Commuting symplectomorphisms on a surface and the flux homomorphism

Let $(S,\omega)$ be a closed connected oriented surface whose genus $l$ is at least two equipped with a symplectic form. Then we show the vanishing of the cup product of the fluxes of commuting symplectomorphisms. This result may be regarded as an obstruction for commuting symplectomorphisms. In particular, the image of an abelian subgroup of $\mathrm{Symp}_0^c(S, \omega)$ under the flux homomorphism is isotropic with respect to the natural intersection form on $H^1(S;\mathbb{R})$. The key to the proof is a refinement of the non-extendability result, previously given by the first-named and second-named authors, for Py's Calabi quasimorphism $\mu_P$ on $\mathrm{Ham}(S, \omega)$.