Test Configurations, Stabilities and Canonical Kahler Metrics

The Yau-Tian-Donaldson conjecture for anti-canonical polarization was recently solved affirmatively by Chen-Donaldson-Sun and Tian. In this book, the unsolved cases of the conjecture will be discussed.It will be shown that the problem is closely related to the geometry of moduli spaces of test configurations for polarized algebraic manifolds.