Real quadrics in C
n
, complex manifolds and convex polytopes |
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Authors: | Frédéric Bosio Laurent Meersseman |
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Institution: | (1) Département de Mathématiques, Université de Poitiers, Boulevard Marie-et-Pierre-Curie Téléport 2, B.P. 30179, FR-86962 Futuroscope Chasseneuil Cedex, France;(2) Institut de Mathématiques de Bourgogne, Université de Bourgogne, B.P. 47870, FR-21078 Dijon Cedex, France |
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Abstract: | In this paper, we investigate the topology of a class of non-Kähler compact complex manifolds generalizing that of Hopf and Calabi-Eckmann manifolds. These manifolds are diffeomorphic to special systems of real quadrics C n which are invariant with respect to the natural action of the real torus (S 1) n onto C n . The quotient space is a simple convex polytope. The problem reduces thus to the study of the topology of certain real algebraic sets and can be handled using combinatorial results on convex polytopes. We prove that the homology groups of these compact complex manifolds can have arbitrary amount of torsion so that their topology is extremely rich. We also resolve an associated wall-crossing problem by introducing holomorphic equivariant elementary surgeries related to some transformations of the simple convex polytope. Finally, as a nice consequence, we obtain that affine non-Kähler compact complex manifolds can have arbitrary amount of torsion in their homology groups, contrasting with the Kähler situation. |
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Keywords: | Topology of non-K?hler compact complex manifolds Affine complex manifolds Combinatorics of convex polytopes Real quadrics Equivariant surgery Subspace arrangements |
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