We study generalized almost contact structures on odd-dimensional manifolds. We introduce a notion of integrability and show that the class of these structures is closed under symmetries of the Courant-Dorfman bracket, including T-duality. We define a notion of geometric type for generalized almost contact structures, and study its behavior under T-duality.

We give several equivalent characterizations of orthogonal subbundles of the generalized tangent bundle defined, up to B-field transform, by almost product and local product structures. We also introduce a pure spinor formalism for generalized CRF-structure and investigate the resulting decomposition of the de Rham operator. As applications we give a characterization of generalized complex manifolds that are locally the product of generalized complex factors and discuss infinitesimal deformations of generalized CRF-structures.

Associated to every generalized complex structure is a differential Gerstenhaber algebra (DGA). When the generalized complex structure deforms, so does the associated DGA. In this paper, we identify the infinitesimal conditions when the DGA is invariant as the generalized complex structure deforms.We prove that the infinitesimal condition is always integrable. When the underlying manifold is a holomorphic Poisson nilmanifolds, or simply a group in the general, and the geometry is invariant, we find a general construction to solve the infinitesimal conditions under some geometric conditions. Examples and counterexamples of existence of solutions to the infinitesimal conditions are given.

Holomorphic Poisson structures arise naturally in the realm of generalized geometry. A holomorphic Poisson structure induces a deformation of the complex structure in a generalized sense, whose cohomology is obtained by twisting the Dolbeault ∂-operator by the holomorphic Poisson bivector field. Therefore, the cohomology space naturally appears as the limit of a spectral sequence of a double complex. The first sheet of this spectral sequence is simply the Dolbeault cohomology with coefficients in the exterior algebra of the holomorphic tangent bundle. We identify various necessary conditions on compact complex manifolds on which this spectral sequence degenerates on the level of the second sheet. The manifolds to our concern include all compact complex surfaces, Kähler manifolds, and nilmanifolds with abelian complex structures or parallelizable complex structures. IntroductionThe algebraic geometry of Poisson brackets over complex manifolds were studied for some time [22]. In recent years, there are significant interests on holomorphic Poisson structures due to their emergence in generalized complex geometry [12] . This observation motivates the authors' desire to find a way to compute the cohomology spaces H k Λ . Since this space is naturally the limit of a spectral sequence of a double complex due to the operators ∂ and [ [Λ, −]], it is an obvious question on when and how fast this spectral sequence could degenerate. We will see that the first sheet of this spectral sequence is the Dolbeault cohomology on the manifold with coefficients in the holomorphic polyvectors. As this is a classical object, it becomes desirable to identify the conditions under which the spectral sequence will degenerate at its second level.In this paper, we identify three situations in which the spectral sequence of the holomorphic Poisson double complex degenerates at its second level. In the next chapter, we will explain the role of holomorphic Poisson structures in generalized geometry, and then set up our computation of cohomology in the context of Lie bi-algebroids and their dual differentials [18]. Our reference for differential calculus of Lie algebroids is the book [19]. We will then set up the holomorphic Poisson double complex and its related spectral sequence.We will make observation on three very different kinds of complex manifolds: complex parallelizable, nilmanifolds with abelian complex structures, and Kählerian manifolds. This theorem will appear as Theorem 6. Note that it is well known that the Frölicher spectral sequence on a compact complex surface always degenerates [1]. The last theorem could be applied to all compact complex surfaces as well as all compact Kählerian manifolds [28]. Holomorphic Poisson Double Complex Holomorphic Poisson Structures as Generalized Complex StructuresA generalized complex manifold [12] [13] is a smooth 2n-dimensional manifold M equipped with a subbundle L of the bundle T = (TM ⊕ T * M) C such that -L and its conjugate bundle L are transversal; -L is maximally isotropic with...