Online Talk

Real toric manifolds are the real loci of nonsingular complete (smooth compact in the usual topology) toric varieties.  A fundamental problem is to determine their integral cohomology. However, unlike the complex case or their mod 2 cohomology, their integral cohomology is usually difficult to compute. In this talk, we describe the additive structure of their integral cohomology in terms of the combinatorial data contained in the underlying simplicial fans, generalizing a formula due to Cai and Choi for real toric maifolds associated to shellable fans. The image of their integral cohomology in the mod 2 cohomology is also determined. As an application, a combinatorial-algebraic criterion is established for when a real toric manifold is spin$^c$. By using an equivariant DAG model, we also present a combinatorial formula to describe the cohomology ring structure, modulo the ideal consisting of elements of order 2, which is analogous to the formula of Choi and Park for $R$-cohomology with $1/2\in R$. Moreover, since the proofs are purely topological, all these results also hold for real topological toric manifolds, a topological generalization of real toric manifolds.