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Journal of the American Mathematical Society

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ISSN 1088-6834 (online) ISSN 0894-0347 (print)

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Gromov invariants for holomorphic maps from Riemann surfaces to Grassmannians
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by Aaron Bertram, Georgios Daskalopoulos and Richard Wentworth PDF
J. Amer. Math. Soc. 9 (1996), 529-571 Request permission


Two compactifications of the space of holomorphic maps of fixed degree from a compact Riemann surface to a Grassmannian are studied. It is shown that the Uhlenbeck compactification has the structure of a projective variety and is dominated by the algebraic compactification coming from the Grothendieck Quot Scheme. The latter may be embedded into the moduli space of solutions to a generalized version of the vortex equations studied by Bradlow. This gives an effective way of computing certain intersection numbers (known as “Gromov invariants”) on the space of holomorphic maps into Grassmannians. We carry out these computations in the case where the Riemann surface has genus one.
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Additional Information
  • Aaron Bertram
  • Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112
  • MR Author ID: 246391
  • Email:
  • Georgios Daskalopoulos
  • Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912
  • MR Author ID: 313609
  • Email:
  • Richard Wentworth
  • Affiliation: Department of Mathematics, University of California, Irvine, California 92717
  • Email:
  • Received by editor(s): June 8, 1993
  • Received by editor(s) in revised form: November 22, 1994, and March 2, 1995
  • Additional Notes: The first author was supported in part by NSF Grant DMS-9218215.
    The second author was supported in part by NSF Grant DMS-9303494.
    The third author was supported in part by NSF Mathematics Postdoctoral Fellowship DMS-9007255.

  • Dedicated: Dedicated to Professor Raoul Bott on the occasion of his 70th birthday
  • © Copyright 1996 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 9 (1996), 529-571
  • MSC (1991): Primary 14C17; Secondary 14D20, 32G13
  • DOI:
  • MathSciNet review: 1320154