On the existence and classification of prequantization spaces over a closed symplectic manifold

Question

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).

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1. Why do we need this rationality condition ?
2. 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 $[\omega] \in H^2(M,\mathbb{Z})$ ?
3. Is there a proof of the existence of a prequantization space under this condition somewhere ?
4. Often the prequantization space is a complex line bundle. Is there an explicit identification between both notions somewhere ?
5. How could one classify all the prequantization spaces over a given closed symplectic manifold ?

Thanks in advance for your help !

Answer 1

There exists an exact sequence $0\rightarrow \mathbb{Z}\rightarrow\mathbb{C}\rightarrow\mathbb{C}^*\rightarrow 1$ defined by the expontial map which induces an isomorphism $H^2(M,\mathbb{Z})\rightarrow H^1(M,\mathbb{C}^*)$, this gives the correspondence between $H^2(M,\mathbb{Z})$ and line bundles which are classify by $H^1(M,\mathbb{C}^*)$ via the identification with Cech cohomology.

The previous correspondence identifies the element of $H^2(M,\mathbb{Z})$ with the Chern class of the line bundle, it is for that reason that the condition is needed. Given a line bundle define by the trivialization $(U_i,g_{ij}$, you can suppose that there exists a local lift $g'_{ij}:U_i\cap U_j\rightarrow \mathbb{C}$ and $c_{ijk}=g'_{ij}g'_{jk}g'_{ki}\in \mathbb{Z}$ is a way to describe the Chern class of the line bundle.

If the class of $\omega$ is not rational, you can associate to it a gerbe (sheaf of categories) I used this approach in this paper.

Aristide, Tsemo. "Gerbes, 2-gerbes and symplectic fibrations." The Rocky Mountain Journal of Mathematics (2008): 727-777.