New aspects of the ddc-lemma

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.

Oxford University D. Phil thesis. 140 pages. This thesis contains material from the papers math.SG/0403067, math.DG/0404451, math.AT/0412053 and `T-duality and generalized complex structures', which is in preparation and is a collaboration with Gualtieri

References in corpus (11)

• Supersymmetric Backgrounds from Generalized Calabi-Yau Manifolds
• Remarks on A-branes, Mirror Symmetry, and the Fukaya category
• Mirror symmetric SU(3)-structure manifolds with NS fluxes
• On the Topology and Flux of T-Dual Manifolds
• T-duality for principal torus bundles
• The Mukai pairing, I: the Hochschild structure
• Mirror Symmetry and Generalized Complex Manifolds
• Generalized geometry and the Hodge decomposition
• Special metric structures and closed forms
• T-duality and Generalized Complex Geometry
• Existence of symplectic structures on torus bundles over surfaces

Cited by in corpus (41)

• SU(3) x SU(3) compactification and mirror duals of magnetic fluxes
• Gauged Supergravities
• Deformations of calibrated D-branes in flux generalized complex manifolds
• Reformulating Supersymmetry with a Generalized Dolbeault Operator
• On moduli and effective theory of N=1 warped flux compactifications
• Leibniz algebroids, twistings and exceptional generalized geometry
• D-brane non-perturbative effects and geometric deformations
• Special metric structures and closed forms
• AdS_5 Solutions of Type IIB Supergravity and Generalized Complex Geometry
• T-duality and Generalized Kahler Geometry
• Stable generalized complex structures
• IIB supergravity on manifolds with SU(4) structure and generalized geometry
• Inequalities à la Frölicher and cohomological decompositions
• Towards mirror symmetry à la SYZ for generalized Calabi-Yau manifolds
• Hodge theory and deformations of SKT manifolds
• Moduli of Coassociative Submanifolds and Semi-Flat Coassociative Fibrations
• On Cohomological Decomposability of Almost-Kähler Structures
• Generalising G${}_{\text{2}}$ geometry: involutivity, moment maps and moduli
• Poisson modules and generalized geometry
• Twistor spaces of generalized complex structures
• Monge-Ampere equations and generalized complex geometry. The two-dimensional case
• Topological T-duality for torus bundles with monodromy
• T-duality and Generalized Complex Geometry
• Hermitian structures on cotangent bundles of four dimensional solvable Lie groups
• Generalized Cohomologies and Supersymmetry
• Variation of Hodge structure for generalized complex manifolds
• Moser Lemma in Generalized Complex Geometry
• The minimal model for the Batalin-Vilkovisky operad
• Deformations of generalized complex and generalized Kahler structures
• Symplectic cohomologies and deformations
• Aeppli and Bott-Chern cohomology for bi-generalized Hermitian manifolds and $d^{\prime }{d}^{\prime \prime }$-lemma
• Invariant generalized complex geometry on maximal flag manifolds and their moduli
• On cohomological decomposition of generalized-complex structures
• The Lefschetz property, formality and blowing up in symplectic geometry
• Cohomologies on almost complex manifolds and the $\partial \bar{\partial }$-lemma
• Symplectic Parabolicity and L^2 Symplectic Harmonic Forms
• From Lie Theory to Deformation Theory and Quantization
• Flux Backgrounds and Exceptional Generalised Geometry
• The basic $d{d}^{\mathcal{J}}$-lemma
• Strings as sigma models and in the tensionless limit
• $E_1$-degeneration and $d^{\prime }{d}^{\prime \prime }$-lemma