Gromov-Witten invariants on Grassmannians

Authors:
Anders Skovsted Buch, Andrew Kresch and Harry Tamvakis

Journal:
J. Amer. Math. Soc. **16** (2003), 901-915

MSC (2000):
Primary 14N35; Secondary 14M15, 14N15, 05E15

DOI:
https://doi.org/10.1090/S0894-0347-03-00429-6

Published electronically:
May 1, 2003

MathSciNet review:
1992829

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Abstract | References | Similar Articles | Additional Information

Abstract: We prove that any three-point genus zero Gromov-Witten invariant on a type Grassmannian is equal to a classical intersection number on a two-step flag variety. We also give symplectic and orthogonal analogues of this result; in these cases the two-step flag variety is replaced by a sub-maximal isotropic Grassmannian. Our theorems are applied, in type , to formulate a conjectural quantum Littlewood-Richardson rule, and in the other classical Lie types, to obtain new proofs of the main structure theorems for the quantum cohomology of Lagrangian and orthogonal Grassmannians.

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Additional Information

**Anders Skovsted Buch**

Affiliation:
Matematisk Institut, Aarhus Universitet, Ny Munkegade, 8000 Århus C, Denmark

Email:
abuch@imf.au.dk

**Andrew Kresch**

Affiliation:
Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, Pennsylvania 19104-6395

Email:
kresch@math.upenn.edu

**Harry Tamvakis**

Affiliation:
Department of Mathematics, Brandeis University - MS 050, P. O. Box 9110, Waltham, Massachusetts 02454-9110

Email:
harryt@brandeis.edu

DOI:
https://doi.org/10.1090/S0894-0347-03-00429-6

Keywords:
Gromov-Witten invariants,
Grassmannians,
Flag varieties,
Schubert varieties,
Quantum cohomology,
Littlewood-Richardson rule

Received by editor(s):
July 18, 2002

Published electronically:
May 1, 2003

Additional Notes:
The authors were supported in part by NSF Grant DMS-0070479 (Buch), an NSF Postdoctoral Research Fellowship (Kresch), and NSF Grant DMS-0296023 (Tamvakis).

Article copyright:
© Copyright 2003
American Mathematical Society