Generalized Contact Structures

We study integrability of generalized almost contact structures, and find conditions under which the main associated maximal isotropic vector bundles form Lie bialgebroids. These conditions differentiate the concept of generalized contact structures from a counterpart of generalized complex structures on odd-dimensional manifolds. We name the latter strong generalized contact structures. Using a Boothby-Wang construction bridging symplectic structures and contact structures, we find examples to demonstrate that, within the category of generalized contact structures, classical contact structures have non-trivial deformations. Using deformation theory of Lie bialgebroids, we construct new families of strong generalized contact structures on the three-dimensional Heisenberg group and its co-compact quotients.Comment: 35 pages. To appear in Journal of LM

Geometry of Hyper-K\"ahler Connections with Torsion

The internal space of a N=4 supersymmetric model with Wess-Zumino term has a connection with totally skew-symmetric torsion and holonomy in \SP(n). We study the mathematical background of this type of connections. In particular, we relate it to classical Hermitian geometry construct homogeneous as well as inhomogeneous examples, characterize it in terms of holomorphic data, develop its potential theory and reduction theory.Comment: 21 pages, LaTe

Holomorphic Poisson Cohomology

A holomorphic Poisson structure induces a deformation of the complex structure as Hitchin's generalized geometry. Its associated cohomology naturally appears as the limit of a spectral sequence of a double complex. The first sheet of this spectral sequence is 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, K\"ahler manifolds, and nilmanifolds with abelian complex structures or complex parallelizable manifolds