Non-Kähler solvmanifolds with generalized Kähler structure
A generalized Kähler structure on a manifold \(M\) is a pair of Hermitian structures \((J_+,g), (J_-,g)\), where \(J_+,J_-\) are complex structures on \(M\)and \(g\) is a Hermitian metric with respect to both \(J_+,J_-\), such that the 3-form \(H=d^c_+F^+=d^c_-F^-\) is closed, \(dH=0\), where \(F^+,F^-\) are fundamental forms associated with \((g,J_{\pm})\) and \(d^c_{\pm}=i({\overline{\partial}}_{\pm}-\partial_{\pm})\). The 3-form \(H\) is called the torsion of the generalized Kähler structure. Such a structure is called twisted if \([H]=0\) and untwisted if \([H]\neq 0\), where \([H]\in H^3(M,\mathbb R)\) is the cohomology class of \(H\). The authors construct a compact 6-dimensional solv-manifold \(M\) with a twisted left-invariant generalized Kähler structure whose first Betti number is \(b_1=1\), and hence \(M\) does not admit any Kähler structure. \(M\) is the total space of a \(T^2\)-bundle over the Inoue surface. The authors also give an example of a non-unimodular 6-dimensional Lie group admitting a generalized Kähler structure, with the corresponding Hermitian structures \((J_{\pm},g)\) being conformally Kähler.
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