Scalar curvature and an infinite-dimensional hyperkähler reduction

We discuss a natural extension of the Kähler reduction of Fujiki and Donaldson, which realises 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. 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 Kähler (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.

Keywords

canonical metrics in Kähler geometry, hyperkähler geometry

2010 Mathematics Subject Classification

32Q15, 32Q60, 53C26, 53C55

Full Text (PDF format)

Received 23 May 2019

Accepted 20 December 2019

Published 18 February 2021