Let $ (M, \omega) $ be a symplectic manifold. The de Rham class of $\omega$ induces a homomorphism $[\omega]: H\_2(M) \to \mathbb{R}$, whose image $\Gamma\_{\omega} \subseteq \mathbb{R}$ is called the group of periods of $\omega$. In the classical theory of prequantization one assumes that $\omega$ is integral, i.e. $\Gamma\_{\omega} \subseteq \mathbb{Z}$. More generally, one can construct a prequantization principal $\mathbb{R} / \Gamma\_\omega$ bundle if $\Gamma\_\omega$ is a discrete subgroup of $\mathbb{R}$. The only other case is that the group of periods is dense. 


Question: What are known examples of closed symplectic manifolds with a dense group of periods? Can one characterize such symplectic manifolds in a different way?



Symplectic manifolds with dense group of periods
The discrete subgroups of $\mathbb{R}$ are the groups $a\mathbb{Z}$ for $a\in\mathbb{R}$; the group of periods of $\omega$ is of this form if and only if some (real) multiple of $\omega $ is integral. Any projective manifold with $h^{1,1}>1$ gives a counter-example. Indeed the Kähler classes form an open convex cone in $H^{1,1}\_{\mathbb{R}}$, and a general line $\mathbb{R}\omega $ of this cone does not meet the integral lattice, so the group of periods of $\omega$ is dense.


