1203.0493 (Gil R. Cavalcanti)

SKT geometry    [PDF]

Gil R. Cavalcanti

SKT structures are closely related to Kaehler structures, the difference being that in the Kaehler case one requires that the Levi--Civita connection has holonomy in U(n), while in the SKT case one requires the existence of a connection with skew-symmetric and closed torsion with holonomy in U(n). The inclusion of the torsion, however leaves several of the usual arguments used in Kaehler geometry without a direct counterpart. We use tools from generalized complex geometry to develop the theory of SKT manifolds. We develop Hodge theory on SKT manifolds and hence prove that their cohomology inherits a decomposition determined by the structure. We study Lie algebroids and differential operators associated to SKT structures and study the deformation theory of these structures. As applications we reobtain a result of Luebke and Teleman regarding the existence of SKT structures on the moduli space of instantons of a bundle over a complex surface and show that even though Kaehler structures are not stable under deformations of the symplectic structure, small deformations are still SKT.

View original: http://arxiv.org/abs/1203.0493