Symplectic harmonicity and generalized coeffective cohomologies

By Raquel Villacampa (CUD, Zaragoza)

Let (M^2n, ω) be a symplectic manifold. The notion of symplectically harmonic form was introduced by Brylinski as a form α such that dα = 0 = d ∗ α, where ∗ is the symplectic star operator. Mathieu [2] proved that every de Rham cohomology class has a symplectically harmonic representative if and only if (M^2n, ω) satisfies the Hard Lefschetz Condition. If not, the number of de Rham cohomology classes containing harmonic representative is not determined by the topology of M. In fact, this number can vary if different symplectic structures are considered on M. When this occurs, the manifold is said to be flexible. Recently, Tseng and Yau have introduced other cohomologies on symplectic manifolds that admit unique harmonic representative within each class and showed that there exist primitive cohomologies associated with them such that their dimensions can vary with the class of the symplectic form, giving rise to another notion of flexibility, [3]. Additional symplectic invariants of cohomological type were introduced by Bouché [1] using coeffective forms, i.e. forms α such that α ∧ ω = 0. In this context it is possible to talk about coeffective flexibility.
In the present talk we relate all these symplectic cohomologies and we give conditions in low dimensions ensuring that all these notions of flexibility are equivalent.

References

[1] T. Bouché: La cohomologie coeffective d’une variété symplectique, Bull. Sci. Math. 114(2), 115–122 (1990).

[2] O. Mathieu: Harmonic cohomology classes of symplectic manifolds, Comment. Math. Helv. 70 1–9 (1995).

[3] L.-S. Tseng, S.-T. Yau: Cohomology and Hodge theory on symplectic manifolds: I and II, J. Differential Geom. 91, 383–416, 417–444 (2012).