Twisted scalar curvature as a moment map

TL;DR

This paper develops a moment map framework for twisted scalar curvature in K"ahler geometry, introducing coupled equations for holomorphic submersions and extending the theory to foliations.

Contribution

It introduces a new coupled system of equations linking base and fiber scalar curvatures, viewed as a moment map, generalizing to foliations.

Findings

Coupled equations produce natural geometry of holomorphic submersions.

System appears canonically as a moment map.

Results extend to foliations.

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.