Examples of Calabi-Yau threefolds parametrised by Shimura varieties

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.