Poisson Structures on Projective Manifolds
Renan Edgard Pereira de Lima
foliations | Poisson structure | Fano manifolds
We are interested to study nondegenerate Poisson structures in Fano manifolds with cyclic Picard group. The singular loci of the Poisson structure in our case are a hypersurface and it is known that this hypersurface is always singular. The idea of this thesis is study the Poisson structures with the simplest singular loci possible. If the irreducible components of the singular loci of the Poisson structure are reduced, smooth and in normal crossing position, the main theorem of the thesis asserts that the Fano manifold is the projective space and the singular loci of the Poisson structure are union of hyperplanes in general position, in other words, the Poisson structure is diagonal. We also proved that small deformations of a generic diagonal Poisson structure are still diagonal Poisson structures.