Contemporary Mathematics Volume 262, 2000 Tores Affines Yves Benoist ABSTRACT. The aim of these lecture notes is to describe the flat affine struc- tures on tori and more generally the flat projective structures on tori. We begin by the very explicit description of the flat affine structures and of the flat projective structures on the 2-dimensional torus due to Nagano, Yagi and Goldman. Then we explain the bricks theorem which roughly asserts that ev- ery flat projective structure with nilpotent holonomy on a compact manifold is obtained by glueing "bricks" . In general, the combinatorics of the bricks and of their glueing is far from being trivial. However, we give a few examples where these combinatorics can be completely described : *For projective structures on filiform nilmanifolds. * For complex projective structures on nilmanifolds. * For projective structures with diagonal holonomy (these structures were first introduced by Smillie). * For projective structures with unipotent holonomy. * For projective structures with cyclic holonomy. * For projective structures with nilpotent holonomy on low dimensional manifolds (dimension 3,4, ... ). As an application we describe the projective structures on 'll'3 . In the appendix we describe all the couples (B, Y), where B is a closed connected subgroup of the group A of diagonal n x n matrices with non zero real coefficients and where Y is an A-invariant open subset of Rn on which B acts properly with compact quotient B\ Y. The description uses "fans" . It is purely topological but has to be compared to the theory of "toric varieties" in algebraic geometry. Introduction Ces notes constituent une introduction a l'etude des structures affi.nes (plates) sur les tores et plus generalement des structures projectives (plates) sur les tores. Pour des introductions a d'autres aspects des varietes affi.nes et projectives, nous renvoyons a (21], (25], (7] et (15]. La premiere partie est consacree au tore 'll'2 : c'est surtout, pour nous, !'occasion de donner les definitions de base, de decrire tres explicitement les structures projectives sur le tore 'll'2 et d'aiguiser notre intuition. Les resultats de cette partie sont dils a Nagano, Yagi et Goldman. 2000 Mathematics Subject Classification. Primary 53A20, 58D27 Secondary 22E40, 57S30. © 2000 Yves Benoist http://dx.doi.org/10.1090/conm/262/04166
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