The HcscK equations in symplectic coordinates

TL;DR

This paper explores the hyperk"ahler reduction of the space of compatible almost complex structures, focusing on abelian varieties and toric manifolds, and establishes new results on existence, uniqueness, and stability related to the HcscK equations.

Contribution

It extends Donaldson's hyperk"ahler reduction framework to abelian and toric manifolds, providing new insights into the equations and their stability properties.

Findings

Decoupling result for the equations

Variational characterization of solutions

Relation to K-stability in toric case

Abstract

The Donaldson-Fujiki K\"ahler reduction of the space of compatible almost complex structures, leading to the interpretation of the scalar curvature of K\"ahler metrics as a moment map, can be lifted canonically to a hyperk\"ahler reduction. Donaldson proposed to consider the corresponding vanishing moment map conditions as (fully nonlinear) analogues of Hitchin's equations, for which the underlying bundle is replaced by a polarised manifold. However this construction is well understood only in the case of complex curves. In this paper we study Donaldson's hyperk\"ahler reduction on abelian varieties and toric manifolds. We obtain a decoupling result, a variational characterisation, a relation to $K$-stability in the toric case, and prove existence and uniqueness under suitable assumptions on the ``Higgs tensor''. We also discuss some aspects of the analogy with Higgs bundles.