Hodge theory and deformations of SKT manifolds

Abstract

We use tools from generalized complex geometry to develop the theory of SKT (a.k.a. pluriclosed Hermitian) manifolds and more generally manifolds with special holonomy with respect to a metric connection with closed skew-symmetric torsion. We develop Hodge theory on such manifolds showing how the reduction of the holonomy group causes a decomposition of the twisted cohomology. For SKT manifolds this decomposition is accompanied by an identity between different Laplacian operators and forces the collapse of a spectral sequence at the first page. Further we study the deformation theory of SKT structures, identifying the space where the obstructions live. We illustrate our theory with examples based on Calabi--Eckmann manifolds, instantons, Hopf surfaces and Lie groups.

46 pages, 9 figures; v5: Added theorem 5.16 and expanded example 5.17 to show that the only Calabi-Eckman manifolds to admit SKT structures are S^1 x S^1, S^1 x S^3 and S^3 x S^3

References in corpus (5)

• The cohomologies of the Iwasawa manifold and of its small deformations
• New aspects of the ddc-lemma
• Generalized geometry and the Hodge decomposition
• Generalized Kahler geometry
• Generalized Kaehler geometry of instanton moduli spaces

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• On generalized Kähler geometry on compact Lie groups
• Bihermitian Geometry and the Holomorphic Sections of Twistor Space
• Strong Kähler with Torsion as Generalised Geometry
• Toric Kato manifolds
• Cohomological properties of Hermitian symplectic threefolds