  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 $\mathrm{Ham}(S, \omega)$.
