Real quadrics in Cn, complex manifolds and convex polytopes

By Frédéric Bosio, L. Meersseman. 2006

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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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/36d70a357871ba802067a10a666ecd21f4f2ff51

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Citations in the corpus

Moment-angle Complexes, Monomial Ideals and Massey Products (2005, result): “As mentioned in §1.5, work of López de Medrano–Verjovsky [39] and Meersseman [38] makes it possible to construct from combinatorial data compact, complex manifolds which are not algebraic, and even not Kähler or symplectic, except in very special cases.”
Toric Topology (2012, background): “[41] Frédéric Bosio and Laurent Meersseman.”
Intersections of Quadrics, Moment-angle Manifolds and Connected Sums (2009, methodology): “Particular members of such families are those moment-angle manifolds for which the result answers the Bosio–Meersseman conjecture in [8].”
Complex-analytic structures on moment-angle manifolds (2010, methodology): “restrict to the polytopal case and relate the complex structure on ZPcomingfrom the normalfan of Pvia the constructionofSection 3to the complex structure coming from identifying ZP as an LVM-manifold [BM]. We also relate the quadratic description of ZP to the quadratic description of LVM-manifolds. ”
Operations on polyhedral products and a new topological construction of infinite families of toric manifolds (2010, background): “nless indicated otherwise, all cohomology rings throughout are considered to be with integral coeﬃcients. The intersections of certain quadrics are known to be diﬀeomorphic to moment-angle manifolds, [4], [11] and implicitly in [5, Construction 3.1.8]. After a ﬁrst draft of this article was written ”
On the cohomology and their torsion of real toric objects (2013, background): “ described. Furthermore, most importantly, PB(K) supports a canonical real toric manifold explained later on. Remark 5.8. Bosio and Meersseman usedtheconstruction of Theorem11.11 and Theorem 11.12 of [2] to show that the cohomology of an LV-M manifold may have arbitrary amount of torsion. One can ea”
Spaces of polytopes and cobordism of quasitoric manifolds (2006, background): “The importance of the real quadratic viewpoint has been emphasised in recent work of Bosio and Meersseman [2], who consider a specific class of complete intersections of real quadrics in Cm, called links.”