A remark on the Brylinski conjecture for orbifolds

TL;DR

This paper extends Mathieu's result on symplectically harmonic forms to foliated manifolds and orbifolds, demonstrating that such representations are always possible for compact Kähler orbifolds, thus advancing the understanding of symplectic cohomology.

Contribution

It reformulates Mathieu's theorem for orbifolds and foliated manifolds, showing the universal existence of symplectically harmonic representatives in these contexts.

Findings

Representation of cohomology classes by harmonic forms is always possible for compact Kähler orbifolds.

Extension of Mathieu's result to foliated manifolds with transversally symplectic structures.

Provides a new perspective on the Brylinski conjecture for orbifolds.

Abstract

We present reformulation of Mathieu's result on representing cohomology classes of symplectic manifold with symplectically harmonic forms. We apply it to the case of foliated manifolds with transversally symplectic structure and to symplectic orbifolds. We obtain in particular that such representation is always possible for compact K\"{a}hler orbifolds.