\(A\)-hypergeometric systems that come from geometry
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 irreducible representation \(\chi\) of \(G\) denote \(H^i_{DR}(X/\mathbb{C}[\lambda])^{\chi}\) the sum of all \(G\) submodules isomorphic to \(\chi\). The \(H^i_{DR}(X/\mathbb{C}[\lambda])^{\chi}\) have a \(\mathcal{D}\)-module structure via Gauss-Manin connection. This \(\mathcal{D}\)-module is called a \(\mathcal{D}\)-module that comes from geometry.NEWLINENEWLINELet \(C(A)\) be a real cone generated by \(A\), let \(l_1,\dots,l_s\) be homogeneous linear forms definig codimension-one faces of \(C(A)\). We say that \(\alpha\) is nonresonant for \(A\) if \(l_i(\alpha)\notin \mathbb{Z}\) for all \(i\).NEWLINENEWLINEThe main result of the paper is the following says that if \(f=x_ng(x_1,\dots,x_{n-1})\), \(\alpha\) is nonresonant for \(A\), \(\alpha\in \mathbb{Q}^n\), then the \(A\)-hypergeometric \(\mathcal{D}\)-module \(\mathcal{M}_{\alpha}\) comes from geometry.NEWLINENEWLINEA relation of the obtained results with results of N. Katz about systems with full set of algebraic solution.
scientific article; zbMATH DE number 6038836
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