Twisted scalar curvature as a moment map

Abstract

We develop the moment map theory of the twisted scalar curvature of a K\"ahler metric. Primarily, we introduce a coupled system of equations on a holomorphic submersion intertwining the twisted scalar curvature of a K\"ahler metric on the base and the fibrewise scalar curvature of a relatively K\"ahler metric on the total space. This resulting system can be viewed as producing the natural coupled metric geometry of holomorphic submersions, and we show that this system appears canonically as a moment map. The approach generalises to foliations, where we prove similar results.