They called this result a global torelli theorem as a

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Unformatted text preview: omology space obtained by integrating a nowhere vanishing holomorphic 2-form REFLECTION GROUPS IN ALGEBRAIC GEOMETRY 3 [94], [101]. They called this result a Global Torelli Theorem. As a corollary of this result they showed that the group of automorphisms of a K3 surface is isomorphic, up to a finite group, to the quotient of the orthogonal group of the integral quadratic form defined on the group of integral algebraic 2-cycles modulo the subgroup Γ generated by reflections in the cohomology classes of smooth rational curves lying on the surface. Thus they reduced the question of finiteness of the automorphism group to the question of finiteness of the volume of the fundamental polyhedron of Γ in a real hyperbolic space. V. Nikulin [87] and E. Vinberg [112] determined which isomorphism types of integral quadratic forms so arise and have the property that the fundamental polyhedron in question has finite volume. This solves, in principle, the problem of classification of fields of algebraic dimension 2 over C whose group of automorphisms over C is infinite. In the 1930s Patrick Du Val [37] found the appearance of Coxeter diagrams in resolution of certain types of singularities on algebraic surfaces (nowadays going under the different names: simple singularities, ADE singularities, Du Val singularities, double rational points, Gorenstein quotient singularities, and Klein singularities). However, Du Val did not find any reflection groups associated to these singularities. A conjectural relation to reflection groups and simple Lie algebras was suggested by A. Grothendieck in the sixties and was confirmed by a construction of E. Brieskorn [16] (full details appeared in [105]). The theory of finite complex reflection groups was developed by G. C. Shephard and A. Todd in 1954 as a follow-up to the classical work on groups of projective transformations generated by homologies (see [102]). Some examples of the arrangements of reflection hyperplanes and the hypersurfaces defined by polynomial invariants of the groups have been known in classical geometry since the 19th century. Infinite reflection groups of finite covolume in complex affine spaces were classified by V. Popov in 1982 [93]. They appear in the theory of compactification of versal deformation of simple elliptic singularities [74] and surface singularities with symmetries [50]. The most spectacular is the appearance of reflection groups in complex hyperbolic spaces of dimension > 1. Extending the work of H. Terada [109], P. Deligne and G. Mostow [28], [80] classified all hypergeometric functions whose monodromy groups Γ are discrete reflection groups of finite covolume in a complex ball Br (complex hyperbolic crystallographic groups, c.h.c. groups for short). The compactified orbit spaces Br /Γ turned out to be isomorphic to some geometric invariant quotients P1 (C)r+3 //PGL(2, C) for r ≤ 9. No other c.h.c. groups in Br had been discovered until a few years ago (except one missed case in Del...
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