The diffeomorphism type of canonical integrations of Poisson tensors on surfaces

A surface $\mathrm{\Sigma }$ endowed with a Poisson tensor $\pi$ is known to admit a canonical integration $\mathcal{G}(\pi )$, which is a 4-dimensional manifold with a (symplectic) groupoid structure. In this short note we show that when $\pi$ is not an area form on the 2-sphere, then $\mathcal{G}(\pi )$ is diffeomorphic to the cotangent bundle $T^{\ast }\mathrm{\Sigma }$, this extending results in \cite{Ma09} and \cite{BCST12}.