LVMB manifolds and simplicial spheres

TL;DR

This paper generalizes the polytopal structures associated with LVM manifolds to LVMB manifolds, demonstrating that their related simplicial spheres form a large class and enabling complex structures on many moment-angle complexes.

Contribution

It extends the connection between LVM manifolds and polytopes to LVMB manifolds, establishing a correspondence with a broad class of simplicial spheres and complex structures.

Findings

LVMB manifolds are associated with a large class of simplicial spheres.

Every sphere in this class can be realized as an LVMB manifold's associated sphere.

Many moment-angle complexes can be endowed with a complex structure.

Abstract

LVM and LVMB manifolds are a large family of examples of non k\"ahler manifolds. For instance, Hopf manifolds and Calabi-Eckmann manifolds can be seen as LVMB manifolds. The LVM manifolds have a very natural action of the real torus and the quotient of this action is a simple polytope. This quotient allows us to relate closely LVM manifolds to the moment-angle manifolds studied (for example) by Buchstaber and Panov. The aim of this paper is to generalize the polytopes associated to LVM manifolds to the LVMB case and study its properties. In particular, we show that it belongs to a very large class of simplicial spheres. Moreover, we show that for every sphere belonging to this class, we can construct a LMVB manifold whose associated sphere is the given sphere. We use this latter result to show that many moment-angle complexes can be endowed with a complex structure (up to product with circles).