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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.
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)
Full Text: DOI
[1] Adolphson, Alan, Hypergeometric functions and rings generated by monomials, Duke Math. J., 73, 2, 269-290 (1994) · Zbl 0804.33013
[2] A2 A. Adolphson, \newblock Composition series for \(M_A(\beta )\), \newblock preliminary report.
[3] Batyrev, Victor V., Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori, Duke Math. J., 69, 2, 349-409 (1993) · Zbl 0812.14035
[4] Bruns, Winfried; Herzog, J\"urgen, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics 39, xii+403 pp. (1993), Cambridge University Press, Cambridge · Zbl 0788.13005
[5] Brylinski, Jean-Luc, Transformations canoniques, dualit\'e projective, th\'eorie de Lefschetz, transformations de Fourier et sommes trigonom\'etriques, G\'eom\'etrie et analyse microlocales (French, with English summary), Ast\'erisque, 140-141, 3-134, 251 (1986) · Zbl 0624.32009
[6] D’Agnolo, Andrea; Eastwood, Michael, Radon and Fourier transforms for \(\mathcal{D} \)-modules, Adv. Math., 180, 2, 452-485 (2003) · Zbl 1051.32008
[7] Fern\'andez-Fern\'andez, Mar\'\ia-Cruz; Walther, Uli, Restriction of hypergeometric \(\mathcal{D} \)-modules with respect to coordinate subspaces, Proc. Amer. Math. Soc., 139, 9, 3175-3180 (2011) · Zbl 1232.13016
[8] Fu, Lei, \( \ell \)-adic GKZ hypergeometric sheaves and exponential sums, Adv. Math., 298, 51-88 (2016) · Zbl 1368.14032
[9] Gel\cprime fand, I. M.; Kapranov, M. M.; Zelevinsky, A. V., Generalized Euler integrals and \(A\)-hypergeometric functions, Adv. Math., 84, 2, 255-271 (1990) · Zbl 0741.33011
[10] Hotta, Ryoshi; Takeuchi, Kiyoshi; Tanisaki, Toshiyuki, \(D\)-modules, perverse sheaves, and representation theory, Progress in Mathematics 236, xii+407 pp. (2008), Birkh\"auser Boston, Inc., Boston, MA · Zbl 1136.14009
[11] Matusevich, Laura Felicia; Miller, Ezra; Walther, Uli, Homological methods for hypergeometric families, J. Amer. Math. Soc., 18, 4, 919-941 (2005) · Zbl 1095.13033
[12] RW T. Reichelt and U. Walther, \newblock Weight filtrations on GKZ-systems, \newblock \href https://arxiv.org/abs/1809.04247https://arxiv.org/abs \href https://arxiv.org/abs/1809.04247/1809.04247.
[13] Steiner, Avi, \(A\)-hypergeometric modules and Gauss-Manin systems, J. Algebra, 524, 124-159 (2019) · Zbl 1408.14032
[14] Schulze, Mathias; Walther, Uli, Hypergeometric D-modules and twisted Gau\ss-Manin systems, J. Algebra, 322, 9, 3392-3409 (2009) · Zbl 1181.13023
[15] Schulze, Mathias; Walther, Uli, Resonance equals reducibility for \(A\)-hypergeometric systems, Algebra Number Theory, 6, 3, 527-537 (2012) · Zbl 1251.13023
[16] Stienstra, Jan, Resonant hypergeometric systems and mirror symmetry. Integrable systems and algebraic geometry, Kobe/Kyoto, 1997, 412-452 (1998), World Sci. Publ., River Edge, NJ · Zbl 0963.14017
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