Line bundles on spectral curves and the generalised Legendre transform construction of hyperkähler metrics

2008/06/30 by Roger Bielawski
Mathematics · #Advanced Algebra and Geometry #Conjugacy class #Curvature #Diagonal #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Hermitian manifold #Hyperkähler manifold #Legendre polynomials #Line (geometry) #Line bundle #Mathematical analysis #Mathematics #Pure mathematics #Ricci curvature #Torus #math.AG #math.DG #msc:14H70 #msc:53C26 #msc:53C28

paper · pdf · doi:10.1016/j.geomphys.2008.11.010

a diagram added; a few clarifications; 25 pages

arxiv created 2008/11/20 · openalex publication_date 2008/12/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

An analogue of the correspondence between GL(k)-conjugacy classes of matricial polynomials and line bundles is given for K-conjugacy classes, where K is one of the following: maximal parabolic, maximal torus, GL(k-1) embedded diagonally. The generalised Legendre transform construction of hyperkaehler metrics is studied further, showing that many known hyperkaehler metrics (including the ones on coadjoint orbits) arise in this way, and giving a large class of new (pseudo-)hyperkaehler metrics, analogous to monopole metrics.

Citations

• Principles of Algebraic Geometry
• Asymptotic Metrics for SU ( N )-Monopoles with Maximal Symmetry Breaking
• Hyper-Kähler Hierarchies and Their Twistor Theory
• The Geometry and Dynamics of Magnetic Monopoles
• A Hyper-KÄHlerian Structure on Coadjoint Orbits of a Semisimple Complex Group
• Supersymmetric σ-models, twistors, and the Atiyah-Hitchin metric
• Determinantal hypersurfaces