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On the better behaved version of the GKZ hypergeometric system

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Abstract

We consider a version of the generalized hypergeometric system introduced by Gelfand, Kapranov and Zelevinsky (GKZ) suited for the case when the underlying lattice is replaced by a finitely generated abelian group. In contrast to the usual GKZ hypergeometric system, the rank of the better behaved GKZ hypergeometric system is always the expected one. We give largely self-contained proofs of many properties of this system. The discussion is intimately related to the study of the variations of Hodge structures of hypersurfaces in algebraic tori.

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Acknowledgments

Upon learning about our construction, in a letter to one of the authors, Alan Adolphson [2] wrote us that he obtained a similar definition for a generalization of the GKZ system in the case of a lattice \(N.\) Hiroshi Iritani informed us that in a recent preprint [14], the better behaved GKZ system appears as a natural ingredient in his study of the quantum \(D\)-module associated to a toric complete intersection and the periods of its mirror. We would also like to thank Vladimir Retakh for a useful reference and the referee for the careful and thoughtful reading of the text. We sketch an alternative point of view on some of the constructions presented in this paper in Remarks 2.2, 2.9 and 2.10, following the referee’s suggestions.

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Author notes

  1. R. Paul Horja

    Present address: Fakultät für Mathematik, Universität Wien, Garnisongasse 3/14, 1090 , Wien, Austria

Authors and Affiliations

  1. Department of Mathematics, Rutgers University, Piscataway, NJ, 08854-8019, USA

    Lev A. Borisov

  2. Department of Mathematics, Oklahoma State University, Stillwater, OK, 74078, USA

    R. Paul Horja

Authors
  1. Lev A. Borisov
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  2. R. Paul Horja
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Correspondence to R. Paul Horja.

Additional information

The first author was partially supported by the NSF Grant DMS-1003445. The second author was partially supported by the NSA Grant MDA904-10-1-0190.

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Borisov, L.A., Paul Horja, R. On the better behaved version of the GKZ hypergeometric system. Math. Ann. 357, 585–603 (2013). https://doi.org/10.1007/s00208-013-0913-6

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  • DOI: https://doi.org/10.1007/s00208-013-0913-6

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