Symplectic area (${\int }_{\Sigma }\omega$)

The symplectic area of a compact symplectic surface $(\Sigma ,\omega )$ is ${\int }_{\Sigma }\omega$, using the orientation determined by the symplectic form.

Two compact connected symplectic surfaces are symplectomorphic exactly when their genus of a surface and symplectic area agree. The sufficiency follows from Moser theorem applied to a positive interpolation of their symplectic forms after an orientation-preserving diffeomorphism.