reflections - BULLETIN(New Series OF THE AMERICAN...

BULLETIN (New Series) OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 45, Number 1, January 2008, Pages 1–60
S 0273-0979(07)01190-1
Article electronically published on October 26, 2007
REFLECTION GROUPS IN ALGEBRAIC GEOMETRY
IGOR V. DOLGACHEV
To Ernest Borisovich Vinberg
Abstract. After a brief exposition of the theory of discrete reflection groups
in spherical, euclidean and hyperbolic geometry as well as their analogs in
complex spaces, we present a survey of appearances of these groups in various
areas of algebraic geometry.
1. Introduction
The notion of a reflection in a euclidean space is one of the fundamental notions
of symmetry of geometric figures and does not need an introduction. The theory of
discrete groups of motions generated by reflections originates in the study of plane
regular polygons and space polyhedra, which goes back to ancient mathematics.
Nowadays it is hard to find a mathematician who has not encountered reflection
groups in his area of research. Thus a geometer sees them as examples of dis-
crete groups of isometries of Riemannian spaces of constant curvature or examples
of special convex polytopes. An algebraist finds them in group theory, especially
in the theory of Coxeter groups, invariant theory and representation theory. A
combinatorialist may see them in the theory of arrangements of hyperplanes and
combinatorics of permutation groups. A number theorist meets them in arithmetic
theory of quadratic forms and modular forms. For a topologist they turn up in
the study of hyperbolic real and complex manifolds, low-dimensional topology and
singularity theory. An analyst sees them in the theory of hypergeometric functions
and automorphic forms, complex higher-dimensional dynamics and ordinary differ-
ential equations. All of the above and much more appears in algebraic geometry.
The goal of this survey is to explain some of “much more”.
One finds an extensive account of the history of the theory of reflection groups in
euclidean and spherical spaces in Bourbaki’s Groupes et Alg` ebres de Lie , Chapters
IV-VI. According to this account the modern theory originates from the works of
geometers A. M¨obius and L. Schl¨ afli in the middle of the 19th century, then was
extended and applied to the theory of Lie algebras in the works of E. Cartan and
W. Killing at the end of the same century, and culminated in the works of H. S. M.
Coxeter [27]. The first examples of reflection groups in hyperbolic plane go back to
F. Klein and H. Poincar´ e at the end of the 19th century.
Received by the editors December 3, 2006, and, in revised form, May 17, 2007.
2000 Mathematics Subject Classification. Primary 20F55, 51F15, 14E02; Secondary 14J28,
14E07, 14H20, 11H55.
The author was supported in part by NSF grant no. 0245203.
c 2007 American Mathematical Society
1

2 IGOR V. DOLGACHEV
It is not general knowledge that reflection groups, finite and infinite, appeared
in 1885-1895 in the works of S. Kantor [63] on classification of subgroups of the
Cremona group of birational transformations of the complex projective plane [63].