Generalized scalar curvature and modified moment map in generalized K\"ahler geometry

Abstract

We introduce a notion of generalized scalar curvature for generalized K\"ahler manifolds using the pure spinor formalism. For an arbitrary compact generalized K\"ahler manifold, we develop an extended action of generalized Hamiltonians involving an abelian Lie algebra. This leads naturally to the notion of a modified moment map, which satisfies the moment-map condition only modulo the infinitesimal action of this abelian Lie algebra. We prove that this modified moment map is given by the generalized scalar curvature. Thus our result extends the theorem of Fujiki and Donaldson, which realizes the scalar curvature as a moment map in ordinary K\"ahler geometry. We also study explicit examples. Although a compact connected Lie group admits a K\"ahler structure only in the torus case, every connected compact even-dimensional Lie group admits generalized K\"ahler structures with constant generalized scalar curvature. In particular, we explicitly construct such structures on the standard Hopf surface.