Scalar curvature and the moment map in generalized Kahler geometry

TL;DR

This paper defines scalar curvature for twisted generalized Kahler manifolds using pure spinors, develops a moment map framework, and constructs examples of such structures with constant scalar curvature on compact Lie groups and Hopf surfaces.

Contribution

It introduces a scalar curvature concept in generalized Kahler geometry and links it to a moment map framework, extending classical results to a broader setting.

Findings

Scalar curvature is expressed via pure spinors formalism.

A moment map framework for generalized Kahler structures is established.

Examples of constant scalar curvature structures on compact Lie groups and Hopf surfaces are provided.

Abstract

We introduce a notion of scalar curvature of a twisted generalized Kahler manifold in terms of pure spinors formalism. A moment map framework with a modified action of generalized Hamiltonians on an arbitrary compact generalized Kahler manifold is developed. Then it turns out that a moment map is given by the scalar curvature, which is a generalization of the result of the scalar curvature as a moment map in the ordinary Kahler geometry, due to Fujiki and Donaldson. A noncommutative compact Lie group $G$ does not have any Kahler structure. However, we show that every compact Lie group admits generalized Kahler structures with constant scalar curvature. In particular, generalized Kahler structures with constant scalar curvature on the standard Hopf surface are explicitly given.