  We define Symplectic cohomology groups for a class of symplectic fibrations
with closed symplectic base and convex at infinity fiber. The crucial geometric
assumption on the fibration is a negativity property reminiscent of negative
curvature in complex vector bundles. When the base is symplectically aspherical
we construct a spectral sequence of Leray-Serre type converging to the
Symplectic cohomology groups of the total space, and we use it to prove new
cases of the Weinstein conjecture.
