The cohomology rings of real toric spaces and smooth real toric varieties

TL;DR

This paper computes the cohomology rings of smooth real toric varieties and real toric spaces, providing an equivariant algebraic model that reveals smooth toric varieties as M-varieties.

Contribution

It introduces an explicit equivariant differential graded algebra model for the cohomology of real toric spaces and varieties, applicable with arbitrary coefficients.

Findings

Cohomology rings of smooth real toric varieties are explicitly computed.

Real toric spaces are shown to have an equivariant dga model.

Smooth toric varieties are established as M-varieties.

Abstract

We compute the cohomology rings of smooth real toric varieties and of real toric spaces, which are quotients of real moment-angle complexes by freely acting subgroups of the ambient 2-torus. The differential graded algebra we present is in fact an equivariant dga model, valid for arbitrary coefficients. We deduce from our description that smooth toric varieties are M-varieties.