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Composition series for GKZ-systems. (English) Zbl 1445.14033
V. V. Batyrev [Duke Math. J. 69, No. 2, 349–409 (1993; Zbl 0812.14035)], J. Stienstra [in: Integrable systems and algebraic geometry. Proceedings of the 41st Taniguchi symposium, Kobe, Japan, June 30–July 4, 1997, and in Kyoto, Japan, July 7–11 1997. Singapore: World Scientific. 412–452 (1998; Zbl 0963.14017)], and A. Adolphson [Duke Math. J. 73, No. 2, 269–290 (1994; Zbl 0804.33013)] considered an increasing filtration \(W_{\bullet}(A,\beta)\) on the GKZ system \(M_{A}(\beta)\). When \(\beta\) is special, this filtration is related to the mixed Hodge structure of the cohomology of hypersurfaces in a toric variety.
In the article under review, the author studies the question that whether the associated graded pieces of this filtration are semisimple \(\mathcal D\)-modules. The author constructs canonical epimorphisms from these associated graded pieces to some \(\mathcal D\)-modules coming from “smaller” GKZ systems, and gives a criterion when these epimorphisms can be made isomorphisms. In particular, he shows that if \(A\) is “simplicial relative to \(\beta\)” and \(\beta\) is “weakly \(A\)-nonresonant” (two conditions that are combinatorial in nature), then these associated graded pieces are indeed semisimple \(\mathcal D\)-modules.
MSC:
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)
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