Symplectic real Bott manifolds

Abstract

A real Bott manifold is the total space of an iterated $\RP ^1$-bundles over a point, where each $\RP^1$-bundle is the projectivization of a Whitney sum of two real line bundles. In this paper, we characterize real Bott manifolds which admit a symplectic form. In particular, it turns out that a real Bott manifold admits a symplectic form if and only if it is cohomologically symplectic. In this case, it admits even a K\"{a}hler structure. We also prove that any symplectic cohomology class of a real Bott manifolds can be represented by a symplectic form. Finally, we study the flux of a symplectic real Bott manifold.