Abstract
These notes are based on a series of talks given by the authors at the CIMPA Summer School on Algebraic Geometry and Hypergeometric Functions held in Istanbul in Summer of 2005. They provide an introduction to recent work on the complex ball uniformization of the moduli spaces of del Pezzo surfaces, K3 surfaces and algebraic curves of lower genus. We discuss the relationship of these constructions with the Deligne-Mostow theory of periods of hypergeometric differential forms. For convenience to a non-expert reader we include an introduction to the theory of periods of integrals on algebraic varieties with emphasis on abelian varieties and K3 surfaces.
Research of the first author is partially supported by NSF grant 0245203.
Research of the second author is partially supported by Grant-in-Aid for Scientific Research A-14204001, Japan.
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References
D. Allcock, J. A. Carlson, D. Toledo, The Complex Hyperbolic Geometry of the Moduli Space of Cubic Surfaces, J. Algebraic Geometry, 11 (2002), 659–724.
P. Deligne, a letter to I. M. Gelfand.
P. Deligne, G. W. Mostow, Monodromy of hypergeometric functions and nonlattice integral monodromy, Publ. Math. IHES, 63 (1986), 5–89.
I. Dolgachev, Mirror symmetry for lattice polarized K3-surfaces, J. Math. Sciences 81 (1996), 2599–2630.
I. Dolgachev, Weighted projective varieties, in “Group actions on algebraic varieties”, Lect. Notes in Math. 956 (1982), 34–71.
I. Dolgachev, B. van Geemen, S. Kondō, A complex ball uniformization of the moduli space of cubic surfaces via periods of K3 surfaces, math.AG/0310342, J. reine angew. Math. (to appear).
Géometrie des surfaces K3: modules et périodes, Astérisque, vol. 126, Soc. Math. France, 1985.
B. van Geemen, Half twists of Hodge structure of CM-type, J. Math. Soc. Japan, 53 (2001), 813–833.
Ph. Griffiths, J. Harris, Principles of Algebraic Geometry, Wiley and Sons, 1994.
G. Heckman, E. Looijenga, The moduli space of rational elliptic surfaces, Adv. Studies Pure Math., 36 (2002), Algebraic Geometry 2000, Azumino, 185–248.
S. Kondō, A complex hyperbolic structure of the moduli space of curves of genus three, J. reine angew. Math., 525 (2000), 219–232.
S. Kondō, The moduli space of curves of genus 4 and Deligne-Mostow’s complex reflection groups, Adv. Studies Pure Math., 36 (2002), Algebraic Geometry 2000, Azumino, 383–400.
S. Kondō, The moduli space of 8 points on ℐ1 and automorphic forms, to appear in the Proceedings of the Conference “Algebraic Geometry in the honor of Professor Igor Dolgachev”.
S. Kondō, The moduli space of 5 points on ℙ1 and K3 surfaces, math.AG/0507006, this volume.
E. Looijenga, Uniformization by Lauricella functions-an overview of the theory of Deligne-Mostow, math.CV/050753, this volume.
K. Matsumoto, T. Terasoma, Theta constants associated to cubic threefolds, J. Alg. Geom., 12 (2003), 741–755.
G. W. Mostow, Generalized Picard lattices arising from half-integral conditions, Publ. Math. IHES, 63 (1986), 91–106.
V. V. Nikulin, Integral symmetric bilinear forms and its applications, Math. USSR Izv., 14 (1980), 103–167.
V. V. Nikulin, Finite automorphism groups of Kähler K3 surfaces, Trans.Moscow Math. Soc., 38 (1980), 71–135.
V. V. Nikulin, Factor groups of groups of automorphisms of hyperbolic forms with respect to subgroups generated by 2-reflections, J. Soviet Math., 22 (1983), 1401–1475.
I. Piatetski-Shapiro, I. R. Shafarevich, A Torelli theorem for algebraic surfaces of type K3, Math. USSR Izv., 5 (1971), 547–587.
G. Shimura, On purely transcendental fields of automorphic functions of several variables, Osaka J. Math., 1 (1964), 1–14.
T. Terada, Fonction hypergéométriques F 1 et fonctions automorphes, J. Math. Soc. Japan 35 (1983), 451–475; II, ibid., 37 (1985), 173–185.
T. Terasoma, Infinitesimal variation of Hodge structures and the weak global Torelli theorem for complete intersections, Ann. Math., 132 (1990), 37 (1985), 213–235.
W. P. Thurston, Shape of polyhedra and triangulations of the sphere, Geometry & Topology Monograph, 1 (1998), 511–549.
A. Varchenko, Hodge filtration of hypergeometric integrals associated with an affine configuration of hyperplanes and a local Torelli theorem, I. M. Gelfand Seminar, Adv. Soviet Math., 16, Part 2, Amer. Math. Soc., Providence, 1993, J. Math. Soc. Japan 35 (1983), pp. 167–177.
C. Voisin, Hodge Theory and Complex Geometry, Cambridge Studies in Adv. Math., vols. 76, 77, Cambridge Univ. Press, 2003.
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Dolgachev, I.V., Kondō, S. (2007). Moduli of K3 Surfaces and Complex Ball Quotients. In: Holzapfel, RP., Uludağ, A.M., Yoshida, M. (eds) Arithmetic and Geometry Around Hypergeometric Functions. Progress in Mathematics, vol 260. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8284-1_3
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