On the integral cohomology of real toric manifolds

Abstract

Real toric manifolds are the real loci of nonsingular complete toric varieties. We compute the integral cohomology groups of real toric manifolds from the combinatorial data encoded by the underlying simplicial fans, generalizing a formula due to Cai and Choi. Furthermore, we determine the image of the integral cohomology in the $\mathbb{Z}/2$-cohomology under the mod $2$ reduction map. As an application, a combinatorial-algebraic criterion is established for when a real toric manifold is $\mathrm{spin}^c$. We also describe the product structure of their integral cohomology rings. In particular, we establish a simple combinatorial formula for the quotient of the cohomology ring by the ideal consisting of elements annihilated by $2$.