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New aspects of the ddc-lemma
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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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Cited by 2 Pith papers
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Symplectic solvmanifolds not satisfying the hard-Lefschetz condition
On completely solvable almost abelian solvmanifolds, a symplectic form satisfies the hard-Lefschetz condition if and only if the defining action is semisimple.
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On the triviality of the generalized tangent bundle
The generalized tangent bundle of a parallelizable manifold is trivial, but the converse does not hold, as shown by the Möbius strip, spheres, and projective spaces; it is also related to generalized geometric structures.
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