A moment map interpretation of the Ricci form, K\"ahler--Einstein structures, and Teichm\"uller spaces

Abstract

This paper surveys the role of moment maps in K\"ahler geometry. The first section discusses the Ricci form as a moment map and then moves on to moment map interpretations of the K\"ahler--Einstein condition and the scalar curvature (Quillen--Fujiki--Donaldson). The second section examines the ramifications of these results for various Teichm\"uller spaces and their Weil--Petersson symplectic forms and explains how these arise naturally from the construction of symplectic quotients. The third section discusses a symplectic form introduced by Donaldson on the space of Fano complex structures.