Symplectic real Bott manifolds

TL;DR

This paper characterizes which real Bott manifolds admit symplectic forms, showing they are exactly the cohomologically symplectic ones, and proves they can support Kähler structures and symplectic forms representing any cohomology class.

Contribution

It provides a complete characterization of symplectic real Bott manifolds and establishes their Kähler structures and the representability of cohomology classes by symplectic forms.

Findings

A real Bott manifold admits a symplectic form iff it is cohomologically symplectic.

Such manifolds also admit Kähler structures.

Any symplectic cohomology class can be represented by a symplectic form.

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.