Let $(M,\omega)$ be a closed symplectic manifold. If the cohomology class $[\omega]$ is rational, that is if it lies in the image of the natural homomorphism $H^2(M,\mathbb{Z}) \to H^2(M,\mathbb{R})$, one can construct a prequantization bundle over $(M,\omega)$, that is a principal $S^1$-bundle $$(V,\alpha) \to (M,\omega),$$ where $\alpha$ is an $S^1$-invariant one-form on $V$ such that $\pi^* \omega = d \alpha$ (it is then a contact form).

- Why do we need this rationality condition ?
- Is there a case where the homomorphism $H^2(M,\mathbb{Z}) \to H^2(M,\mathbb{R})$ is not an embedding ? Otherwise, isn't the rationality condition equivalent to $[\omega] \in H^2(M,\mathbb{Z})$ ?
- Is there a proof of the existence of a prequantization space under this condition somewhere ?
- Often the prequantization space is a complex line bundle. Is there an explicit identification between both notions somewhere ?
- How could one classify all the prequantization spaces over a given closed symplectic manifold ?

Thanks in advance for your help !