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Special metrics in hypercomplex geometry

Elia Fusi, Giovanni Gentili

Abstract

We provide a detailed treatment of special hyperhermitian metrics on hypercomplex manifolds. The quaternionic Gauduchon and quaternionic balanced conditions are investigated at length: we describe their properties and characterize their existence. We also consider strong HKT metrics proving an incompatibility result with balanced hyperhermitian metrics. Additionally, we introduce an Einstein-type condition and determine basic properties and obstructions. Finally, we collect several examples and counterexamples regarding the existence of these types of special metrics as well as two constructions that allow to build new examples.

Special metrics in hypercomplex geometry

Abstract

We provide a detailed treatment of special hyperhermitian metrics on hypercomplex manifolds. The quaternionic Gauduchon and quaternionic balanced conditions are investigated at length: we describe their properties and characterize their existence. We also consider strong HKT metrics proving an incompatibility result with balanced hyperhermitian metrics. Additionally, we introduce an Einstein-type condition and determine basic properties and obstructions. Finally, we collect several examples and counterexamples regarding the existence of these types of special metrics as well as two constructions that allow to build new examples.
Paper Structure (28 sections, 52 theorems, 250 equations)

This paper contains 28 sections, 52 theorems, 250 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hyperhermitian structures
  4. Positive forms and currents
  5. Useful formulae
  6. Canonical forms in hyperhermitian geometry
  7. The forms $\alpha$ and $\beta$
  8. Holonomy of the Obata connection
  9. Scalar curvatures
  10. Special metrics
  11. The first quaternionic Bott-Chern class
  12. Quaternionic Gauduchon metrics
  13. Quaternionic balanced metrics
  14. Strong HKT metrics
  15. Chern-Einstein hyperhermitian metrics
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $(M^n, \mathsf{H})$ be a compact hypercomplex manifold and $\Omega_{\mathrm{G}}$ a hyperhermitian metric compatible with $\mathsf{H}$ which is Gauduchon in the complex sense, i.e. $\partial \bar{\partial} \omega_{\mathrm{G}}^{2n-1}=0$, where $\omega_{\mathrm{G}}$ is the corresponding $(1,1)$-for and Moreover, the existence of a unique quaternionic balanced metric of unit volume in the conform

Theorems & Definitions (117)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Lemma 2.3: Alesker-Verbitsky (2006)
  • Lemma 2.4
  • ...and 107 more