Symmetries in generalized Kähler geometry
Generalized complex geometry, as developed by Hitchin, interpolates between complex and symplectic manifolds. On the other hand, Marco Gualtieri showed that generalized Kähler geometry is equivalent to a bi-Hermitian geometry with torsion, first discovered by physicists. The article under review defines momentum map and reduction in both, generalized complex geometry and generalized Kähler geometry. They obtain, as applications, explicit constructions of bi-Hermitian structures on manifolds which can be obtained as the symplectic quotient of \(\mathbb{C}^N\), i.e., \(\mathbb{C}\mathbb{P}^N\), Hirzebruch surfaces, the blow up of \(\mathbb{C}\mathbb{P}^N\) at arbitrarily many points, and other toric varieties, as well as complex Grassmannians. Their constructions carry over to the twisted case. The paper under review also compares their results with others in this field.
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