Generalized scalar curvature and modified moment map in generalized Kähler geometry

Abstract

We introduce a notion of generalized scalar curvature for generalized Kähler manifolds using the pure spinor formalism. For an arbitrary compact generalized Kähler 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 Kähler geometry. We also study explicit examples. Although a compact connected Lie group admits a Kähler structure only in the torus case, every connected compact even-dimensional Lie group admits generalized Kähler structures with constant generalized scalar curvature. In particular, we explicitly construct such structures on the standard Hopf surface.

Substantially revised version with a modified title. We introduce a modified moment-map framework for the extended generalized Hamiltonian action and explain its relation to the ordinary moment-map framework on the adapted-volume locus. The exposition and several proofs have been clarified, and the introduction and examples have been revised. 65 pages