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New families of Calabi-Yau threefolds without maximal unipotent monodromy. (English) Zbl 1260.14047

In this paper, five new families of Calabi-Yau threefolds without maximal unipotent monodromy are constructed. For these families, the variation of their Hodge structures and their parametrizing spaces are described, and their Picard-Fuchs differential equations are explicitly determined. Since these families have no boundary points with maximal unipotent monodromy, the mirror families cannot be readily defined within the usual mirror symmetry framework.
The construction of these new families of Calabi-Yau threefolds are done as follows: Start with \(K3\) surfaces and elliptic curves admitting non-symplectic automorphisms (of order \(4\)). Consider a product of such a \(K3\) surface \(S\) and such an elliptic curve \(E\). Next take the quotient of the product \(E\times S\) by the non-symplectic automorphism of order \(4\). Resolving singularities, one obtains a family of Calabi-Yau threefolds.
The \(K3\) surfaces used in this construction are rather special. Indeed, a \(K3\) surface is realized as the quotient of the product of an elliptic curve \(E\) (admitting a non-symplectic automorphism \(\alpha_E\) of order \(4\)) with a curve \(C\) of genus \(g\in\{1,2,3\}\) admitting the automorphism \(\alpha_C\) with a specific property. Then the quotient of the product \(E\times C\) by the automorphism \(<\alpha_E, \alpha_C>\) admits a desingularization \(S\), which is a \(K3\) surface with non-symplectic automorphism \(\alpha_S\) of order \(4\). This yields five families of \(K3\) surfaces with non-symplectic automorphisms \(\alpha_S\) of order \(4\).
Next consider the product \(E\times S\) of an elliptic curve \(E\) and a K3 surface \(S\), where \(S\) and \(E\) are one of the \(K3\) surfaces and elliptic curves, respectively, considered above. Then the desingularization of its quotient by the automorphism \(\alpha_E^3\times \alpha_S\) gives rise to five families of Calabi-Yau threefolds with Hodge numbers \((h^{1,1},h^{2,1}) \in\{(90,0), (73,1), (56,2), (61,1), (39,3)\}\).
For these five families of Calabi-Yau threefolds, the parametrizing spaces are also determined. Also the Picard-Fuchs differential equations are computed, based on the fact that the variation of the Hodge structures depends only on the variation of the Hodge structures of the elliptic curve component. In particular, this results in the assertion that these families of Calabi-Yau threefolds admit no maximal unipotent monodromy.

MSC:

14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14J28 \(K3\) surfaces and Enriques surfaces
14J33 Mirror symmetry (algebro-geometric aspects)
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
14D07 Variation of Hodge structures (algebro-geometric aspects)
14J10 Families, moduli, classification: algebraic theory
14J50 Automorphisms of surfaces and higher-dimensional varieties
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References:

[1] Artebani M.: A one dimensional family of K3 surfaces with a $${\(\backslash\)mathbb{Z}/4\(\backslash\)mathbb{Z}}$$ action. Can. Math. Bull. 52, 493–510 (2009) · Zbl 1180.14038
[2] Borcea C.: K3 surfaces with involution and mirror pairs of Calabi-Yau manifolds, in: mirror symmetry, II. AMS/IP Stud. Adv. Math. 1, 717–743 (1997) · Zbl 0939.14021
[3] Cynk S., Hulek K.: Higher dimensional modular Calabi-Yau manifolds. Can. Math. Bull. 50, 486–503 (2007) · Zbl 1141.14009
[4] Dolgachev, I.V., Kondo, S.: Moduli of K3 surfaces and complex ball quotients. In: progress in mathematics. Arithmetic and Geometry Around Hypergeometric Functions, vol. 260, pp. 43–100. Birkhäuser, Basel (2007) · Zbl 1124.14032
[5] Garbagnati A., van Geemen B.: The Picard-Fuchs equation of a family of Calabi-Yau threefolds without maximal unipotent monodromy. Int. Math. Res. Notices 16, 3134–3143 (2010) · Zbl 1207.14042
[6] van Geemen, B.: Projective models of Picard modular varieties. In: Classification of Irregular Varieties, vol. 1515, pp. 68–99. Springer LNM, York (1992) · Zbl 0793.14010
[7] Miranda, R.: The basic theory of elliptic surfaces, Università à di Pisa, Dottorato di ricerca in matematica, ETS Editrice Pisa (1988)
[8] Nikulin V.V.: Finite groups of automorphisms of Kählerian K3 surfaces (Russian). Trudy Moskov. Mat. Obshch. 38, 75–137 (1979) · Zbl 0433.14024
[9] Nikulin V.V.: Finite groups of automorphisms of Kählerian K3 surfaces. Trans. Moscow Math.Soc. 38, 71–135 (1980) · Zbl 0454.14017
[10] Nikulin V.V.: Factor groups of groups of the automorphisms of hyperbolic forms with respect to subgroups generated by 2-reflections. J. Sov. Math. 22, 1401–1475 (1983) · Zbl 0508.10020
[11] Rohde, J.C.: Cyclic Coverings, Calabi-Yau Manifolds and Complex Multiplication. LNM 1975. Springer, Berlin (2009) · Zbl 1168.14001
[12] Rohde J.C.: Maximal automorphisms of Calabi-Yau manifolds versus maximally unipotent monodromy. Manuscr. Math. 131, 459–474 (2010) · Zbl 1189.14014
[13] Shioda, T., Inose, H.: On Singular K3 surfaces. In: Complex Analysis and Algebraic Geometry, pp. 119–136. Iwanami Shoten, Tokyo (1977) · Zbl 0374.14006
[14] Voisin, C.: Miroirs et involutions sur les surfaces K3. Journées de Géométrie Algébrique d’Orsay (Orsay, 1992). Astérisque 218, 273–323 (1993)
[15] Voisin, C.: Théorie de Hodge et Géométrie Algébrique Complexe, Cours Spécial 10. Société Mathematique de France, Paris (2002)
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