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The monodromy of the period map of a 4-parameter family of K3 surfaces and the hypergeometric function of type (3,6). (English) Zbl 0763.32016

There are several ways to consider a \(K3\)-surface as a 2-dimensional analogue of an elliptic curve. This paper gives another example of this kind: If \(\eta\) denotes a nonzero holomorphic 1-form of an elliptic curve \(E\), and \(\gamma_ 1\), \(\gamma_ 2\) a basis of \(H_ 1(E,\mathbb{Z})\), the integrals \(\omega_ i=\int_{\gamma_ i}\eta\) give periods of the curve \(E\). Under suitable normalization \(\omega_ 1\) and \(\omega_ 2\) form a set of linearly independent solutions of the hypergeometric differential equation \(E\bigl({1\over 2},{1\over 2},1\bigr)\). Moreover the projectivization of the period map \(z\mapsto\omega_ 1(z)/\omega_ 2(z)\) gives a multivalued map \(\mathbb{C}-\{0,1\}\to{\mathfrak H}=\{z>0\}\) for which there is a single-valued inverse map that gives rise to an isomorphism \({\mathfrak H} \bmod \Gamma(2)@>\sim>>\mathbb{C}-\{0\}\).
The present paper presents an exact 2-dimensional analogue of these results. The authors consider a 4-dimensional family of \(K3\)-surfaces, namely the double cover of the projective plane branched along 6 lines. The configuration space \(X\) of the 6 lines may be considered as the algebraic moduli space of this family. On the other hand, the space of periods of the \(K3\)-surfaces is the transcendental moduli space of the family, which is isomorphic to a bounded symmetric domain \(D\) modulo a discrete subgroup \(\Gamma_ A(2)\). The projectivization of the period map gives an isomorphism between the algebraic moduli space and the transcendental moduli space \(D/\Gamma_ A(2)\). Moreover the period map is described by the solutions of the hypergeometric system of type (3,6). Finally it is shown that a compactification \(\overline X\) of \(X\) is isomorphic to the Satake compactification of the quotient space \(\Gamma_ A(2)\backslash D\) and that the family of curves of genus 2 can be considered as a 3-dimensional subfamily of \(X\).
Reviewer: H.Lange (Erlangen)

MSC:

32G20 Period matrices, variation of Hodge structure; degenerations
33C70 Other hypergeometric functions and integrals in several variables
14J25 Special surfaces
32J05 Compactification of analytic spaces
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