Existence of fiberwise symplectic form on symplectic fibration

Question

In this question, symplectic fibration is as usual, locally trivial fiber bundle $M$ over some smooth manifold $B$ whose fiber $(F, \omega)$ is a symplectic manifold.

Via its construction, we know there exists a $2$-form $\sigma_b$ for each fiber, which is symplectomorphic to $\omega$.

However, I've got confused why it's not obvious that there is a $2$-form $\sigma \in \Omega^2(M)$ such that $\sigma_b = \sigma|F_b$. Since we could pull back $\omega$ along its trivialization and they coincide at each points on overlapping, couldn't we conclude that there is a fiberwise symplectic $2$-form $\sigma \in \Omega^2(M)$?

I don't mind that $\sigma$ is symplectic, closed nor etc, but I think there is such $\sigma$. What am I missing? Since Thurston and several authors consider as there is no such $\sigma$. So, I would like to clarify why this happen.

Answer 1

Such a form exists, but your argument for its existence is flawed. As a concrete example, consider the map $\varphi:\mathbb{R}^3\to\mathbb{R}^3$ given by $$ \varphi\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}\cos(z)x+\sin(z)y\\\cos(z)y-\sin(z)x\\z\end{pmatrix} $$ For fixed $z$ this map is a symplectomorphism w.r.t. the standard symplectic structure $dx\wedge dy$ on $\mathbb{R}^2$, but if we treat $dx\wedge dy$ as a form on $\mathbb{R}^3$, we find that $\varphi^*(dx\wedge dy)\neq dx\wedge dy$. This is what is happening with your transition functions: the symplectomorphism condition is not enough to ensure that the forms induced by different trivializations agree on their common domain.

To describe things more abstractly, let $\pi:M\to B$ by a symplectic fibration with typical fiber $(F,\omega)$ and let $VM:=\ker(d\pi)$ be the vertical subbundle of $TM$. Given a neighborhood $U\subseteq B$ and a trivialization $\theta:\pi^{-1}(U)\to U\times F$ we can define a form $\omega_\theta\in\Omega^2(\pi^{-1}(U))$ by $\omega_{\theta}=\theta^*p_2^*\omega$, where $p_2:U\times F\to F$ is the projection onto the second factor.

The key point is that the $\omega_\theta$ need not agree on their common domain when considered as forms on $TM$. When restricted to $VM$, however, they do agree due to the following straightforward lemma.

Lemma: Let $\pi:M\to B$ be a smooth fiber bundle, and let $\iota_x:\pi^{-1}(x)\to M$ be the inclusion of the fiber above $x\in M$, and let $\alpha,\beta$ be covariant tensors on $VM$. $\alpha=\beta$ if and only if $\iota_x^*\alpha=\iota_x^*\beta$ for each $x\in M$.

This means that even if the $\omega_\theta$ do not agree on the horizontal part of $TM$, it suffices to construct a global form that agrees with $\omega_\theta$ on $VM$. This can be done e.g. with a partition of unity.