Scalar Curvature, Moment Maps, and the Deligne Pairing *

Let X be a compact complex manifold and L → X a positive holomorphic line bundle. Assume that Aut(X,L)/C× is discrete, where Aut(X,L) is the group of holomorphic automorphisms of the pair (X,L). Donaldson [D] has recently proved that if X admits a metric ω ∈ c1(L) of constant scalar curvature, then (X,L) is Hilbert-Mumford stable for k sufficiently large. Since Kähler-Einstein metrics have constant scalar curvature, this confirms in one direction the well-known conjecture of Yau [Y1-4] which asserts that the existence of a Kähler-Einstein metric is equivalent to stability in the sense of geometric invariant theory. Additional evidence for Yau’s conjecture had been provided earlier by Tian [T2-4], who showed that the existence of constant scalar curvature metrics implies K-stability and CM -stability.