Journal of Symplectic Geometry

Volume 18 (2020)

Number 1

Scalar curvature as moment map in generalized Kähler geometry

Pages: 147 – 190



Ryushi Goto (Department of Mathematics, Graduate School of Science, Osaka University Toyonaka, Osaka, Japan)


It is known that the scalar curvature arises as the moment map in Kähler geometry. In pursuit of the analogy, we develop the moment map framework in generalized Kähler geometry of symplectic type. Then we establish the definition of the scalar curvature on a generalized Kähler manifold of symplectic type from the moment map view point. We also obtain the generalized Ricci form which is a representative of the first Chern class of the anticanonical line bundle. We show that infinitesimal deformations of generalized Kähler 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 Kähler manifold of type $(0, 0)$. Poisson structures constructed from a Kähler action of $T^m$ on a Kähler–Einstein manifold give rise to intriguing deformations of generalized Kähler–Einstein structures. In particular, the anticanonical divisor of three lines on $\mathbb{C}P^2$ in a general position yields nontrivial examples of generalized Kähler–Einstein structures

Received 22 March 2018

Accepted 21 November 2018

Published 25 March 2020