Critical Set of the Master Function and Characteristic Variety of the Associated Gauss–Manin Differential Equations
Critical Set of the Master Function and Characteristic Variety of the Associated Gauss–Manin Differential Equations
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相关高斯-马南微分方程主函数的临界集和特征多样性
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发表时间：
2014
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通讯作者：
A. Varchenko
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作者：
A. Varchenko
我们考虑 k 维仿射空间中 n 个平行传输超平面的加权族，并描述相关超几何积分的高斯-马宁微分方程的特征变化。特征变化以洛朗多项式的零集给出，其系数由权重和格拉斯曼 Gr(k, n) 中的关联点确定。洛朗多项式是对合的。这些陈述是概括性的（Varchenko，数学 2：218-231，2014），其中这样的描述是针对平行传输超平面的加权通用族获得的。微分方程和特征簇之间的中间对象是相关主函数的临界集上的函数代数。我们在高斯-马宁微分方程的向量空间和函数代数之间构造了线性同构。同构使我们能够描述特征多样性。它还允许我们定义代数向量空间上的积分结构以及此类代数族上的相关（组合）连接。
We consider a weighted family of n parallelly transported hyperplanes in a k-dimensional affine space and describe the characteristic variety of the Gauss–Manin differential equations for associated hypergeometric integrals. The characteristic variety is given as the zero set of Laurent polynomials, whose coefficients are determined by weights and the associated point in the Grassmannian Gr(k, n). The Laurent polynomials are in involution. These statements generalize (Varchenko, Mathematics 2:218–231, 2014), where such a description was obtained for a weighted generic family of parallelly transported hyperplanes. An intermediate object between the differential equations and the characteristic variety is the algebra of functions on the critical set of the associated master function. We construct a linear isomorphism between the vector space of the Gauss–Manin differential equations and the algebra of functions. The isomorphism allows us to describe the characteristic variety. It also allowed us to define an integral structure on the vector space of the algebra and the associated (combinatorial) connection on the family of such algebras.