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Balanced metrics for extremal Kähler metrics and Fano manifolds

Yoshinori Hashimoto

TL;DR

The paper surveys the author's work on balanced metrics and stability notions in algebraic geometry, linking the existence of balanced and anticanonically balanced metrics to algebro-geometric stability citations such as Chow stability, K-stability, and Ding stability. It shows that for large $k$ the existence of these metrics is equivalent to positivity of asymptotic slopes of corresponding energy functionals, connecting differential-geometric quantisations to stability invariants via the Chow weight, Donaldson-Futaki invariant, and the Ding+Chow sum; it further extends the framework to extremal metrics through relative stability and higher Futaki invariants. The geodesic-convex analysis of balancing energies and a Hopf--Rinow based argument yields a general criterion: a geodesically convex function on a complete length space has a critical point exactly when its asymptotic slope at infinity is positive, a result that unifies the metric and variational perspectives. Overall, the work provides a coherent bridge between finite-dimensional quantisations, stability notions in algebraic geometry, and intrinsic metric geometry, with concrete criteria for the existence of canonical metrics on Fano and more general polarised manifolds.

Abstract

The first three sections of this paper are a survey of the author's work on balanced metrics and stability notions in algebraic geometry. The last section is devoted to proving the well-known result that a geodesically convex function on a complete Riemannian manifold admits a critical point if and only if its asymptotic slope at infinity is positive, where we present a proof which relies only on the Hopf--Rinow theorem and extends to locally compact complete length metric spaces.

Balanced metrics for extremal Kähler metrics and Fano manifolds

TL;DR

The paper surveys the author's work on balanced metrics and stability notions in algebraic geometry, linking the existence of balanced and anticanonically balanced metrics to algebro-geometric stability citations such as Chow stability, K-stability, and Ding stability. It shows that for large the existence of these metrics is equivalent to positivity of asymptotic slopes of corresponding energy functionals, connecting differential-geometric quantisations to stability invariants via the Chow weight, Donaldson-Futaki invariant, and the Ding+Chow sum; it further extends the framework to extremal metrics through relative stability and higher Futaki invariants. The geodesic-convex analysis of balancing energies and a Hopf--Rinow based argument yields a general criterion: a geodesically convex function on a complete length space has a critical point exactly when its asymptotic slope at infinity is positive, a result that unifies the metric and variational perspectives. Overall, the work provides a coherent bridge between finite-dimensional quantisations, stability notions in algebraic geometry, and intrinsic metric geometry, with concrete criteria for the existence of canonical metrics on Fano and more general polarised manifolds.

Abstract

The first three sections of this paper are a survey of the author's work on balanced metrics and stability notions in algebraic geometry. The last section is devoted to proving the well-known result that a geodesically convex function on a complete Riemannian manifold admits a critical point if and only if its asymptotic slope at infinity is positive, where we present a proof which relies only on the Hopf--Rinow theorem and extends to locally compact complete length metric spaces.
Paper Structure (4 sections, 16 theorems, 39 equations, 1 figure)

This paper contains 4 sections, 16 theorems, 39 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Balanced metrics for extremal metrics
  3. Anticanonically balanced metrics
  4. Geodesically convex functions on complete Riemannian manifolds

Key Result

Theorem 1.2

(Bouche Bouche, Tian tian90, Yau yauberg, Zelditch zelditch, Catlin catlin, Ruan ruan, Lu lu, Ma--Marinescu mm, amongst many others) The Bergman function admits an asymptotic expansion when $k \in \mathbb{N}$ is sufficiently large, where $S( \omega_h )$ is the scalar curvature of $\omega_h$ defined by where $\mathrm{Ric} (\omega_h )$ is the Ricci curvature of $\omega_h$, defined locally as

Figures (1)

  • Figure 1: Asymptotic behaviours of convex functions

Theorems & Definitions (35)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Remark 1.8
  • Definition 1.9
  • Definition 1.10
  • ...and 25 more