New aspects of the ddc-lemma

TL;DR

This paper explores new generalized complex structures on manifolds, classifies invariant structures on nilmanifolds, and investigates the implications of the generalized dd^c-lemma on cohomology and spectral sequences.

Contribution

It generalizes the dd^c-lemma to the setting of generalized complex geometry, providing classifications and analyzing its effects on cohomology and manifold structures.

Findings

Constructed examples of generalized complex structures on manifolds.

Classified invariant structures on 6-nilmanifolds.

Showed the generalized dd^c-lemma affects cohomology decomposition.

Abstract

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.