) 2 is generated by the automorphisms β f × α E × 1 E and 1 Cf × α E ×α −1E of C f × E× E. This shows that the variation of the Hodge structures H 3 (X f ,Q) isentirely coming from the variation of the Hodge structures of the curves C f . Note thatH 3,0 (X f ) ∼ = H 1,0 (C f ,Q) ξ⊗ H 1 (E,Q) ξ ⊗ H 1 (E,Q) ξ .The Picard–Fuchs equations for the variation of Hodge structures of the curvesC f is explicitly given in [14]. One can parametrise P 1 in such a way that g(t)= t(t−1) and h(t) = (t − λ) (and the other zero of h is at ∞), thus C f∼ = Cλ with definingequation v 3 = t(t− 1)(t− λ) 2 . The holomorphic one forms on this curve are dt/v and(t−λ)dt/v 2 , note that they have distinct eigenvalues ξ,ξ for the automorphism β f . ThePicard–Fuchs equation for η := dt/v∈H 1 (C f ,Q) ξturns out to be:(λ(1−λ) ∂2∂λ 2 +(1−2λ) ∂ ∂λ − 2 )η = 0.3This is also the Picard–Fuchs equation for the holomorphic three form on the correspondingfamily of CY threefolds.Rohde computes the Hodge numbers of these CY threefolds X f and finds:dimH 1,1 (X f ) = 73, dimH 2,1 (X f ) = 1.Any CY threefold Y from the Mirror family, if it exists, should thus have h 1,1 (Y)=1and h 2,1 (Y)=73. At least three families of CY threefolds with these Hodge numbersare known: the complete intersections of type (3,3) in P 5 , (2,2,3) in P 6 and (4,4) inthe weighted projective space P 5 (1,1,1,1,2,2). But in these cases the Mirror familiesare known and they have maximally unipotent monodromy (cf. [8]), hence they cannotbe the Mirrors of the family of the X f .

284 A. Garbagnati and B. van Geemen3.4. The case q>1In case q≤5, we again find that the K3 surface S has an isotrivial fibration with smoothfibers isomorphic to E, but we could not find such a fibration in case q=6. The CYthreefold X S is then again a desingularization of a quotient of the product of a curve Cwith two copies of the fixed elliptic curve E. The variation of Hodge structures of theX S is obtained from the deformations of C.For q≤3, we consider the surface (cf. [14])S f : y 2 = x 3 + f(t) 2 , f = gh 2 ,deg( f)=6,such that g and h have no common zeros and no multiple zeros. The curve C f : v 3 =f(t) has the automorphism β f :(t,v)→(t,ξv). As in 3.3, Rohde’s CY threefold X f isthe desingularization of(C f × E× E)/H, where H =〈β f × α E × 1 E ,1 Cf × α E × α −1E 〉(see [14, Remark 1.3]). The Hodge numbers of X f and the genus g(C f ) of C f are asfollows:deg(g) deg(h) g(C f ) q=h 2,1 (X f ) h 1,1 (X f )6 0 4 3 514 1 3 2 622 2 2 1 730 3 1 0 84The last line corresponds to a rigid CY threefold X f where C f ≃ E, and X f isthe desingularization of the quotient E× E× E by 〈α −1E × α E × 1 E , 1 E × α E × α −1E 〉.In this case, the K3 surface S f is described in [26].In case q=4, we consider the curveC l : v 6 = l(t)deg(l)=12such that l(t) has 5 double zeros. It admits the automorphism β l :(t,v)↦→(t,ξv). Thequotient(C l × E)/(β l × α E ) has a desingularization S l which is a K3 surface having anelliptic fibration with Weierstrass equation Y 2 = X 3 + l(t), where X := v 2 x, Y := v 3 y.The surface S l admits an automorphism α l of order 3 induced by α E . The fixed locus ofα f consists of 2 rational curves and 5 points. Applying Rohde’s construction to the K3surface S l one obtains a CY threefold X such that h 2,1 (X)=q=4 and h 1,1 (X)=40.In case q = 5, one needs a K3 surface S with an automorphism α S of order 3which fixes one rational curve and 4 points (cf. [25]). In [1] a projective model of sucha surface is given: it is a (singular) complete intersection inP 4 with equations{F2 (x 0 ,...,x 3 )=0,G 3 (x 0 ,...,x 3 ) = x 3 4 ,

Examples of Calabi–Yau threefolds parametrised by Shimura varieties 285where F 2 and G 3 are homogeneous polynomials of degree 2 and 3 respectively. Moreover,the curve V(F 2 )∩V(G 3 ) has 4 singular points of type A 1 . The surface S is clearlya triple cover of the quadric defined F 2 = 0 in P 3 branched over the curve which isthe intersection of this quadric with the cubic surface defined by G 3 = 0 in P 3 . Theinverse image in S of a line in a ruling of the quadric in P 3 is an elliptic curve with acovering automorphism of order three which fixes the ramification points. Hence suchan elliptic curve is isomorphic to E. Thus S admits an isotrivial fibration (in generalwithout section) with general fiber isomorphic to E. In this case Rohde’s CY threefoldhas h 1,1 (X)=29.3.5. The complex ballWe briefly recall why the CY-type Hodge structures H 3 (X S ,Z) are parametrised by acomplex q-ball. More generally, with the notation from Section 1.3, consider polarizedweight three Hodge structures on(V Z∼ =Z 2(1+q) ,Q) of CY type which, moreover, admitan automorphism of order three φ:φ : V Z −→ V Z , Q(φx,φy) = Q(x,y), φ C (V p,q ) = V p,q , φ 3 = 1 VZ .Then we have a decomposition of V C into φ-eigenspaces, and we assume, asin the examples above, that the eigenspace of φ with eigenvalue ξ is exactly F 2 , soF 2 = F 2 ξ . Then the V p,q are also φ-eigenspaces:V C = V ξ ⊕ V ξ= V 3,0ξ⊕ V 2,1ξ⊕ V 2,1ξ⊕ V 0,3 .ξIn particular, the subspace F 2 , being an eigenspace of the fixed automorphism φ of V Z ,is now fixed in V C . It remains to find the moduli of V 3,0 inside F 2 = V 3,0 ⊕V 2,1 . Recallthe Hermitian form H on V C which is positive definite on V 3,0 and negative definite onV 2,1 . These two subspaces are perpendicular for H. Thus the unitary group of H |F 2is isomorphic to the group U(1,q). It is well-known that this group acts transitivelyon the orthogonal decompositions F 2 = W ⊕W ⊥ with H |W > 0 (and thus H |W ⊥ < 0).The stabiliser of a given decomposition is the subgroup U(1)×U(q), hence the modulispace of these decompositions is the Hermitian symmetric domainU(1,q)/(U(1)×U(q)) ∼ = B q = {w∈C q : ||w|| 0, that is, |w 0 | 2 > ∑ n j=1 |w j| 2 . In particular,w 0 ≠ 0 and so we may assume that w 0 = 1. Then w ′ is determined by the pointw :=(w 1 ,...,w q )∈C q with ∑ n j=1 |w j| 2 < 1, that is, a point of the q-ball. Conversely,

286 A. Garbagnati and B. van Geemengiven w ∈ B q , let w ′ = (1,w) and define V 3,0 = Cw ′ , V 2,1 = (V 3,0 ) ⊥ , the orthogonalcomplement, w.r.t. H, in F 2 of V 3,0 and define V 1,2 ,V 0,3 using V p,q = V q,p . One easilychecks that this gives a polarized Hodge structure on (V Z ,Q) which admits the automorphismφ.As we observed before in Sections 3.3, 3.4, the ball also parametrises familiesof curves, like the C f , and K3 surfaces, like the S f . Equivalently, it also parametrisescertain Hodge structures of weight one and two. The relation between these Hodgestructures is given by the “half twist” construction, see [27, 10].References[1] ARTEBANI M. AND SARTI A. Non-symplectic automorphisms of order 3 on K3 surfaces.Math. Ann. 342, 4 (2008), 903–921.[2] BINI G. Quotients of hypersurfaces in weighted projective space. arXiv:0905.2099, toappear in Adv. Geom.[3] BINI G., VAN GEEMEN B. AND KELLY T. L. Mirror quintics, discrete symmetries andShioda maps. arXiv:0809.1791, to appear in J. Algebraic Geom.[4] BORCEA C. Calabi-Yau threefolds and complex multiplication. In Essays on mirror manifolds.Int. Press, Hong Kong, 1992, pp. 489–502.[5] BORCEA C. K3 surfaces with involution and mirror pairs of Calabi-Yau manifolds. InMirror symmetry, II, vol. 1 of AMS/IP Stud. Adv. Math. Amer. Math. Soc., Providence, RI,1997, pp. 717–743.[6] CANDELAS P., DERRICK E. AND PARKES L. Generalized Calabi-Yau manifolds and themirror of a rigid manifold. Nuclear Phys. B 407, 1 (1993), 115–154.[7] CARLSON J., GREEN M. AND GRIFFITHS P. Variations of Hodge structure considered asan exterior differential system: old and new results. SIGMA Symmetry Integrability Geom.Methods Appl. 5 (2009), Paper 087, 40.[8] CHEN Y.-H., YANG Y. AND YUI N. Monodromy of Picard-Fuchs differential equationsfor Calabi-Yau threefolds. J. Reine Angew. Math. 616 (2008), 167–203. With an appendixby Cord Erdenberger.[9] COX D. A. AND KATZ S. Mirror symmetry and algebraic geometry, vol. 68 of MathematicalSurveys and Monographs. American Mathematical Society, Providence, RI, 1999.[10] DOLGACHEV I. V. AND KONDŌ S. Moduli of K3 surfaces and complex ball quotients.In Arithmetic and geometry around hypergeometric functions, vol. 260 of Progr. Math.Birkhäuser, Basel, 2007, pp. 43–100.[11] DORAN C., GREENE B. AND JUDES S. Families of quintic Calabi-Yau 3-folds withdiscrete symmetries. Comm. Math. Phys. 280, 3 (2008), 675–725.[12] DORAN C. F. AND MORGAN J. W. Mirror symmetry and integral variations of Hodgestructure underlying one-parameter families of Calabi-Yau threefolds. In Mirror symmetry.V, vol. 38 of AMS/IP Stud. Adv. Math. Amer. Math. Soc., Providence, RI, 2006, pp. 517–537.[13] GARBAGNATI A. New families of Calabi-Yau 3-folds without maximal unipotent monodromy.arXiv:1005.0094.

Examples of Calabi–Yau threefolds parametrised by Shimura varieties 287[14] GARBAGNATI A. AND VAN GEEMEN B. The Picard-Fuchs equation of a family of Calabi-Yau threefolds without maximal unipotent monodromy. Int. Math. Res. Not. IMRN, 16(2010), 3134–3143.[15] GERKMANN R., MAO S. AND ZUO K. Disproof of modularity of moduli space of CY3-folds of double covers of P3 ramified along eight planes in general positions. arXiv:0709.[16] GREENE B. R., PLESSER M. R. AND ROAN S.-S. New constructions of mirror manifolds:probing moduli space far from Fermat points. In Essays on mirror manifolds. Int. Press,Hong Kong, 1992, pp. 408–448.[17] GROSS B. H. A remark on tube domains. Math. Res. Lett. 1, 1 (1994), 1–9.[18] GROSS M., HUYBRECHTS D. AND JOYCE D. Calabi-Yau manifolds and related geometries.Universitext. Springer-Verlag, Berlin, 2003. Lectures from the Summer School heldin Nordfjordeid, June 2001.[19] GUKOV S. AND VAFA C. Rational conformal field theories and complex multiplication.Comm. Math. Phys. 246, 1 (2004), 181–210.[20] KREUZER M. AND SKARKE H. Complete classification of reflexive polyhedra in fourdimensions. Adv. Theor. Math. Phys. 4, 6 (2000), 1209–1230.[21] LIU K., SUN X. AND YAU S.-T. Recent development on the geometry of the Teichmüllerand moduli spaces of Riemann Surfaces and polarized Calabi-Yau manifolds.arXiv:0912.5471.[22] MEYER C. Modular Calabi-Yau threefolds, vol. 22 of Fields Institute Monographs. AmericanMathematical Society, Providence, RI, 2005.[23] ROHDE J. C. Calabi-Yau manifolds and generic Hodge groups. arXiv:1001.4239.[24] ROHDE J. C. Cyclic coverings, Calabi-Yau manifolds and complex multiplication,vol. 1975 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2009.[25] ROHDE J. C. Maximal automorphisms of Calabi-Yau manifolds versus maximally unipotentmonodromy. Manuscripta Math. 131, 3-4 (2010), 459–474.[26] SHIODA T. AND INOSE H. On singular K3 surfaces. In Complex analysis and algebraicgeometry. Iwanami Shoten, Tokyo, 1977, pp. 119–136.[27] VAN GEEMEN B. Half twists of Hodge structures of CM-type. J. Math. Soc. Japan 53, 4(2001), 813–833.[28] VAN GEEMEN B. AND TOP J. An isogeny of K3 surfaces. Bull. London Math. Soc. 38, 2(2006), 209–223.[29] VOISIN C. Miroirs et involutions sur les surfaces K3. Astérisque, 218 (1993), 273–323.Journées de Géométrie Algébrique d’Orsay (Orsay, 1992).AMS Subject Classification: 14J32, 14J33, 14D07Alice GARBAGNATI and Bert VAN GEEMEN,Dipartimento di Matematica, Università di Milano,Via Saldini 50, 20133 Milano, ITALIAe-mail: alice.garbagnati@unimi.it, lambertus.vangeemen@unimi.itLavoro pervenuto in redazione il 04.05.2010 e, in forma definitiva, il 15.09.2010