\(A\)-hypergeometric systems that come from geometry
type: Item
instance of: scholarly article
zbMATH Open document ID: 1257.14016
DOI: 10.1090/S0002-9939-2011-11073-6
author: Alan Adolphson
author: Steven Sperber
published in: Proceedings of the American Mathematical Society
publication date: 2012-05-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/1007.4030
Mathematics Subject Classification ID: 14F10
Mathematics Subject Classification ID: 33C70
Mathematics Subject Classification ID: 14F40
Mathematics Subject Classification ID: 52B20
zbMATH DE Number: 6038836
zbMATH Keywords: A-hypergeomeric system
zbMATH Keywords: \(\mathcal{D}\)-module
zbMATH Keywords: Gauss-Manin connection
zbMATH Keywords: torus
MaRDI profile type: Publication
arXiv ID: 1007.4030
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cites work: Algebraic solutions of differential equations (p-curvature and the Hodge filtration)
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cites work: Q3739243
cites work: Hypergeometric D-modules and twisted Gauß-Manin systems
OpenAlex ID: W2592530723
title: \(A\)-hypergeometric systems that come from geometry
review text: In recent work, \textit{F. Beukers} [Invent. Math. 180, No. 3, 589--610 (2010; Zbl 1251.33011)] characterized \(A\)-hypergeometric systems having a full set of algebraic solutions. He determined which \(A\)-hypergeometric systems have a full set of solutions modulo \(p\) for almost all primes \(p\) and showed that these systems come from geometry. Then he applied a theorem of N. Katz which says the such systems have a full set of algebraic solutions.NEWLINENEWLINEIn the present paper a connection between nonresonant \(A\)-hypergeometric systems and de Rham-type complexes is established, this leads to a determination of which \(A\)-hypergeometric systems come from geometry.NEWLINENEWLINEConsider a \(A\)-hypergeomeric system determined by \(A=\{a^{(1)},\dots,a^{(N)}\}\subset \mathbb{Z}^n\) and \(\alpha=(\alpha_1\dots,\alpha_n)\in \mathbb{C}^n\). Let \(L=\{(l_1,\dots,l_N)\in \mathbb{Z}^N| \sum_{j=1}^N l_ja^{(j)}=0\}\). An \(A\)-hypergeometric system if defined as a system of equationsNEWLINENEWLINENEWLINE\[NEWLINE\square_lf=\prod_{l_j>0}(\frac{\partial }{\partial \lambda_j})^{l_j}f-\prod_{l_j<0}(\frac{\partial }{\partial \lambda_j})^{-l_j}f=0,NEWLINE\]NEWLINENEWLINE\[NEWLINEZ_{i,\alpha}f=\sum_{j=1}^N a_i^{(j)}\lambda_j\frac{\partial }{\partial \lambda_j}f-\alpha_if=0. NEWLINE\]NEWLINENEWLINENEWLINELet \(\mathcal{D}=\mathbb{C}(\lambda_1,\dots,\lambda_N,\frac{\partial}{\partial \lambda_1},\dots,\frac{\partial}{\partial \lambda_N})\) be the rign of differential operators. Denote as \(\mathcal{M}_{\alpha}=\mathcal{D}/(\sum_l\mathcal{D}\square_lf+\sum_{i=1}^n\mathcal{D}Z_{i,\alpha})\) the corresponing \(\mathcal{D}\)-module. It is called \(A\)-hypergeometric \(\mathcal{D}\)-module.NEWLINENEWLINENow let us define the class of \(\mathcal{D}\)-modules that come from geometry. Let \(X\) be a smooth variety over \(\mathbb{C}[\lambda]=\mathbb{C}[\lambda_1,\dots,\lambda_n]\). Let \(G\) be a group acting on \(X\), then it acts on \(H^i_{DR}(X/\mathbb{C}[\lambda])\) also. For an irr…
reviewed by: Valentina Golubeva
description: scientific article; zbMATH DE number 6038836