Acta Mathematica

Real quadrics in Cn, complex manifolds and convex polytopes

Frédéric Bosio and Laurent Meersseman

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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 Cn which are invariant with respect to the natural action of the real torus (S1)n onto Cn. 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.


Dedicated to Alberto Verjovsky on his 60th birthday.

Article information

Acta Math., Volume 197, Number 1 (2006), 53-127.

Received: 5 October 2005
Revised: 3 April 2006
First available in Project Euclid: 31 January 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32Q55: Topological aspects of complex manifolds
Secondary: 32M17: Automorphism groups of Cn and affine manifolds 52B05: Combinatorial properties (number of faces, shortest paths, etc.) [See also 05Cxx] 52C35: Arrangements of points, flats, hyperplanes [See also 32S22]

Topology of non-Kähler compact complex manifolds Affine complex manifolds Combinatorics of convex polytopes Real quadrics Equivariant surgery Subspace arrangements

2006 © Institut Mittag-Leffler


Bosio, Frédéric; Meersseman, Laurent. Real quadrics in C n , complex manifolds and convex polytopes. Acta Math. 197 (2006), no. 1, 53--127. doi:10.1007/s11511-006-0008-2.

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