Real Toric Varieties: Interactions between their Geometry and their Topology

TL;DR

This paper explores the topology of real toric varieties, focusing on non-split cases, by analyzing their fibrations, cohomology, Betti numbers, and classification of embeddings, revealing deep geometric-topological interactions.

Contribution

It provides new insights into the topology of non-split real toric varieties, including cohomology properties and classification results, extending previous understanding in the field.

Findings

Computed Betti numbers for certain real toric varieties

Established that their cohomology is totally algebraic

Classified equivariant embeddings of non-split 3-dimensional tori

Abstract

In the present article, we investigate the topology of real toric varieties, especially those whose torus is not split over the field of real numbers. We describe some canonical fibrations associated to their real loci. Then, we establish various properties of their cohomology provided that their real loci are compact and smooth. For instance, we compute their Betti numbers, show that their cohomology is totally algebraic, and extend a criterion of orientability. In addition, we provide the topological classification of equivariant embeddings of non-split tridimensional tori.