Fibered Symplectic Cohomology and the Leray-Serre Spectral Sequence

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.