Banca examinadora: Henrique Bursztyn - Orientador - IMPA Roberto Rubio - Coorientador - Univ. Autonoma de Barcelona Reimundo Heluani - IMPA Cristian Ortiz - USP Alejandro Cabrera - UFRJ Thiago Linhares - Suplente - UFRJ.

Abstract: This thesis studies complex Dirac structures (i.e., Dirac structures in the complexification of the generalized tangent bundle of a manifold) with constant real index. These objects extend generalized complex structures, which arise when the real index is zero, and encode geometric structures such as presymplectic, transverse holomorphic and CR structures. We introduce a new invariant that we call order, which is an nonnegative integer that allows us to obtain a classification of complex Dirac structures at the linear-algebraic level. We prove that complex Dirac structures with constant real index and order carry a presymplectic foliation which comes from an underlying (real) Dirac structure (generalizing the Poisson structures associated with generalized complex structures). We prove a local splitting theorem for complex Dirac structures with constant real index and order which extends the Abouzaid-Boyarchenko's splitting theorem for generalized complex structures. Finally we focus on complex Dirac structures with real index one; we study a pairing , analogous to the Chevalley-Mukai pairing, which gives information about the dimension of the intersection of the annihilators of two pure spinors. We use it to give a spinorial description of complex Dirac structure with real index one .