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Formal structure of scalar curvature in generalized Kähler geometry

Vestislav Apostolov, Jeffrey Streets, Yury Ustinovskiy

TL;DR

The paper extends Calabi's canonical metric program to generalized Kähler geometry by proving that Goto's scalar curvature is the moment map for the action of generalized Hamiltonian automorphisms subject to an adapted volume form, and by deriving an explicit curvature formula $\mathrm{Gscal}=\tfrac14\big(R-\tfrac{1}{12}|H|^2+2\Delta f-|df|^2\big)$. It places generalized Kähler-Ricci solitons squarely within this variational framework, showing they have constant Goto scalar curvature and identifying their adapted volumes and symmetries. In the generically symplectic case, the generalized Kähler class is realized as a complexified orbit, enabling a Mabuchi-type geometry with a nonpositive curvature and a GK Calabi problem. The work culminates in a GK Calabi–Lichnerowicz–Matsushima theory and a Futaki invariant tailored to GK classes, providing obstructions and stability criteria for the existence of extremal GK structures. Together, these results extend the scalar-curvature–stability paradigm of Kähler geometry to the broader GK setting, with applications to non-symplectic GK examples and soliton geometries.

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 Kähler-Ricci solitons. Restricting to the generically symplectic type case, we realize the generalized Kähler 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.

Formal structure of scalar curvature in generalized Kähler geometry

TL;DR

The paper extends Calabi's canonical metric program to generalized Kähler geometry by proving that Goto's scalar curvature is the moment map for the action of generalized Hamiltonian automorphisms subject to an adapted volume form, and by deriving an explicit curvature formula . It places generalized Kähler-Ricci solitons squarely within this variational framework, showing they have constant Goto scalar curvature and identifying their adapted volumes and symmetries. In the generically symplectic case, the generalized Kähler class is realized as a complexified orbit, enabling a Mabuchi-type geometry with a nonpositive curvature and a GK Calabi problem. The work culminates in a GK Calabi–Lichnerowicz–Matsushima theory and a Futaki invariant tailored to GK classes, providing obstructions and stability criteria for the existence of extremal GK structures. Together, these results extend the scalar-curvature–stability paradigm of Kähler geometry to the broader GK setting, with applications to non-symplectic GK examples and soliton geometries.

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 Kähler-Ricci solitons. Restricting to the generically symplectic type case, we realize the generalized Kähler class as the complexified orbit of the Hamiltonian action above. This leads to a natural extension of Mabuchi's metric and -energy, implying a conditional uniqueness result. Finally, in this setting we derive a Calabi-Matsushima-Lichnerowicz obstruction and a Futaki invariant.
Paper Structure (34 sections, 43 theorems, 254 equations)

This paper contains 34 sections, 43 theorems, 254 equations.

Table of Contents

  1. Introduction
  2. Scalar curvature as moment map
  3. Scalar curvature and generalized Kähler-Ricci solitons
  4. Complexified orbit and Calabi problem
  5. Obstruction theory
  6. Background
  7. Generalized geometry
  8. Divergence of generalized complex structures
  9. Generalized Kähler structures
  10. Scalar curvature as moment map
  11. Definition and basic properties
  12. A formal moment map setup
  13. The generically symplectic case
  14. Generalized Kähler-Ricci solitons as cscGK structures
  15. An explicit formula
  16. ...and 19 more sections

Key Result

Theorem 1.1

goto-21 (Theorem t:gotomoment) The natural action of $\mathfrak{ham}^{\mathbb T}({\mathbb J})$ on $\mathcal{AGK}({\mathbb J}, \mathrm{vol})$ is Hamiltonian with moment map given by the generalized scalar curvature. More concretely, for any one-parameter family ${\mathbb I}_t\in \mathcal{AGK}({\mathbb J}, \mathrm{vol})$.

Theorems & Definitions (133)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Definition 2.2: GRFbook
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Definition 2.7
  • ...and 123 more