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Period- and mirror-maps for the quartic \(K3\). (English) Zbl 1303.14044
The paper gives a detailed description of mirror symmetry, in terms of Hodge structures, between quartic \(K3\) surfaces in projective space and the Dwork family of \(K3\) surfaces. More precisely, given a \(K3\) surface, we can endow integral cohomology with two different Hodge structures. The first one \(H_{A}\) depends only on the choice of Kähler form on X; the second \(H_{B}\) depends on its complex structure. P. S. Aspinwall and D. R. Morrison [AMS/IP Stud. Adv. Math. 1, 703–716 (1997; Zbl 0931.14020)] have described how mirror symmetry between two varieties \(X\) and \(X'\) can be seen as a Hodge isometry between \(H_A(X)\) and \(H_B(X')\). In this paper, \(X\) and \(X'\) are as follows: \(X_p\) is a family of quartic surfaces in projective space whose volume form is parametrized by a multiplicative factor \(p\); \(X'_t\) is the \(K3\) surface obtained as a fiber of the Dwork family \[ \left\{\sum_{i=0}^3 z_i^4 - 4 t z_0 z_1 z_2 z_3=0\right\}_{t\in\mathbb{C}} \] after taking a suitable resolution of a quotient by \(\mu_4\).
The author provide an explicit condition on \(t\) and \(p\) for \(X_{p}\) and \(X'_{t}\) to be mirrors, namely \[ \exp(2\pi p) = w + 104 w ^2 + 15188 w ^3 + 2585184 w ^4 + 480222434 w ^5 + \ldots \] where \(w = \frac{1}{(4t)^{4}}\). To establish the correspondence, the author computes the period maps for both families of surfaces and picks isomorphism between the Hodge structures at mirror points. He then proceeds to show that near points of maximal unipotent monodromy, this map is compatible with the Morrison mirror map [D. R. Morrison, in: Essays on mirror manifolds. Cambridge, MA: International Press. 241–264 (1992; Zbl 0841.32013)].

14J28 \(K3\) surfaces and Enriques surfaces
14J33 Mirror symmetry (algebro-geometric aspects)
53D37 Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
14D07 Variation of Hodge structures (algebro-geometric aspects)
Full Text: DOI arXiv
[1] Aspinwall, P.S., Morrison, D.R.: String theory on \(K\)3 surfaces. In: Mirror Symmetry II, vol. 1 of AMS/IP Studies in Advanced Mathematics, pp. 703-716. American Mathematical Society, Providence (1997) · Zbl 0931.14020
[2] Auroux, D.; Katzarkov, L.; Orlov, D., Mirror symmetry for del Pezzo surfaces: vanishing cycles and coherent sheaves, Invent. Math., 166, 537-582, (2006) · Zbl 1110.14033
[3] Bailey W.N.: Generalized Hypergeometric Series. Cambridge University Press, Cambridge (1935) · Zbl 0011.02303
[4] Beauville, A., Bourguignon, J.-P., Demazure, M.: Géométrie des surfaces \(K\)3: modules et périodes. Société Mathématique de France, Paris (1985) · Zbl 0947.14017
[5] Belcastro, S.-M., Picard lattices of families of \(K\)3 surfaces, Commun. Algebra, 30, 61-82, (2002) · Zbl 1053.14045
[6] Beukers, F.: Gauss’ hypergeometric function. In: Arithmetic and Geometry Around Hypergeometric Functions, vol. 260 of Progress in Mathematics, pp. 23-42. Birkhäuser, Basel (2007) · Zbl 1118.14012
[7] Candelas, P.; Ossa, X.; Green, P.S.; Parkes, L., A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nucl. Phys. B, 359, 21-74, (1991) · Zbl 1098.32506
[8] Carlson, J., Müller-Stach, S., Peters, C.: Period Mappings and Period Domains, vol. 85 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2003) · Zbl 1030.14004
[9] Dolgachev, I.V., Mirror symmetry for lattice polarized \(K\)3 surfaces, J. Math. Sci., 81, 2599-2630, (1996) · Zbl 0890.14024
[10] Efimov, A.I., A remark on mirror symmetry for curves, Uspekhi Mat. Nauk, 65, 191-192, (2010)
[11] Fukaya, K., Mirror symmetry of abelian varieties and multi-theta functions, J. Algebraic Geom., 11, 393-512, (2002) · Zbl 1002.14014
[12] Griffiths, P.A., On the periods of certain rational integrals. I, II, Ann. Math., 90, 496-541, (1969) · Zbl 0215.08103
[13] Griffiths P., Harris J.: Principles of Algebraic Geometry. Wiley Classics Library. Wiley, New York (1978) · Zbl 0408.14001
[14] Gross, M.; Siebert, B., From real affine geometry to complex geometry, Ann. Math.(2), 174, 1301-1428, (2011) · Zbl 1266.53074
[15] Gross, M., Katzarkov, L., Ruddat, H.: Towards mirror symmetry for varieties of general type. Preprint arXiv:1202.4042 (2012) · Zbl 1371.14046
[16] Huybrechts, D.: Moduli spaces of hyperkähler manifolds and mirror symmetry. In: Intersection Theory and Moduli. ICTP Lecture Notes XIX, pp. 185-247. The Abdus Salam International Centre for Theoretical Physics, Trieste (2004) · Zbl 1266.53074
[17] Huybrechts, D., Generalized Calabi-Yau structures. \(K\)3 surfaces and \(B\)-fields, Int. J. Math., 16, 13-36, (2005) · Zbl 1120.14027
[18] Ince E.L.: Ordinary Differential Equations. Dover Publications, New York (1944) · Zbl 0063.02971
[19] Iritani, H.: An integral structure in quantum cohomology and mirror symmetry for toric orbifolds. Preprint arXiv/0903.1463 (2009) · Zbl 1190.14054
[20] Kapustin, A.; Katzarkov, L.; Orlov, D.; Yotov, M., Homological mirror symmetry for manifolds of general type, Cent. Eur. J. Math., 7, 571-605, (2009) · Zbl 1200.53079
[21] Katzarkov, L.: Generalized homological mirror symmetry and rationality questions. In: Cohomological and Geometric Approaches to Rationality Problems, vol. 282 of Progress in Mathematics, pp. 163-208. Birkhäuser, Boston (2010) · Zbl 1210.14043
[22] Katzarkov, L., Kontsevich, M., Pantev, T.: Hodge theoretic aspects of mirror symmetry. In: From Hodge Theory to Integrability and TQFT tt*-Geometry, vol. 78 of Proceedings of Symposia in Pure Mathematics, pp. 87-174. American Mathematical Society, Providence (2008) · Zbl 1206.14009
[23] Klein F.: Vorlesungen über Hypergeometrische Funktionen. Springer, Berlin (1933) · Zbl 0007.12202
[24] Kontsevich, M.: Homological algebra of mirror symmetry. In: Proceedings of the International Congress of Mathematicians, vols. 1, 2 (Z ürich, 1994), pp. 120-139. Birkhäuser, Basel (1995) · Zbl 0846.53021
[25] Kontsevich, M., Soibelman, Y.: Homological mirror symmetry and torus fibrations. In: Symplectic Geometry and Mirror Symmetry (Seoul, 2000), pp. 203-263. World Scientific Publishing, River Edge (2001) · Zbl 1072.14046
[26] Lian B., H.; Yau, S.-T., Arithmetic properties of mirror map and quantum coupling, Commun. Math. Phys, 176, 163-191, (1996) · Zbl 0867.14017
[27] Morrison, D. R.: Picard-Fuchs equations and mirror maps for hypersurfaces. In: Essays on Mirror Manifolds, pp. 241-264. International Press, Hong Kong (1992) · Zbl 0841.32013
[28] Moser, J., On the volume elements on a manifold, Trans. Am. Math. Soc., 120, 286-294, (1965) · Zbl 0141.19407
[29] Nagura, M.; Sugiyama, K., Mirror symmetry of the \(K\)3 surface, Int. J. Mod. Phys. A, 10, 233-252, (1995) · Zbl 1044.14509
[30] Narumiya, N., Shiga, H.: The mirror map for a family of \(K\)3 surfaces induced from the simplest 3-dimensional reflexive polytope. In: Proceedings on Moonshine and Related Topics (Montréal, QC, 1999), vol. 30 of CRM Proceedings and Lecture Notes, pp. 139-161. American Mathematical Society, Providence (2001) · Zbl 1082.14519
[31] Nikulin V., V., Finite groups of automorphisms of K ählerian surfaces of type \(K\)3, Uspehi Mat. Nauk, 31, 223-224, (1976) · Zbl 0331.14019
[32] Nikulin, V.V.: Integral symmetric bilinear forms and some of their geometric applications. Izv. Akad. Nauk SSSR Ser. Mat. 43(1), 111-177, 238 (1979) · Zbl 0408.10011
[33] Nohara, Y., Ueda, K.: Homological mirror symmetry for the quintic 3-fold. Preprint arXiv:1103.4956 (2011)
[34] Orlov, D.O., Equivalences of derived categories and \(K\)3 surfaces, J. Math. Sci. (N.Y.), 84, 1361-1381, (1997) · Zbl 0938.14019
[35] Peters, C., Monodromy and Picard-Fuchs equations for families of \(K\)3-surfaces and elliptic curves, Ann. Sci. École Norm. Sup. (4), 19, 583-607, (1986) · Zbl 0612.14006
[36] Polishchuk, A.; Zaslow, E., Categorical mirror symmetry: the elliptic curve, Adv. Theor. Math. Phys, 2, 443-470, (1998) · Zbl 0947.14017
[37] Rohsiepe, F.: Lattice polarized toric K3 surfaces, arxiv:hep-th/0409290v1. Preprint arXiv:hep-th/0409290v1 (2004)
[38] Seidel, P.: Homological mirror symmetry for the quartic surface. Preprint arXiv/math.SG/0310414 (2003) · Zbl 1334.53091
[39] Seidel, P., Homological mirror symmetry for the genus two curve, J. Algebraic Geom., 20, 727-769, (2011) · Zbl 1226.14028
[40] Sheridan, N.: Homological mirror symmetry for Calabi-Yau hypersurfaces in projective space. Preprint arXiv:1111.0632 (2011) · Zbl 1344.53073
[41] Smith, J.P.: Picard-Fuchs Differential Equations for Families of K3 Surfaces. PhD thesis, University of Warwick, 2007 · Zbl 0612.14006
[42] Verrill, H., Yui, N.: Thompson series, and the mirror maps of pencils of \(K\)3 surfaces. In: The Arithmetic and Geometry of Algebraic Cycles (Banff., AB, 1998), vol. 24 of CRM Proceedings and Lecture Notes, pp. 399-432. American Mathematical Society, Providence (2000) · Zbl 0996.14018
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