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Projective bundles that admit coupled Kähler-Einstein metrics but no Kähler-Einstein metrics

Naoto Yotsutani

Abstract

Using Hultgren's polytope formulation of the existence of coupled Kähler-Einstein (cKE) metrics on toric Fano manifolds, we construct explicit higher-dimensional toric Fano manifolds that admit two coupled Kähler-Einstein metrics but no ordinary Kähler-Einstein metrics. In particular, we exhibit such examples among certain projective bundles over products of projective spaces. Motivated by these constructions, we conjecture that examples of this type exist in all dimensions $n\geq 4$.

Projective bundles that admit coupled Kähler-Einstein metrics but no Kähler-Einstein metrics

Abstract

Using Hultgren's polytope formulation of the existence of coupled Kähler-Einstein (cKE) metrics on toric Fano manifolds, we construct explicit higher-dimensional toric Fano manifolds that admit two coupled Kähler-Einstein metrics but no ordinary Kähler-Einstein metrics. In particular, we exhibit such examples among certain projective bundles over products of projective spaces. Motivated by these constructions, we conjecture that examples of this type exist in all dimensions .
Paper Structure (7 sections, 10 theorems, 78 equations)

This paper contains 7 sections, 10 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Reductivity of Automorphism Groups of toric Fano varieties
  3. Demazure Roots and Reductivity
  4. Five-dimensional construction
  5. Five-dimensional cKE projective bundle
  6. Six-dimensional cKE projective bundle construction
  7. Conjectures and future works

Key Result

Proposition 1.1

Let Then $X$ does not admit a KE metric, but $\mathop{\mathrm{Aut}}\nolimits(X)$ is reductive. Moreover, there exists a decomposition such that $(\alpha_1,\alpha_2)$ admits a two-coupled KE metric.

Theorems & Definitions (18)

  • Proposition 1.1: see Proposition \ref{['prop:D5B']}
  • Theorem 1.2: see Theorem \ref{['thm:D6']}
  • Conjecture 1.3: see Conjecture \ref{['conj:Higher']}
  • Theorem 2.1: Ni06
  • Proposition 2.2: Y17, Proposition 4.3
  • proof
  • Remark 2.3
  • Lemma 2.4
  • Lemma 2.5: Ni06, Theorem 5.2
  • Lemma 3.1
  • ...and 8 more