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 \(\textrm{Symp}^c_0(S, \omega )\) 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 \(\textrm{Ham}^c(S, \omega )\).

设\((S,\omega )\)为闭连通定向曲面，其亏格l至少为两个配备辛形式的曲面。然后我们展示了通勤辛同态通量杯积的消失。这个结果可以被视为对辛同态交换的阻碍。特别是，在通量同态下\(\textrm{Symp}^c_0(S, \omega )\)的阿贝尔子群的图像相对于\(H^1(S;{ \mathbb {R}})\)。证明的关键是对不可扩展性结果的细化，该结果由第一作者和第二作者先前给出，对于 Py 的 Calabi 拟同构\(\mu _P\) on \(\textrm{Ham}^c( S, Ω )\)。