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Quantum cohomology of the Grassmannian and alternate Thom–Sebastiani

Part of: Families, fibrations Projective and enumerative geometry Special varieties Symplectic geometry, contact geometry Singularities

Published online by Cambridge University Press:  01 January 2008

Bumsig Kim  and
Claude Sabbah
Bumsig Kim
Affiliation:
School of Mathematics, Korea Institute for Advanced Study, 207-43 Cheongnyangni 2-dong, Dongdaemun-gu, Seoul, 130-722 Korea (email: bumsig@kias.re.kr)
Claude Sabbah
Affiliation:
UMR 7640 du C.N.R.S., Centre de mathématiques Laurent Schwartz, École polytechnique, F-91128 Palaiseau cedex, France (email: sabbah@math.polytechnique.fr)
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Abstract

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We introduce the notion of an alternate product of Frobenius manifolds and we give, after Ciocan-Fontanine et al., an interpretation of the Frobenius manifold structure canonically attached to the quantum cohomology of G(r,n+1) in terms of alternate products. We also investigate the relationship with the alternate Thom–Sebastiani product of Laurent polynomials.

Keywords

Frobenius manifoldGrassmannianThom–Sebastiani sumalternate productGauss–Manin system

MSC classification

Secondary: 53D45: Gromov-Witten invariants, quantum cohomology, Frobenius manifolds 14D05: Structure of families (Picard-Lefschetz, monodromy, etc.) 32S40: Monodromy; relations with differential equations and $D$-modules 14M15: Grassmannians, Schubert varieties, flag manifolds 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants
Type
Research Article
Information
Compositio Mathematica , Volume 144 , Issue 1 , January 2008 , pp. 221 - 246
Copyright
Copyright © Foundation Compositio Mathematica 2008

References

The first author is supported by KOSEF grant R01-2004-000-10870-0.