Scalar curvature and an infinite-dimensional hyperkähler reduction

We discuss a natural extension of the Kaehler reduction of Fujiki and Donaldson, which realises the scalar curvature of Kaehler metrics as a moment map, to a hyperkaehler reduction. Our approach is based on an explicit construction of hyperkaehler metrics due to Biquard and Gauduchon. This extension is reminiscent of how one derives Hitchin’s equations for harmonic bundles, and yields real and complex moment map equations which deform the constant scalar curvature Kaehler (cscK) condition. In the special case of complex curves we recover previous results of Donaldson. We focus on the case of complex surfaces. In particular we show the existence of solutions to the moment map equations on a class of ruled surfaces which do not admit cscK metrics.