Canonical measures and Kähler-Ricci flow

By Jian Song, G. Tian. 2008

247 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/a7ff1852f2c7a18edd7da39db0e335a50a15ea41

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Metrics with cone singularities along normal crossing divisors and holomorphic tensor fields (2011, background): “Kołodziej [24] (see also [16], [36], [31], [6], [33], [34] for substantial improvements and interesting applications of this result):”
On the Kahler Ricci flow on projective manifolds of general type (2015, methodology): “The analytic Minimal Model program, introduced in [19, 20], aims to find the minimal model of an algebraic variety, by running the Kähler-Ricci flow.”
Monge–Ampère equations in big cohomology classes (2008, background): “It is also shown in [38] that φKE is an analytic Zariski decomposition in the sense of Tsuji, which means that every pluricanonical section σ∈H(mKX) is L∞ with respect to the metric induced by φKE.”
The Kähler–Ricci flow through singularities (2009, background): “In [32,33], we conjecture that the Ricci flow performs an analytic/geometric surgery equivalent to an algebraic surgery such as a divisorial contraction or a flip, and the flow can be continued on a new projective variety X ′ with mild singularities.”
Infinite time singularities of the Kähler-Ricci flow (2014, methodology): “The second setup is as follows (see Song and Tian [20; 22; 21], Fong and Zhang [5] and Tosatti, Weinkove and Yang [27]).”
The Kähler-Ricci flow, Ricci-flat metrics and collapsing limits (2014, methodology): “According to the analytic minimal model program [39, 50, 40, 42] the Kähler-Ricci flow will perform a surgery at each finite time singularity, corresponding to a birational operation of the minimal model program from algebraic geometry, and arrive at the minimal model Xmin with nef canonical bundle ”
Uniform stability of twisted constant scalar curvature K\"ahler metrics (2014, background): “Song-Tian [39, 40] answered this question affirmatively by showing that each variety with semi-ample canonical class admits a twisted Kähler-Einstein metric of the form Ricω − α = −ω, (6) where roughly α comes from a Weil-Petersson type metric on the moduli space of Calabi-Yau manifolds.”
On the evolution of a Hermitian metric by its Chern-Ricci form (2011, methodology): “The Kähler-Ricci flow has a deep connection to the Minimal Model Program in algebraic geometry, as demonstrated by the work of SongTian and others [14, 26, 34, 35, 36, 37, 38, 39, 40, 41, 50, 51, 56, 63].”