Harmonic cohomology of symplectic manifolds
by
Stefan Haller
Department of Mathematics, University of Vienna, Austria

On symplectic manifolds one can speak of (symplectically) harmonic differential forms and thus of harmonic cohomology classes. These were introduced by Brylinski who also asked whether every cohomology class had a harmonic representative (`symplectic Hodge theory'). This turned out to be false in general. According to a theorem of Mathieu this is the case iff the manifold satisfies the `Hard Lefschetz Theorem'. We will see that one can explicitly compute the harmonic cohomology of a symplectic manifold in terms of its cohomology ring and the cohomology class of the symplectic form. Similar methods can be used to show that a class of symplectic manifolds (satisfying a weakened Lefschetz condition) has the c--splitting property. That is every Hamiltonian fiber bundle with such a manifold as typical fiber c--splits, i.e. the cohomology of the total space is additively the same as the cohomology of the product (trivial fiber bundle).

Date received: April 9, 2003