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Hermite--Einstein metrics in singular settings

Junyan Cao, Patrick Graf, Philipp Naumann, Mihai Paun, Thomas Peternell, Xiaojun Wu

Abstract

In this article we pursue the following main goals. In the first place, we establish the existence of "estimable" Hermite--Einstein metrics for stable reflexive coherent sheaves on compact normal Kähler spaces. If moreover the background variety has klt singularities, we obtain a much more precise result.

Hermite--Einstein metrics in singular settings

Abstract

In this article we pursue the following main goals. In the first place, we establish the existence of "estimable" Hermite--Einstein metrics for stable reflexive coherent sheaves on compact normal Kähler spaces. If moreover the background variety has klt singularities, we obtain a much more precise result.
Paper Structure (25 sections, 30 theorems, 251 equations)

This paper contains 25 sections, 30 theorems, 251 equations.

Table of Contents

  1. Preliminaries and main Results
  2. Abstract
  3. Acknowledgements
  4. Hermitian metrics on reflexive sheaves
  5. Mean-value inequality
  6. Desingularisation of sheaves
  7. Connection with a theorem of Rossi
  8. Mean-value inequality
  9. Approximate stability and uniformity properties
  10. Proof of Theorem \ref{['HE']}
  11. Canonical constructions
  12. End of the proof
  13. The analysis of the first case
  14. The analysis of the second case
  15. Applications: a few intermediate results
  16. ...and 10 more sections

Key Result

Theorem 1.1

The bundle $E$ admits a Hermitian metric $h_E$ such that there exists a constant $C> 0$ for which the inequality is satisfied. Moreover, the restriction of the metric $h_E$ to the $\pi$-inverse image of $X_0$ is induced by a metric $h_{\mathcal{F}}$ on $\mathcal{F}$ constructed by a similar procedure as above.

Theorems & Definitions (62)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Theorem 1.9
  • Lemma 2.1
  • ...and 52 more