Between generalized complex and Poisson geometry: Elliptic symplectic structures, boundary fibrations and homogeneous vector fields

This thesis studies two specific geometric structures: generalized complex structures and Poisson structures. In the first part the class of self-crossing stable generalized complex structures is introduced. We describe how these structures are determined by their underlying Poisson structures, which belong to the class of self-crossing elliptic symplectic structures. These are symplectic structures on the Lie algebroid, called the self-crossing elliptic tangent bundle. We show how to construct self-crossing elliptic symplectic structures using connected sums and how in four dimensions they can be deformed into Poisson structures with embedded degeneracy locus (called smooth elliptic symplectic structures). Moreover, we introduce the notion of boundary Lefschetz fibrations, which are singular fibrations used to better understand these structures. We furthermore make the first steps into describing their deformations, by studying the Lie algebroid cohomology of the self-crossing elliptic tangent bundle. In the second part we study (homogeneous) vector fields on vector bundles. We carefully describe the underlying filtered Gerstenhaber structure, and describe relations with representations up to homotopy. These structures give us some insight in normal forms result in Poisson geometry, which is the final chapter in this thesis.