TITLE: A symplectic form on a symplectic vector bundle
QUESTION [6 upvotes]: 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).

REPLY [9 votes]: 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.