Cohomologies of generalized complex manifolds and nilmanifolds

TL;DR

This paper investigates the cohomological properties of generalized complex structures, especially on nilmanifolds and their deformations, using symplectic fiber bundles over complex manifolds.

Contribution

It introduces new methods to compute generalized complex cohomologies on nilmanifolds and analyzes their deformation spaces.

Findings

Cohomologies are explicitly computed for certain nilmanifolds.

Deformation spaces of generalized complex structures are characterized.

New relations between symplectic fiber bundles and generalized complex cohomology are established.

Abstract

We study generalized complex cohomologies of generalized complex structures constructed from certain symplectic fibre bundles over complex manifolds. We apply our results in the case of left-invariant generalized complex structures on nilmanifolds and to their space of small deformations.