[Paper Review] Convergence of weak Kähler-Ricci Flows on minimal models of positive Kodaira dimension

Philippe Eyssidieux, Vincent Guedj+1 more|arXiv (Cornell University)|Apr 24, 2016

Geometry and complex manifolds18 references3 citations

TL;DR

This paper establishes the convergence of the normalized weak Kähler-Ricci flow on minimal models of positive Kodaira dimension with terminal singularities, using viscosity solutions to degenerate parabolic complex Monge-Ampère equations. It proves that the flow converges in capacity to a canonical current $ T_{\text{can}} $, generalizing Song-Tian's results to singular varieties via a canonical twisted Kähler-Einstein metric on the Iitaka fibration base.

ABSTRACT

Studying the behavior of the Kähler-Ricci flow on mildly singular varieties, one is naturally lead to study weak solutions of degenerate parabolic complex Monge-Ampère equations. In this article, the third of a series on this subject, we study the long term behavior of the normalized Kähler-Ricci flow on mildly singular varieties of positive Kodaira dimension, generalizing results of Song and Tian who dealt with smooth minimal models.

Motivation & Objective

• To generalize Song-Tian's convergence results for the normalized Kähler-Ricci flow on smooth minimal models to singular varieties with terminal singularities.
• To establish the existence of a canonical twisted Kähler-Einstein metric $ \theta_{\text{can}} $ on the Iitaka fibration base $ J $ for klt pairs $ (X,\Delta) $ with semi-ample canonical bundle.
• To prove that the normalized Kähler-Ricci flow converges in capacity to the pullback current $ T_{\text{can}} = j^*\theta_{\text{can}} $ on minimal models of positive Kodaira dimension.
• To develop a viscosity solution approach for degenerate parabolic complex Monge-Ampère equations arising in the flow, avoiding regularization techniques used in prior works.

Proposed method

• Employing the parabolic viscosity method to analyze weak solutions of degenerate parabolic complex Monge-Ampère equations modeling the Kähler-Ricci flow.
• Constructing a supersolution $ v_\varepsilon $ using a perturbed potential involving $ \varphi_\infty $, $ \rho $, $ \log|s|_h $, and a time-dependent ODE $ g(t) $.
• Using Păun’s Laplacian estimate and Kawamata’s canonical bundle formula to establish the generic smoothness of the canonical metric $ \theta_{\text{can}} $ on $ J $.
• Reducing the convergence problem to a regularity statement for the semi-flat current $ \omega_{\text{SF}} $, defined as $ \omega_0 + dd^c\rho $ on fibers of the Iitaka fibration.
• Applying the comparison principle and $ L^1 $-compactness to show that any limit of the flow potentials $ \varphi_t $ equals the limit potential $ \varphi_\infty $, implying convergence in capacity.
• Using uniform bounds and the viscosity supersolution construction to control the long-time behavior of the flow in $ ]T_0, \infty[ \times \Omega_r $.

Experimental results

Research questions

• RQ1 Does the normalized Kähler-Ricci flow converge to a canonical current on minimal models of positive Kodaira dimension with terminal singularities?
• RQ2 Can the canonical volume form and twisted Kähler-Einstein metric of Song-Tian be generalized to klt pairs with semi-ample canonical bundles?
• RQ3 Is the convergence of the flow to $ T_{\text{can}} $ stable under singularities, and does it hold in capacity rather than just pointwise?
• RQ4 Can the viscosity solution method replace regularization techniques in the study of weak Kähler-Ricci flows on singular varieties?
• RQ5 What is the regularity of the canonical metric $ \theta_{\text{can}} $ on the base of the Iitaka fibration for singular minimal models?

Key findings

• The normalized Kähler-Ricci flow converges in capacity to the canonical current $ T_{\text{can}} = j^*\theta_{\text{can}} $ on minimal models of positive Kodaira dimension with terminal singularities.
• A canonical twisted Kähler-Einstein metric $ \theta_{\text{can}} $ exists on the Iitaka fibration base $ J $, with continuous potentials and smoothness on a Zariski open subset when $ X $ is projective.
• The canonical metric $ \theta_{\text{can}} $ is constructed via viscosity methods and Păun’s Laplacian estimate, relying on Kawamata’s canonical bundle formula.
• The flow potentials $ \varphi_t $ converge locally uniformly to $ \varphi_\infty $ on $ X \setminus D $, and in capacity on $ X $, as $ t \to \infty $.
• The convergence is established via a viscosity supersolution construction involving $ v_\varepsilon(t,x) = [1+\varepsilon A]\varphi_\infty(x) + e^{-t}\rho(x) - \varepsilon\log|s|_h(x) + Ce^{-t} + g(t) $, with $ g(t) $ solving a specific ODE.
• The key estimate $ \limsup_{t\to\infty} \varphi(t,x) \leq \varphi_\infty(x) $ holds for all $ x \in X^{\text{reg}} \setminus D $, and by density and comparison, $ \varphi_t \to \varphi_\infty $ in capacity.