The HcscK equations in symplectic coordinates

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.