New aspects of the ddc-lemma

Details New aspects of the ddc-lemma New aspects of the ddc-lemma

2005-01-24

arxiv.org/abs/math/0501406v1

Mathematics

Differential Geometry

Oxford University D. Phil thesis. 140 pages. This thesis contains material from the papers math.SG/0403067, math.DG/0404451, m

Scientific paper

We produce examples of generalized complex structures on manifolds by generalizing results from symplectic and complex geometry. We produce generalized complex structures on symplectic fibrations over a generalized complex base. We study in some detail different invariant generalized complex structures on compact Lie groups and provide a thorough description of invariant structures on nilmanifolds, achieving a classification on 6-nilmanifolds. We study implications of the `dd^c-lemma' in the generalized complex setting. Similarly to the standard dd^c-lemma, its generalized version induces a decomposition of the cohomology of a manifold and causes the degeneracy of the spectral sequence associated to the splitting d = \del + \delbar at E_1. But, in contrast with the dd^c-lemma, its generalized version is not preserved by symplectic blow-up or blow-down (in the case of a generalized complex structure induced by a symplectic structure) and does not imply formality.

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