[Paper Review] Recent progress in Kähler geometry

Xiuxiong Chen|ArXiv.org|Apr 18, 2003

Geometry and complex manifolds14 references3 citations

TL;DR

This paper surveys recent advances in Kähler geometry, focusing on the existence, uniqueness, and obstructions to extremal Kähler metrics, particularly through the study of the infinite-dimensional space of Kähler potentials and the Kähler-Ricci flow. A key contribution is the proof that Kähler-Ricci flow converges exponentially fast to a Kähler-Einstein metric on manifolds with positive bisectional curvature and non-negative curvature, establishing path-connectedness of the space of such metrics.

ABSTRACT

In recent years, there are many progress made in Kähler geometry. In particular, the topics related to the problems of the existence and uniqueness of extremal Kähler metrics, as well as obstructions to the existence of such metrics in general Kähler manifold. In this talk, we will report some recent developments in this direction. In particular, we will discuss the progress recently obtained in understanding the metric structure of the infinite dimensional space of Kaehler potentials, and their applications to the problems mentioned above. We also will discuss some recent on Kaehler Ricci flow.

Motivation & Objective

• To summarize recent breakthroughs in the existence and uniqueness of extremal Kähler metrics on compact Kähler manifolds.
• To investigate obstructions to the existence of extremal metrics, particularly through generalized Futaki invariants and holomorphic invariants.
• To analyze the metric structure of the infinite-dimensional space of Kähler potentials and its implications for geometric flows.
• To establish convergence results for the Kähler-Ricci flow under curvature conditions, especially in the positive scalar curvature case.
• To extend the understanding of stability conditions in Kähler geometry, linking analytic and algebraic stability.

Proposed method

• The paper employs the study of functionals $ E_k $ and $ J_k $ on the space of Kähler potentials $ ilde{\cal H} $, which generalize the Mabuchi K-energy and are used to analyze the flow of metrics.
• It uses the Kähler-Ricci flow as a geometric evolution process, showing that the functionals $ E_k $ decrease along the flow, enabling uniform lower bounds on volume forms.
• The method involves proving that the derivative of $ E_k $ along a curve of metrics yields holomorphic invariants $ \Im_k $, which vanish on Kähler-Einstein metrics, allowing for normalization of the flow.
• A bootstrapping argument is applied after establishing a uniform lower bound on the volume form, leading to higher-order estimates and global convergence.
• The closure property of geodesic solutions in the space of Kähler potentials is established via Fredholm theory of holomorphic discs with totally real boundary conditions.
• The proof of exponential convergence relies on Tian’s inequality and the effective use of curvature assumptions, particularly non-negative bisectional curvature with positivity at one point.

Experimental results

Research questions

• RQ1 Under what conditions does the Kähler-Ricci flow converge to a Kähler-Einstein metric on a compact Kähler manifold with positive scalar curvature?
• RQ2 How do holomorphic invariants $ \Im_k $ derived from the functionals $ E_k $ relate to the existence and uniqueness of extremal metrics?
• RQ3 What is the role of the metric structure of the space of Kähler potentials in resolving existence problems for extremal metrics?
• RQ4 Can the closure property of geodesic solutions in the space of Kähler potentials be established under weak regularity assumptions?
• RQ5 How do generalized Futaki invariants and curvature conditions interact to obstruct or allow the existence of extremal metrics?

Key findings

• The Kähler-Ricci flow converges exponentially fast to a Kähler-Einstein metric on a compact Kähler-Einstein manifold with positive scalar curvature if the initial metric has non-negative bisectional curvature and positive curvature at at least one point.
• The space of Kähler metrics with non-negative bisectional curvature is path-connected, as shown by the convergence of the flow between any two such metrics.
• The functionals $ E_k $ are shown to decrease along the Kähler-Ricci flow, and their derivatives yield holomorphic invariants $ \Im_k $, which vanish identically on Kähler-Einstein metrics.
• A uniform lower bound on the volume form of the evolving metric is established via Tian’s inequality and the effective use of curvature assumptions, enabling higher-order estimates.
• The closure property of geodesic solutions in the space of Kähler potentials is proven under the condition that solutions are smooth almost everywhere, extending the openness result of Donaldson.
• The generalized Futaki invariants and holomorphic invariants $ \Im_k $ provide necessary obstructions to the existence of extremal metrics, particularly in the case of non-constant scalar curvature.