Real quadrics in Cn, complex manifolds and convex polytopes

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 in 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.

Topology of moment-angle manifolds arising from flag nestohedra

“…To prove the statement (b), note, that a nerve complex of P e n is a barycentric subdivision of a boundary of ∆ n , and the nerve complex of St n+1 contains a cone over the nerve complex of P e n as an induced subcomplex (see Figure 2.3 for the case n = 2). Therefore, if K has a torsion element in H * (K), then the same is true for its triangulation, and by Theorem 1.1, the statement holds with K being a minimal triangulation of RP 2 on 6 vertices, as its barycentric subdivision is an induced subcomplex in a nerve complex of any P e n , n ≥ 5 (see also [3]). The final part follows from the Alexander duality and the Hochster theorem, and holds for any simple polytope of dimension less than 5.…”

“…Then P Γ is either a point, a segment, a pentagon or a hexagon. The corresponding moment-angle manifold Z P is either a disk D 2 , a sphere S 3 , or a connected sum of products of spheres, respectively ([20], [3]). These manifolds are formal spaces, therefore, there are no nontrivial higher Massey products in H * (Z P ).…”

Complex manifolds with maximal torus actions

“…Momentangle manifolds are topological manifolds equipped with actions of compact tori, constructed from simplicial complexes. In [4], the smooth manifolds underlying a large class of LVM manifolds are shown to be equivariantly homeomorphic to moment-angle manifolds coming from simplicial polytopal sphere. In [20] and [21], it is shown that even dimensional moment-angle manifolds coming from star-shaped simplicial spheres carry complex structures independently.…”

Биградуированные Числа Бетти Некоторых Простых Многогранников

“…Наконец, мы применяем наши вычисления к известному описанию с точностью до диффеоморфизма многообразий для многогранников усечения, см. [6].…”

“…Так как ( ) 3, любые диагонали ∈ 2 и ∈ 3 пересекают . Поэтому, и исходят из общей верши-ны многоугольника 6 . Мы получили противоречие с тем, что ( 2 ) и ( 3 ) не пересекаются (см.…”