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Affine tori. (Tores affines.) (French) Zbl 0990.53053

Igodt, Paul (ed.) et al., Crystallographic groups and their generalizations. II. Proceedings of the workshop, Katholieke Universiteit Leuven, Campus Kortrijk, Belgium, May 26-28, 1999. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 262, 1-37 (2000).
In these lecture notes, the flat affine and projective structures on tori are considered and described, leading to the following main results (1)–(4):
(1) a very explicit description of the (flat) affine and projective structures on the 2-dimensional torus – due to T. Nagano and K. Yagi [Osaka J. Math. 11, 181-210 (1974; Zbl 0285.53030)] and W. Goldman (1988/1990);
(2) the “bricks theorem” (Y. Benoist, 1994) which roughly asserts that every flat projective structure with nilpotent holonomy on a compact manifold is obtained by glueing “bricks” (= certain affine or projective, compact manifolds having vertices);
(3) the (non-trivial) combinatorics of the bricks and of their glueing is described completely for complex projective structures on nilmanifolds, for the projective structures on the 3-dimensional nilmanifold of Heisenberg, and for projective structures with diagonal/unipotent/cyclic or (in the case of dimension 2,3,4) nilpotent holonomy;
(4) the description of all projective structures on the torus \(\mathbb{T}^3\).
Finally, an appendix “Actions de matrices diagonales” provides a description – which is purely topological but has to be compared to the theory of “toric varieties” in algebraic geometry – of all couples \((B, Y)\), where \(B\) is a closed connected subgroup of the group \(A\) of diagonal \(n \times n\) matrices with non-zero real coefficients and where \(Y\) is an \(A\)-invariant open subset of \(\mathbb{R}^n\) on which \(B\) acts properly with compact quotient \(B\setminus Y\).
For the entire collection see [Zbl 0947.00024].

MSC:

53C29 Issues of holonomy in differential geometry
57S30 Discontinuous groups of transformations
58D27 Moduli problems for differential geometric structures
57N16 Geometric structures on manifolds of high or arbitrary dimension
22E40 Discrete subgroups of Lie groups
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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