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

Hodge theory and deformations of SKT manifolds