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.

中文翻译：

---

标量曲率和无限维hyperkähler约简

我们讨论了Fujiki和Donaldson的Kähler约简的自然扩展，它实现了将Kähler度量的标量曲率作为矩图映射到hyperkähler约简。我们的方法基于Biquard和Gauduchon的hyperkähler指标的显式构造。这一扩展使人想起了如何推导谐波束的希钦方程，并产生了使不变的标量曲率Kähler（cscK）条件变形的实矩和复矩矩图方程。在复杂曲线的特殊情况下，我们恢复了唐纳森的先前结果。我们专注于复杂表面的情况。特别是，我们显示了一类不包含cscK度量的直纹曲面上矩映射图方程的解的存在。