Suppose $E \to B$ is a symplectic vector bundle, i.e. it possesses a fibrewise linear symplectic form $\omega\_F$. Further, suppose $\omega\_B$ is a symplectic form on $B$. 


Question: is there a symplectic form $\omega$ on $E$, such that (a) on any fibre of $E$, it restricts to $\omega\_F$, and (b) on the zero section it restricts to $\omega\_B$?


Remark: A closed 2-form can be obtained on $E$ that satisfies the first condition -- this can be done by choosing a connection on the associated principal bundle, and using Weinstein's theorem (Theorem 6.17 in McDuff-Salamon).



A symplectic form on a symplectic vector bundle
Such a symplectic form does not exist in general. In my paper (theorem 8.2). Gerbes, 2-gerbes and symplectic fibrations, I have shown that the obstruction of the existence of such a symplectic form on $E$ can geometrically be represented by a $2$-gerbes. My paper is available at 


<http://lanl.arxiv.org/pdf/math/0504274>


Tsemo Aristide


Gerbes 2-gerbes and symplectic fibrations.


Rocky Mountain J. Math. Volume 38, Number 3 (2008), 727-777.


