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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Nondegenerate multidimensional matrices and instanton bundles

Authors: Laura Costa and Giorgio Ottaviani
Journal: Trans. Amer. Math. Soc. 355 (2003), 49-55
MSC (2000): Primary 14D21, 14J60; Secondary 15A72
Published electronically: September 6, 2002
MathSciNet review: 1927201
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Abstract: In this paper we prove that the moduli space of rank $2n$ symplectic instanton bundles on ${\mathbb{P} ^{2n+1}}$, defined from the well-known monad condition, is affine. This result was not known even in the case $n=1$, where by Atiyah, Drinfeld, Hitchin, and Manin in 1978 the real instanton bundles correspond to self-dual Yang Mills $Sp(1)$-connections over the $4$-dimensional sphere. The result is proved as a consequence of the existence of an invariant of the multidimensional matrices representing the instanton bundles.

References [Enhancements On Off] (What's this?)

  • [AO94] V. Ancona and G. Ottaviani.
    Stability of special instanton bundles on $\mathbb{P} ^{2n+1}$.
    Trans. Amer. Math. Soc., 341, (1994), 677-693. MR 94d:14017
  • [AO99] V. Ancona and G. Ottaviani.
    Unstable hyperplanes for Steiner bundles and multidimensional matrices.
    Advances in Geometry, 1, (2001), 165-192.
  • [AO00] V. Ancona and G. Ottaviani.
    On the irreducible components of the moduli space of instanton bundles on $\mathbb{P} ^5$.
    Geometry Seminars 1998-1999 (S. Coen, ed.), 95-100, Bologna 2000. MR 2001h:14055
  • [ADHM78] M. F. Atiyah, V. G. Drinfeld, N. J. Hitchin, and Yu. I. Manin.
    Construction of instantons,
    Phys Lett., A65, (1978), 185 -187. MR 82g:81049
  • [BH78] W. Barth and K. Hulek.
    Monads and moduli of vector bundles.
    Manuscripta Math., 25, (1978), 323 -347. MR 80f:14005
  • [BS] D. Bayer and M. Stillman.
    Macaulay, a computer algebra system for algebraic geometry ( bayer/Macaulay.html).
  • [GKZ94] I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky.
    Discriminants, Resultants and Multidimensional Determinants,
    Birkhäuser, Boston, 1994. MR 95e:14045
  • [HH86] A. Hirschowitz and K. Hulek.
    Complete families of stable vector bundles over ${\mathbb{P} ^2}$.
    Complex analysis and algebraic geometry, Proc. Conf. Göttingen 1985, Lect. Notes Math., 1194, 1986, 19-40. MR 87j:14019
  • [KO99] P. I. Katsylo and G. Ottaviani.
    Regularity of the moduli space of instanton bundles $MI_{\mathbb{P} ^3}(5)$.
    AG/ 9911184, 1999.
  • [OS86] C. Okonek and H. Spindler.
    Mathematical instanton bundles on $\mathbb{P} ^{2n+1}$.
    J. Reine Angew. Math., 364, (1986), 35-50. MR 87e:14016
  • [MO97] R. M. Miró-Roig and X. Orus-Lacort.
    On the smoothness of the moduli space of mathematical instanton bundles.
    Comp. Math., 105, (1997), 109-119. MR 97m:14011
  • [PV89] V. L. Popov and E. B. Vinberg.
    Invariant theory.
    Algebraic geometry. IV: Linear algebraic groups, invariant theory, Encycl. Math. Sci. 55, (1994), 123-278; translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 55, (1989), 137-309.
  • [Sal84] S. Salamon.
    Quaternionic structures and twistor spaces.
    Global Riemannian Geometry (eds. N. J. Willmore and N. J. Hitchin), Ellis Horwood, London 1984.
  • [WZ96] J. Weyman and A. Zelevinsky.
    Singularities of hyperdeterminants.
    Ann. Inst. Fourier, Grenoble, 46, (1996), 3, 591-644. MR 97m:14050

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Additional Information

Laura Costa
Affiliation: Departament Algebra i Geometria, Universitat de Barcelona, Gran Via, 585, 08007 Barcelona, Spain

Giorgio Ottaviani
Affiliation: Dipartimento di Matematica “U. Dini", Università di Firenze, viale Morgagni 67/A, I 50134 Firenze, Italy

Received by editor(s): October 23, 2001
Published electronically: September 6, 2002
Additional Notes: The first author was partially supported by DGICYT BFM2001-3584
The second author was partially supported by Italian MURST
Article copyright: © Copyright 2002 American Mathematical Society

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