Holomorphic Poisson structures on complex manifolds are defined by holomorphic sections of the second exterior power of the tangent bundle subject to an integrability condition (vanishing of Schouten bracket of the section with itself). They are a common generalization of holomorphic symplectic structures and two-dimensional foliations with trivial canonical bundles.
The study of the global structure of holomorphic Poisson structures is still in its infancy. Except for papers by Polishchuck, Druel, Gualtieri-Pym, Lima-Pereira, and Pym, not much is known on the subject. Moreover, with the exception of Druel's results, they all concern Poisson structures on projective spaces or Fano manifolds. A systematic study of the structure of varieties carrying Poisson structures remains to be done. The goal of this Research in Pairs is to start carrying out such systematic study.
Stéphane Druel (CNRS, Université Grenoble Alpes)
Jorge Vitório Pereira (IMPA Rio & Aix-Marseille Université)
Brent Pym (University of Edinburgh)