Scalar curvature and moment map in Generalized Kähler geometry

Abstract: We introduce a notion of scalar curvature of a twisted generalized Kähler manifold in terms of pure spinors formalism. A moment map framework on an arbitrary compact twisted generalized Kähler manifold is provided and then it turns out that a moment map is given by the scalar curvature under the certain condition, which is a generalization of the result of the scalar curvature as a moment map in the ordinary Kähler geometry, due to Fujiki and Donaldson. A noncommutative compact Lie group G does not have any Kähler structure. However, we show that a compact Lie group has a family of generalized Kähler structures twisted by the Cartan 3-form, which is constructed by the action of the real Pin group of the double of Cartan subalgebra. Then we show that an arbitrary compact Lie group admits generalized Kähler structures with constant scalar curvature. In particular, generalized Kähler structures with constant scalar curvature on the standard Hopf surface are explicitly given.

algebraic geometryalgebraic topologycomplex variablesdifferential geometrygeometric topologysymplectic geometry

Audience: researchers in the topic

2021 Pacific Rim Complex & Symplectic Geometry Conference