Generalized complex structure on certain principal torus bundles

Poddar M., Thakur A.S.

AbstractWe give a construction of integrable complex structures on the total space of a smooth principal bundle over a complex manifold, with an even dimensional compact Lie group as structure group, under certain conditions. This generalizes the constructions of complex structure on compact Lie groups by Samelson and Wang, and on principal torus bundles by Calabi-Eckmann and others. It also yields large classes of new examples of non-Kähler compact complex manifolds. Moreover, under suitable restrictions on the base manifold, the structure group, and characteristic classes, the total space of the principal bundle admits SKT metrics. This generalizes recent results of Grantcharov et al. We study the Picard group and the algebraic dimension of the total space in some cases. We also use a slightly generalized version of the construction to obtain (non-Kähler) complex structures on tangential frame bundles of complex orbifolds.

Angella D., Calamai S., Kasuya H.

We study generalized complex cohomologies of generalized complex structures constructed from certain symplectic fiber 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.

Gualtieri M.

Generalized complex geometry encompasses complex and symplectic ge- ometry as its extremal special cases. We explore the basic properties of this geometry, including its enhanced symmetry group, elliptic deforma- tion theory, relation to Poisson geometry, and local structure theory. We also dene and study generalized complex branes, which interpolate be- tween at bundles on Lagrangian submanifolds and holomorphic bundles on complex submanifolds.

Cavalcanti G.R.

We study the decomposition of forms induced by a generalized complex structure giving a complete description of the bundles involved and, around regular points, of the operators ∂ and ∂ ¯ associated to the generalized complex structure. We prove that if the generalized ∂ ∂ ¯ -lemma holds then the decomposition of forms gives rise to a decomposition of the cohomology of the manifold, H • ( M ) = ⊕ − n n G H k ( M ) , and the canonical spectral sequence degenerates at E 1 . We also show that if the generalized ∂ ∂ ¯ -lemma holds, any generalized complex submanifold can be associated to a Poincaré dual cohomology class in the middle cohomology space G H 0 ( M ) .

Hitchin N.

Using the idea of a generalized Kähler structure, we construct bihermitian metrics on CP2 and CP1×CP1, and show that any such structure on a compact 4-manifold M defines one on the moduli space of anti-self-dual connections on a fixed principal bundle over M. We highlight the role of holomorphic Poisson structures in all these constructions.

Cavalcanti G.R., Gualtieri M.