Real structures of models of arrangements

We will deal with two “hidden” real structures in the theory of models of subspace arrangements. Given a real subspace arrangement A and its complexification A_C, the first structure is a real De Concini-Procesi model that can be seen as the manifold Y_A of (canonical) real points inside the complex De Concini-Procesi model Y_{A_C}. We will study its combinatorial properties by describing it as a quotient of a real model with corners CY_A introduced in 2003. A second structure arises, on the contrary, as an “extension” of CY_A, when A is a Coxeter arrangement. We will “add faces” to CY_A and obtain a convex body (or even a polytope); this gives rise to an interesting new family of “realized” posets which includes for instance Kapranov's permutoassociahedra.