Abstract: A central theme in complex geometry is the study of canonical metrics. Such metrics are solutions to PDEs, for example the constant scalar curvature Kahler (cscK) equation, Hermite--Einstein equation, and J-equation. These equations in turn arise from moment maps on infinite dimensional spaces of metrics, suggesting links between the existence of solutions and stability conditions. In recent work with Ruadhaí Dervan, we show that the moment map property can be viewed from an equivariant cohomological perspective, namely by integrating equivariant forms over the fibres of a fibration. The construction requires only a choice of topological data as input, and produces naturally a moment map as a result. In this talk, I will explain this perspective, its advantages, and show how it can be used to recover the above equations, as well as other important information.