Codimension-one Symplectic Foliations: Constructions and Examples

Abstract

This thesis addresses the question: which compact manifolds admit codimension-one symplectic foliations? It develops a method to construct such symplectic foliations on compact manifolds, called the “turbulisation method”. This method is applied then for the construction of such symplectic foliations on manifolds admitting certain type of open book decompositions and on products of the circle with manifolds admitting achiral Lefschetz fibrations. This applications allow us to enlarge the class of manifolds for which the answer to the main question is positive. The method relies on the existence of certain symplectic structures that are “constant” around the boundary. These symplectic structures are related with a special type of Poisson structures, called the log-symplectic structures. In the last part of the thesis, we study and characterise the space of deformations of log-symplectic structures.