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Microscopic stability thresholds and constant scalar curvature Kähler metrics

Takahiro Aoi

Abstract

In this paper, we directly prove that if the limit of microscopic stability thresholds introduced by Berman for a polarized manifold satisfies some condition, then there exists a unique constant scalar curvature Kähler metric. This is an analogue of K.Zhang's result which is proved by the delta-invariant introduced by Fujita-Odaka. This work is motivated by Berman's result which shows that if a Fano manifold is uniformly Gibbs stable, then there exists a unique Kähler-Einstein metric, without uniform K-stability. We also give some sufficient conditions of the existence of a constant scalar curvature Kähler cone metric.

Microscopic stability thresholds and constant scalar curvature Kähler metrics

Abstract

In this paper, we directly prove that if the limit of microscopic stability thresholds introduced by Berman for a polarized manifold satisfies some condition, then there exists a unique constant scalar curvature Kähler metric. This is an analogue of K.Zhang's result which is proved by the delta-invariant introduced by Fujita-Odaka. This work is motivated by Berman's result which shows that if a Fano manifold is uniformly Gibbs stable, then there exists a unique Kähler-Einstein metric, without uniform K-stability. We also give some sufficient conditions of the existence of a constant scalar curvature Kähler cone metric.

Paper Structure

This paper contains 3 sections, 15 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Energy functionals and coercivity
  3. Proof

Key Result

Theorem 1.1

We fix a positive number $c$ such that $c \, \omega + {\rm Ric}\,dV \geq 0$. Fix $\gamma >0$ and $\tau >0$. We take a positive integer $k$ such that $f \in L^{k\tau / (k\tau-\gamma(1+\tau))} (dV)$. Then, there exists a positive constant $C_1$ independent of $k, \gamma, \tau$ such that we have for any $\phi \in \mathcal{H}(L)$, where $\gamma^\prime := \gamma (1-ck^{-1})(1-C_1 k^{-1})$.

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3: Be2FO
  • Corollary 1.4
  • Remark 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Definition 2.1
  • Lemma 2.2
  • proof
  • ...and 19 more