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Twisted scalar curvature as a moment map

Ruadhaí Dervan, Thomas Murphy, Julius Ross, Lars Martin Sektnan, Xiaowei Wang

TL;DR

The paper develops a moment-map framework for twisted scalar curvature $S(\omega) - \Lambda_{\omega}\alpha$ across maps, submersions, and foliations, uncovering a natural coupled system that intertwines fibrewise (or leafwise) geometry with base (or transverse) geometry. It provides both finite-dimensional and infinite-dimensional formulations, connecting equivariant differential geometry with the Donaldson–Fujiki program to obtain moment maps whose zeros correspond to fibrewise cscK metrics and base (or transversal) twisted scalar curvature constraints. A key construct is the Weil–Petersson type twisting form $\alpha$ and its role in the coupled equations, with adiabatic (large-k) analysis yielding a transverse Futaki invariant as an obstruction and clarifying the base–fibre coupling. The results advance the understanding of stability and moduli in submersion and foliation settings, offering a robust geometric framework for coupled canonical metrics and their obstructions with potential connections to stability notions in algebraic geometry and moduli theory.

Abstract

We develop the moment map theory of the twisted scalar curvature of a Kähler metric. Primarily, we introduce a coupled system of equations on a holomorphic submersion intertwining the twisted scalar curvature of a Kähler metric on the base and the fibrewise scalar curvature of a relatively Kähler metric on the total space. This resulting system can be viewed as producing the natural coupled metric geometry of holomorphic submersions, and we show that this system appears canonically as a moment map. The approach generalises to foliations, where we prove similar results.

Twisted scalar curvature as a moment map

TL;DR

The paper develops a moment-map framework for twisted scalar curvature across maps, submersions, and foliations, uncovering a natural coupled system that intertwines fibrewise (or leafwise) geometry with base (or transverse) geometry. It provides both finite-dimensional and infinite-dimensional formulations, connecting equivariant differential geometry with the Donaldson–Fujiki program to obtain moment maps whose zeros correspond to fibrewise cscK metrics and base (or transversal) twisted scalar curvature constraints. A key construct is the Weil–Petersson type twisting form and its role in the coupled equations, with adiabatic (large-k) analysis yielding a transverse Futaki invariant as an obstruction and clarifying the base–fibre coupling. The results advance the understanding of stability and moduli in submersion and foliation settings, offering a robust geometric framework for coupled canonical metrics and their obstructions with potential connections to stability notions in algebraic geometry and moduli theory.

Abstract

We develop the moment map theory of the twisted scalar curvature of a Kähler metric. Primarily, we introduce a coupled system of equations on a holomorphic submersion intertwining the twisted scalar curvature of a Kähler metric on the base and the fibrewise scalar curvature of a relatively Kähler metric on the total space. This resulting system can be viewed as producing the natural coupled metric geometry of holomorphic submersions, and we show that this system appears canonically as a moment map. The approach generalises to foliations, where we prove similar results.
Paper Structure (21 sections, 29 theorems, 148 equations)

This paper contains 21 sections, 29 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Preliminary Material
  3. Equivariant differential geometry
  4. The scalar curvature as a moment map
  5. The space of almost complex structures
  6. Maps
  7. Maps in finite dimensions
  8. Returning to the space of almost complex structures
  9. Submersions: a finite-dimensional framework
  10. The transverse Futaki invariant
  11. Transverse Kähler geometry
  12. Transverse holomorphy potentials
  13. The transverse Futaki invariant
  14. The transverse Futaki invariant through an adiabatic expansion
  15. The transverse Futaki invariant through moment maps
  16. ...and 6 more sections

Key Result

Theorem 1.1

A moment map $\mu: \mathscr J^{\mathop{\mathrm{Int}}\nolimits}_{\pi} \to \mathop{\mathrm{Lie}}\nolimits \mathcal{G}^*$ for the $\mathcal{G}$-action on $(\mathscr J_{\pi},\Omega)$ is given by

Theorems & Definitions (73)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Example 2.5
  • Definition 2.6
  • Theorem 2.7
  • proof
  • ...and 63 more