Foliations modeling nonrational simplicial toric varieties

TL;DR

This paper explores a new correspondence between nonrational simplicial fans and complex manifolds called LVMB-manifolds, extending the understanding of toric varieties beyond the rational case.

Contribution

It establishes a unified framework linking rational and nonrational toric varieties via foliated complex manifolds and computes their basic Betti numbers.

Findings

Computed basic Betti numbers for shellable fans

Proved basic cohomology is generated in degree two for polytopal fans

Unified rational and nonrational toric variety constructions

Abstract

We establish a correspondence between simplicial fans, not necessarily rational, and certain foliated compact complex manifolds called LVMB-manifolds. In the rational case, Meersseman and Verjovsky have shown that the leaf space is the usual toric variety. We compute the basic Betti numbers of the foliation for shellable fans. When the fan is in particular polytopal, we prove that the basic cohomology of the foliation is generated in degree two. We give evidence that the rich interplay between convex and algebraic geometries embodied by toric varieties carries over to our nonrational construction. In fact, our approach unifies rational and nonrational cases.