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istanbul - Moduli of K3 Surfaces and Complex Ball Quotients...

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Moduli of K3 Surfaces and Complex Ball Quo- tients Igor V. Dolgachev and Shigeyuki Kond¯ o 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 a 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. Mathematics Subject Classification (2000). Primary 14J10; Secondary 14J28, 14H15. Keywords. Hodge structure, Periods, Moduli, Abelian varieties, arrangements of hyperplanes, K 3 surfaces, Complex ball. 1. Introduction These notes are based on a series of talks at the CIMPA Summer School on Al- gebraic Geometry and Hypergeometric Functions held in Istanbul in Summer of 2005. The topic of the talks was an introduction to a recent work of various people on the complex ball uniformization of the moduli spaces of Del Pezzo surfaces and algebraic curves of lower genus ([ ? ], [DGK], [K1]-[K4], [HL]). Keeping in mind the diverse background of the audience we include a general introduction to the theory of Hodge structures and period domains with more emphasis on abelian varieties and K3 surfaces. So, an expert may start reading the notes from section 6. 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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2 Igor V. Dolgachev and Shigeyuki Kond¯ o It has been known for more than a century that a complex structure on a Riemann surface of genus g is determined up to isomorphism by the period ma- trix Π = ( γ j ω i ), where ( γ 1 , . . . , γ 2 g is a basis of 1-homology and ( ω 1 , . . . , ω g ) is a basis of holomorphic 1-forms. It is possible to choose the bases in a such a way that the matrix Π has the form ( Z I g ), where Z is a symmetric complex matrix of size g with positive definite imaginary part. All such matrices are parametrized by a complex domain Z g in C g ( g +1) / 2 which is homogeneous with respect to the group Sp(2 g, R ). In fact, it represents an example of a Hermitian symmetric space of non-compact type, a Siegel half-plane of degree g . A different choice of bases with the above property of the period matrix corresponds to a natural action of the group Γ g = Sp(2 g, Z ) on Z g . In this way the moduli space M g of complex structures on Riemann surfaces of genus g admits a holomorphic map to the orbit space Γ g \Z g which is called the period map. The fundamental fact is the Torelli Theorem which asserts that this map is an isomorphism onto its image. This gives a moduli theoretical interpretation of some points of the orbit space. All points
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