Cohomological properties of ruled symplectic structures

Donaldson’s work on Lefschetz pencils has shown that, after a slight perturbation of the symplectic form and a finite number of blow-ups, any closed symplectic manifold (M,ω) can be expressed as a singular fibration with generic fiber a smooth codimension 2 symplectic submanifold. Thus fibrations play a fundamental role in symplectic geometry. It is then natural to study smooth (nonsingular) ruled symplectic manifolds (P,Ω), i.e. symplectic manifolds where P is the total space of a smooth fiber bundle M →֒ P → B and the symplectic form Ω is such that its restriction ω to each M -fiber is nondegenerate. In this survey, we present some of the recent results obtained by the authors and Leonid Polterovich in [9, 11, 10] that show that bundles endowed with such structures have interesting stability and cohomological properties. For instance, under certain topological conditions on the base, the rational cohomology of P necessarily splits as the tensor product of the cohomology of B with that of M . As we describe below in § 2, the bundle M →֒ P → B corresponding to a ruled symplectic manifold, has the group Ham(M,ω) of Hamiltonian diffeomorphisms of M for structure group if the base B is simply connected. They can therefore be divided into two classes: those whose structural group belongs to a finite dimensional Lie subgroup of the group Ham(M) of Hamiltonian diffeomorphisms of M , and those whose structural group is genuinely infinite dimensional. The first case belongs to the realm of classical symplectic geometry