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On Kleinian Singularities and Quivers

  • Chapter
Singularities

Part of the book series: Progress in Mathematics ((PM,volume 162))

Abstract

Starting from McKay’s observation on the description of (an essential part of) the representation theory of binary polyhedral groups Γ in terms of extended Coxeter-Dynkin-Witt diagrams \(\tilde \Delta (\Gamma )\) and working in the differential geometric framework of Hyper-Kähler-quotients P.B. Kronheimer was able to give a new construction of the semiuniversal deformations of the Kleinian singularities X = ℂ2/Γ as well as of their simultaneous resolutions ([24], [25], [26]). As far as the deformations were concerned, he already gave a purely algebraic geometric formulation of his results in terms of representations of certain quivers naturally attached to the diagrams \(\tilde \Delta (\Gamma )\). By making use of the invariant-theoretic notion of “linear modification” (cf. Section 6, below) and applying it to Kronheimer’s quiver construction we show here how to obtain a purely algebraic geometric simultaneous resolution as well (Section 7). On the way, we shall take the opportunity to remind the reader of various facts about

  • Kleinian singularities (Section 1),

  • McKay’s observation (Section 2),

  • Symplectic geometry (Section 3),

  • Kronheimer’s work (Section 4), and

  • Quivers (Section 5).

This article covers the main results of the doctoral dissertation [9] written at Hamburg university under the guidance of the second named author and supported by a DFG-grant (Ri 303/3-2). More details and worded out examples may be found there.

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Authors and Affiliations

  1. Reuters A.G., Graf-Adolf-Str. 35-37, 40210, Düsseldorf, Germany

    Heiko Cassens

  2. Mathematisches Seminar, Universität Hamburg, Bundesstraße 55, 20146, Hamburg, Germany

    Peter Slodowy

Authors
  1. Heiko Cassens
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  2. Peter Slodowy
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Editor information

Editors and Affiliations

  1. Department of Geometry and Topology, Steklov Mathematical Institute, 8, Gubkina Stree, 117966, Moscow GSP-1, Russia

    V. I. Arnold

  2. CEREMADE, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, Postfach 3049, F-75775, Paris Cedex 16e, France

    V. I. Arnold

  3. Fachbereich Mathematik, Universität Kaiserslautern, D-67653, Kaiserslautern, Germany

    G.-M. Greuel

  4. Subfaculteit Wiskunde, Katholieke Universiteit Nijmegen Toernooiveld, NL-6525 ED, Nijmegen, The Netherlands

    J. H. M. Steenbrink

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Cassens, H., Slodowy, P. (1998). On Kleinian Singularities and Quivers. In: Arnold, V.I., Greuel, GM., Steenbrink, J.H.M. (eds) Singularities. Progress in Mathematics, vol 162. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8770-0_14

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  • DOI: https://doi.org/10.1007/978-3-0348-8770-0_14

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9767-9

  • Online ISBN: 978-3-0348-8770-0

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