zbMATH — the first resource for mathematics

Irregular hypergeometric \(\mathcal D\)-modules. (English) Zbl 1236.14018
The paper under review studies the irregularity of the hypergeometric \(\mathcal{D}\)-modules \(\mathcal{M}_A(\beta)\) introduced by I. M. Gel’fand, M. I. Graev, M. M. Kapranov and A. V. Zelevinsky using explicit construction of Gevrey series along coordinate subspaces of \(\mathbb{C}^n\). Here \(A\) denotes a full rank \(d\times n\)-matrix with integer entries and \(\beta=(\beta_1,\dots,\beta_d)\) are complex parameters. The module \(\mathcal{M}_A(\beta)\) is the quotient \(\mathcal{D}/H_A(\beta)\) where \(H_A(\beta)\) is made of toric operators and Euler operators.
After reviewing facts on Gevrey series and slopes of \(\mathcal{D}\)-modules, the author explicitly constructs Gevrey series solutions of \(\mathcal{M}_A(\beta)\) along coordinate subspaces. Using those series, ideas from M. Schulze and U. Walther [Duke Math. J. 142, No. 3, 465–509 (2008; Zbl 1144.13012)] and the Comparison Theorem of the slopes of Y. Laurent and Z. Mebkhout [Ann. Sci. Éc. Norm. Supér. (4) 32, No. 1, 39–69 (1999; Zbl 0944.14007)], she then gives in Theorem 5.9 a combinatorial description of the set of slopes of \(\mathcal{M}_A(\beta)\) along coordinate hyperplanes. This generalizes the result of Schulze and Walther (in the case of coordinate hyperplanes) because there are no further assumptions on \(A\) (for instance, pointedness).
The author also gives lower bounds for the dimensions of the Gevrey solution spaces, which are the actual dimensions for very generic parameters. Under some conditions on \(A\) and \(\beta\), she finally obtains a basis of the stalk at a generic point of the irregularity sheaves of \(\mathcal{M}_A(\beta)\) along a coordinate hyperplane.

14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
32C38 Sheaves of differential operators and their modules, \(D\)-modules
13N10 Commutative rings of differential operators and their modules
33C99 Hypergeometric functions
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
Full Text: DOI
[1] Adolphson, A., A-hypergeometric functions and rings generated by monomials, Duke math. J., 73, 2, 269-290, (1994) · Zbl 0804.33013
[2] Berkesch, Ch., The rank of a hypergeometric system · Zbl 1214.33009
[3] Castro-Jiménez, F.J.; Takayama, N., Singularities of the hypergeometric \(\mathcal{D}\)-module associated with a monomial curve, Trans. amer. math. soc., 355, 9, 3761-3775, (2003) · Zbl 1060.33023
[4] Fernández-Fernández, M.C.; Castro-Jiménez, F.J., Gevrey solutions of irregular hypergeometric systems in two variables · Zbl 1239.32008
[5] M.C. Fernández-Fernández, F.J. Castro-Jiménez, Gevrey solutions of the irregular hypergeometric system associated with an affine monomial curve, Trans. Amer. Math. Soc., in press, arXiv:0811.3392v1 [math.AG]
[6] Gel’fand, I.M.; Graev, M.I.; Zelevinsky, A.V., Holonomic systems of equations and series of hypergeometric type, Dokl. akad. nauk SSSR, 295, 1, 14-19, (1987) · Zbl 0661.22005
[7] Gel’fand, I.M.; Zelevinsky, A.V.; Kapranov, M.M., Hypergeometric functions and toral manifolds, Funktsional. anal. i prilozhen., 23, 2, 12-26, (1989) · Zbl 0721.33006
[8] Grothendieck, A., Cohomologie des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux, Sga, vol. 2, (1962), Publications de l’IHES · Zbl 0159.50402
[9] Hartillo, M.I., Hypergeometric slopes of codimension 1, Rev. mat. iberoam., 19, 2, 455-466, (2003) · Zbl 1058.35008
[10] Hartillo, M.I., Irregular hypergeometric systems associated with a singular monomial curve, Trans. amer. math. soc., 357, 11, 4633-4646, (2004) · Zbl 1082.32004
[11] Hotta, R., Equivariant \(\mathcal{D}\)-modules, (1998)
[12] Laurent, Y., Calcul d’indices et irrégularité pour LES systèmes holonomes, Differential systems and singularities, luminy, 1983, Astérisque, 130, 352-364, (1985) · Zbl 0569.58031
[13] Laurent, Y., Polygône de Newton et b-fonctions pour LES modules micro-différentiels, Ann. sci. ecole norm. sup. (4), 20, 3, 391-441, (1987) · Zbl 0646.58021
[14] Laurent, Y.; Mebkhout, Z., Pentes algébriques et pentes analytiques d’un \(\mathcal{D}\)-module, Ann. sci. ecole norm. sup. (4), 32, 1, 39-69, (1999) · Zbl 0944.14007
[15] Matusevich, L.F.; Miller, E.; Walther, U., Homological methods for hypergeometric families, J. amer. math. soc., 18, 4, 919-941, (2005) · Zbl 1095.13033
[16] Mebkhout, Z., Sur le théorème de semi-continuité des équations différentielles, Astérisque, 130, 365-417, (1985) · Zbl 0592.32009
[17] Mebkhout, Z., Le théorème de positivité de l’irrégularité pour LES \(\mathcal{D}_X\)-modules, (), 83-131
[18] Mebkhout, Z., Le théorème de positivité, le théorème de comparaison et le théorème d’existence de rieman, Semin. congr., 8, 163-310, (2004) · Zbl 1082.32006
[19] Ohara, K.; Takayama, N., Holonomic rank of A-hypergeometric differential-difference equations, J. pure appl. algebra, 213, 1501-1642, (2009)
[20] Passare, M.; Sadykov, T.; Tsikh, A., Singularities of hypergeometric functions in several variables, Compos. math., 141, 787-810, (2005) · Zbl 1080.33012
[21] Saito, M., Logarithm-free A-hypergeometric functions, Duke math. J., 115, 1, (2002) · Zbl 1031.33011
[22] Saito, M.; Sturmfels, B.; Takayama, N., Gröbner deformations of hypergeometric differential equations, Algorithms comput. math., vol. 6, (2000), Springer · Zbl 0946.13021
[23] Schulze, M.; Walther, U., Irregularity of hypergeometric systems via slopes along coordinate subspaces, Duke math. J., 142, 3, 465-509, (2008) · Zbl 1144.13012
[24] Sturmfels, B., Gröbner bases and convex polytopes, Univ. lecture ser., vol. 8, (1995), American Mathematical Society Providence
[25] Takayama, N., Modified A-hypergeometric systems, Kyushu J. math., 63, 113-122, (2009) · Zbl 1180.33021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.