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Gromov invariants for holomorphic maps
from Riemann surfaces to Grassmannians

Authors: Aaron Bertram, Georgios Daskalopoulos and Richard Wentworth
Journal: J. Amer. Math. Soc. 9 (1996), 529-571
MSC (1991): Primary 14C17; Secondary 14D20, 32G13
MathSciNet review: 1320154
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Abstract: 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

Georgios Daskalopoulos
Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912

Richard Wentworth
Affiliation: Department of Mathematics, University of California, Irvine, California 92717

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
Article copyright: © Copyright 1996 American Mathematical Society