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Unformatted text preview: omology space obtained by integrating a nowhere vanishing holomorphic 2form 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 2cycles 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 followup 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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This note was uploaded on 02/24/2012 for the course MATH 285 taught by Professor Igordolgachev during the Fall '04 term at University of MichiganDearborn.
 Fall '04
 IgorDolgachev
 Algebra, Geometry, The Land

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