Poisson structures on smooth 4-manifolds
  We show that every closed oriented smooth 4-manifold admits a complete singular Poisson structure in each homotopy class of maps to the 2-sphere. The rank of this structure is 2 outside a small singularity set, which consists of finitely many circles and isolated points. The Poisson bivector has rank 0 on the singularities, where we give its local form explicitly. 
Luis C. Garc\'ia-Naranjo, Pablo Su\'arez-Serrato, and Ram\'on Vera
math.DG math-ph math.MP math.SG