This expository article provides nice introduction to some current trends of research in stability in geometric invariant theory (GIT) and problems of Kähler metrics of constant curvature. It was partially motivated by a conjecture of S.T. Yau which says that the existence of a Kähler form \(\omega\in c_1(L)\) with constant curvature should be equivalent to the stability of \(c_1(L)\) in the sense of GIT, where \(L\) is is a positive line bundle over a compact complex manifold \(X\).
Hence, besides the classical notions of Chow-Mumford (Chap. 6.1), and Hilbert-Mumford stability, they also discuss the notions of analytic and algebraic \(K\)-stability due to Tian and Donaldson. Also included in this discussion are: Donaldson’s infinite-dimensional GIT (Chap. 6.2) and stability conditions (B) and (S) introduced by the authors themselves (Chap. 6.3) which arise in the context of Kähler Ricci flows.
Several analytic methods are also discussed: the Tian-Yau-Zelditch approximation theorem (Chap. 4.2), the degenerate complex Monge-Ampère equation (Chap. 12.3.1), estimates for moment maps, etc.…Detailed references (168 items) on the subject are provided.
For the entire collection see [Zbl 1166.00310].