A remark on the c--splitting conjecture

Abstract

Let $M$ be a closed symplectic manifold and suppose $M\to P\to B$ is a Hamiltonian fibration. Lalonde and McDuff raised the question whether one always has $H^{\ast }(P;\mathbb{Q})=H^{\ast }(M;\mathbb{Q})\otimes H^{\ast }(B;\mathbb{Q})$ as vector spaces. This is known as the c--splitting conjecture. They showed, that this indeed holds whenever the base is a sphere. Using their theorem we will prove the c--splitting conjecture for arbitrary base $B$ and fibers $M$ which satisfy a weakening of the Hard Lefschetz condition.