Topology of real toric surfaces

We determine the homeomorphism type of the set of real points of a smooth projective toric surface. The purpose of this note is to prove the following topological classification of real toric surfaces. Theorem [De1, Proposition 4.1.2.] Let X = X(∆) be a smooth projective toric surface. If X is isomorphic to an even Hirzebruch surface F2a, for some integer a ≥ 0, then X(R) is homeomorphic to S × S. Otherwise, X(R) is homeomorphic to a connect sum of #∆(1)− 2 copies of RP. Here, ∆(1) denotes the set of 1-dimensional cones in ∆. We begin with a few preliminaries on the topology of real projective toric varieties in general, roughly following [GKZ, Chapter 11, Section 5]. Let X = X(∆) be a projective toric variety, and let P ⊂ MR be the polytope corresponding to an ample toric divisor on X . For each group homomorphism ǫ : M → {±1}, define Tǫ := {t ∈ T (R) ⊂ X(R) : sgn(χ (t)) = ǫ(u) for all u ∈ M}. Let Xǫ ⊂ X(R) be the closure of Tǫ in the real analytic topology. Let μ : X(R) → P be the moment map, and μǫ the restriction of μ to Xǫ. We claim that μǫ is a homeomorphism. The case ǫ0(M) = +1 is proved in [Ful, Section 4.2]. For general ǫ, the semigroup homomorphism ǫ : M → R corresponds to a point tǫ ∈ T (R). Translation by tǫ takes Xǫ0 to Xǫ and commutes with the moment map, so the claim follows. Now X(R) = ⋃ ǫ Xǫ. The following proposition describes Xǫ ∩Xǫ′ and the induced construction of X(R) by gluing 2 copies Pǫ of P , indexed by the group homomorphisms ǫ : M → {±1}. For a face F ⊂ P , let Fǫ denote the When this note was prepared and submitted to the arXiv, the author was not aware that these results (and much more) had already appeared in C. Delaunay’s work on real toric varieties [De1] [De2]. We hope that this note may serve as an expository introduction to some of the ideas and techniques in Delaunay’s work.