A second cohomology class of the group cohomology of the symplectomorphism group is defined for a symplectic manifold with first Chern class proportional to the class of symplectic form and with trivial first real cohomology. Some properties of it are studied. In particular, it is characterized in terms of cohomology classes of the universal symplectic fiber bundle over the classifying space of the symplectomorphism group with the discrete topology.
"A second cohomology class of the symplectomorphism group with the discrete topology." Algebr. Geom. Topol. 18 (1) 187 - 219, 2018. https://doi.org/10.2140/agt.2018.18.187