Poisson cohomology of plane Poisson structures with isolated singularities revisited

Continuing the work of Monnier [Israel J. Math. 129, 189-207 (2002)], we determine the Gerstenhaber algebra structure over the Poisson cohomology groups for a large class of Poisson structures with isolated singularities over the plane. It reveals that there exists a GAGA-type phenomenon. We give an explicit description via generators and relations for the case of simple singularities.