Canonical measures and Kahler-Ricci flow

TL;DR

This paper proves that the Kahler-Ricci flow on certain algebraic manifolds converges to a unique canonical metric and introduces a canonical measure related to Zariski decomposition, invariant under birational transformations.

Contribution

It establishes the convergence of the Kahler-Ricci flow to a canonical metric and constructs a unique canonical measure on algebraic manifolds with positive Kodaira dimension.

Findings

Kahler-Ricci flow converges to a canonical metric on the canonical model.

Existence of a canonical measure of analytic Zariski decomposition.

Canonical measure is unique and birationally invariant under certain conditions.

Abstract

We show that the Kahler-Ricci flow on an algebraic manifold of positive Kodaira dimension and semi-ample canonical line bundle converges to a unique canonical metric on its canonical model. It is also shown that there exists a canonical measure of analytic Zariski decomposition on an algebraic manifold of positive Kodaira dimension. Such a canonical measure is unique and invariant under birational transformations under the assumption of the finite generation of canonical rings.