Formal structure of scalar curvature in generalized K\"ahler geometry

TL;DR

This paper explores the geometric and algebraic structures underlying generalized K"ahler geometry, focusing on scalar curvature as a moment map, and extends classical concepts like Mabuchi's metric and Futaki invariant to this setting.

Contribution

It provides an explicit formula for Goto's scalar curvature, interprets it as a moment map, and extends classical K"ahler geometry tools to the generalized K"ahler context.

Findings

Goto's scalar curvature is the moment map for generalized Hamiltonian automorphisms.

Constant scalar curvature corresponds to generalized K"ahler-Ricci solitons.

Extension of Mabuchi's metric and K-energy to generalized K"ahler geometry.

Abstract

Building on works of Boulanger and Goto, we show that Goto's scalar curvature is the moment map for an action of generalized Hamiltonian automorphisms of the associated Courant algebroid, constrained by the choice of an adapted volume form. We derive an explicit formula for Goto's scalar curvature, and show that it is constant for generalized K\"ahler-Ricci solitons. Restricting to the generically symplectic type case, we realize the generalized K\"ahler class as the complexified orbit of the Hamiltonian action above. This leads to a natural extension of Mabuchi's metric and $K$-energy, implying a conditional uniqueness result. Finally, in this setting we derive a Calabi-Matsushima-Lichnerowicz obstruction and a Futaki invariant.