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Einstein metrics on bundles over hyperKähler manifolds

Udhav Fowdar

Abstract

We construct explicit examples of quaternion-Kähler and hypercomplex structures on bundles over hyperKähler manifolds. We study the infinitesimal symmetries of these examples and the associated Galicki-Lawson quaternion-Kähler moment map. By performing the QK reduction we produce several explicit QK metrics. Moreover we are led to a new proof of a hyperKähler/quaternion-Kähler type correspondence. We also give examples of other Einstein metrics and balanced Hermitian structures on these bundles.

Einstein metrics on bundles over hyperKähler manifolds

Abstract

We construct explicit examples of quaternion-Kähler and hypercomplex structures on bundles over hyperKähler manifolds. We study the infinitesimal symmetries of these examples and the associated Galicki-Lawson quaternion-Kähler moment map. By performing the QK reduction we produce several explicit QK metrics. Moreover we are led to a new proof of a hyperKähler/quaternion-Kähler type correspondence. We also give examples of other Einstein metrics and balanced Hermitian structures on these bundles.

Paper Structure

This paper contains 34 sections, 18 theorems, 157 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hermitian structures
  4. Quaternionic structures
  5. Proofs of Theorems \ref{['maintheorem']}, \ref{['hypercomplextheorem']} and \ref{['extendedhypercomplextheorem']}
  6. Examples
  7. Examples with $M=\mathbb{T}^4$
  8. Examples from the Gibbons-Hawking ansatz
  9. Symmetries and moment maps
  10. The QK reduction
  11. Reduction using tri-holomorphic Killing vector fields
  12. Reduction using permuting vector fields
  13. Reduction using homothetic Killing vector fields
  14. An explicit example
  15. $3$-Sasakian manifolds
  16. ...and 19 more sections

Key Result

Theorem 1.1

The manifold $P^{4n+2}=\mathbb{R}_t \times M^{4n+1}_\alpha$ admits a Kähler-Einstein metric with associated Kähler $2$-form given by The manifold $L^{4n+3}=\mathbb{R}_t \times M^{4n+2}_{\alpha, \xi}$ admits an Einstein metric The manifold $N^{4n+4}=\mathbb{R}_t \times M^{4n+3}_{\alpha,\xi,\eta}$ admits a quaternion-Kähler metric with the associated quaternion-Kähler $4$-form given by where F

Theorems & Definitions (44)

  • Theorem 1.1
  • Theorem 1.2
  • proof : Proof of Theorem \ref{['maintheorem']}
  • Remark 3.1
  • proof : Proof of Theorem \ref{['hypercomplextheorem']}
  • Remark 3.2
  • Theorem 3.3
  • proof : Proof of Theorem \ref{['extendedhypercomplextheorem']}
  • Remark 3.4
  • Remark 4.1
  • ...and 34 more