Non-kähler Solvmanifolds with Generalized Kähler Structure

The generalized Kähler structures were introduced and studied by M. Gualtieri in his PhD thesis [16] in the more general context of generalized geometry started by N. Hitchin in [20]. There are many explicit constructions of non-trivial generalized-Kähler structures [1, 2, 21, 24, 25, 4, 7]. For instance Gualtieri proved that all compact-even dimensional semisimple Lie groups are generalized Kähler. In [25] the generalized Kähler quotient construction is considered in relation with the hyperkähler quotient construction and generalized Kähler structures are given on CP, on some toric varieties and on the complex Grassmannian. Some obstructions and conditions on the underlying complex manifolds were found (see [2, 5, 16] and related references). By [16, 2] it turns out that a generalized Kähler structure on a 2n-dimensional manifoldM is equivalent to a pair of Hermitian structures (J+, g) and (J−, g), where J± are two integrable almost complex structures on M and g is a Hermitian metric with respect to J±, such that the 3-form H = d c +F+ = −d c −F− is closed, where F±(·, ·) = g(J±·, ·) are the fundamental 2-forms associated with the Hermitian structures (J±, g) and d c ± = i(∂± − ∂±). In particular, any Kähler metric (J, g) gives rise to a generalized Kähler structure by taking J+ = J and J− = ±J . In the context of Hermitian geometry, the closed 3-form H is called the torsion of the generalized Kähler structure and it can be also identified with the torsion of the Bismut connection associated with the Hermitian structure (J±, g) (see [3, 14]). The generalized Kähler structure is called untwisted or twisted according to the fact that the cohomology class [H ] ∈ H(M,R) vanishes or not. In this paper we will give a homogeneous example of twisted generalized Kähler manifold which does not admit any Kähler structure. If (J±, g) is a generalized Kähler structure, then the fundamental 2-forms F± are ∂±∂±-closed. Therefore the Hermitian structures (J±, g) are strong Kähler