Scalar curvature as moment map in generalized Kahler geometry

TL;DR

This paper extends the concept of scalar curvature as a moment map from Kahler to generalized Kahler geometry, introducing a generalized scalar curvature and exploring its properties and deformations.

Contribution

It defines a generalized scalar curvature via a moment map in generalized Kahler geometry and studies its properties, including deformations and explicit examples.

Findings

Generalized scalar curvature is defined as a moment map in generalized Kahler geometry.

Infinitesimal deformations with constant scalar curvature are finite dimensional on compact manifolds.

Explicit descriptions and examples of generalized Kahler-Einstein structures are provided.

Abstract

It is known that the scalar curvature arises as the moment map in Kahler geometry. In pursuit of this analogy, we introduce the notion of a moment map in generalized Kahler geometry which gives the definition of a generalized scalar curvature on a generalized Kahler manifold. From the viewpoint of the moment map, we obtain the generalized Ricci form which is a representative of the first Chern class of the anticanonical line bundle. It turns out that infinitesimal deformations of generalized Kahler structures with constant generalized scalar curvature are finite dimensional on a compact manifold. Explicit descriptions of the generalized Ricci form and the generalized scalar curvature are given on a generalized Kahler manifold of type $(0,0)$. Poisson structures constructed from a Kahler action of $T^m$ on a Kahler-Einstein manifold give intriguing deformations of generalized Kahler-Einstein structures. In particular, the anticanical divisor consists of three lines on $C P^2$ in general position yields nontrivial examples of generalized Kahler-Einsein structures