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Quantum hyperplane section principle for concavex decomposable vector bundles. (English) Zbl 0986.14035

From the introduction: The Lefschetz hyperplane section theorem says that there is an intimate relationship between the cohomology group of an ambient space and that of a smooth zero locus of a positive line bundle over the ambient space. Roughly speaking, its quantum version says that there is an intimate relationship between quantum cohomology rings of the ambient space and that of a smooth zero locus of the decomposable spanned vector bundle.
In a previous paper [B. Kim, Acta Math. 183, 71-99 (1999; Zbl 1023.14028)], we proved the quantum analog when the ambient space is a generalized flag manifold and the decomposable vector bundle is convex. We claimed that the quantum analog can be generalized to the case when the bundle is concavex and decomposable. We explain the claim in this paper. As an application we reprove the multiple cover formula. This work is originally motivated by the paper in Asian J. Math. 1, 729-763 (1997; Zbl 0953.14026) by B. H. Lian, K. Liu and S.-T. Yau.

MSC:

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14M17 Homogeneous spaces and generalizations
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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