Canonical measures and Kähler-Ricci flow (Q2879886)

The authors study the obtaining of canonical measures on projective varieties of positive Kodaira dimension. Such a canonical measure can be considered as a birational invariant and it induces a canonical singular metric on the canonical model, generalizing the notion of Kähler-Einstein metrics. After presenting some well-known results in algebraic geometry and complex Monge-Ampère equations, the authors prove the existence and uniqueness for canonical metrics of Kähler-Einstein type on projective manifolds with semi-ample canonical bundle. Then they prove the existence of canonical measures for projective manifolds of positive Kodaira dimension. They prove the convergence of the normalized Kähler-Ricci flow on projective manifolds with semi-ample canonical bundle. Finally, they propose a generalized constant scalar curvature equation and also give an adjunction formula for the Mabuchi energy, which could be helpful for understanding the collapsing behaviour of normalized Kähler-Ricci flow.