The Hitchin-cscK system: an infinite-dimensional hyperkähler reduction

We discuss a natural extension of the Kähler reduction of Fujiki and Donaldson, which realizes the scalar curvature of Kähler metrics as a moment map, to a hyperkähler reduction. Our approach is based on an explicit construction of hyperkähler metrics due to Biquard and Gauduchon. The construction is motivated by how one can derive Hitchin’s equations for harmonic bundles from the Hermitian Yang-Mills equations, and yields real and complex moment map equations which deform the constant scalar curvature Kähler (cscK) condition by adding a “Higgs field” to the cscK equation. In the special case of complex curves, we recover previous results of Donaldson, while for higher-dimensional manifolds the system of equations has not yet been studied. We study the existence of solutions to the system in some special cases. On a Riemann surface, we extend an existence result for Donaldson’s equation to our system. We then showthe existence of solutions to the moment map equations on a class of ruled surfaces which do not admit cscK metrics, showing that adding a suitable Higgs termto the cscK equation might stabilize the manifold. Lastly, we study the system on equations on an abelian surface and a complex torus, taking advantage of a description of the system in symplectic coordinates analogous to Abreu’s formula for the scalar curvature.