A remark on the c--splitting conjecture

TL;DR

This paper proves the c--splitting conjecture for Hamiltonian fibrations over arbitrary bases with fibers satisfying a weakened Hard Lefschetz condition, extending previous results limited to sphere bases.

Contribution

It generalizes the c--splitting conjecture to broader classes of bases and fibers, building on Lalonde and McDuff's work.

Findings

Proves c--splitting conjecture for arbitrary base B.

Extends known results beyond sphere bases.

Uses a weakened Hard Lefschetz condition for fibers.

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 $H^*(P;\mathbb Q)=H^*(M;\mathbb Q)\otimes H^*(B;\mathbb Q)$ 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.