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Algebraic aspects of hypergeometric differential equations. (English) Zbl 1468.14042

GKZ systems are certain modules over the ring of differential operators \(\mathcal{D}=\mathbb{C}[x_1,\ldots,x_n]\langle\partial_1,\ldots,\partial_n\rangle\) that generalise the hypergeometric differential equations that trace back to Euler and Gauss and encompass information of both combinatorial and algebro-geometrical nature. Therefore, they have become a common and useful object when working in several branches of algebraic geometry. The article under review provides a good survey of the modern study of such objects and their application to mirror symmetry. After all, its authors have contributed in several significant works to such study.
The paper begins with a so-called introduction, in which the authors deal with somewhat older aspects of GKZ systems, such as their relation with the more classical hypergeometric functions and differential equations and the solutions of such systems. In the next section, they introduce the Euler-Koszul complex (cf. [L. F. Matusevich et al., J. Am. Math. Soc. 18, No. 4, 919–941 (2005; Zbl 1095.13033)]) and discuss some algebraic notions such as the Fourier transform of a GKZ system and results on the holonomy or rank of such module.
The last three sections are devoted to more concrete topics: in section three, the focus in on the irregularity of GKZ systems following [M. Schulze and U. Walther, Duke Math. J. 142, No. 3, 465–509 (2008; Zbl 1144.13012)]. The next one covers the fact that GKZ-systems can be seen, under certain conditions, as (irregular) mixed Hodge modules, and as such, can be endowed with a Hodge filtration, regular or irregular, depending on the nature of the \(\mathcal{D}\)-module itself (cf. [Th. Reichelt and Ch. Sevenheck, Algebr. Geom. 7, No. 3, 263–345 (2020; Zbl 1453.14058)] and [the reviewer et al., Algebra Number Theory 13, No. 6, 1415–1442 (2019; Zbl 1440.14099)]), and a weight filtration in the regular case [Th. Reichelt and U. Walther, “Weight filtrations on GKZ-systems”, Preprint, arXiv:1809.04247]. The survey finishes with the application to Landau-Ginzburg models and mirror symmetry of smooth toric varieties (or subvarieties of them), after [Reichelt and Sevenheck, loc. cit.].

MSC:

14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
33C70 Other hypergeometric functions and integrals in several variables
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
32C38 Sheaves of differential operators and their modules, \(D\)-modules
14J33 Mirror symmetry (algebro-geometric aspects)
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
13N10 Commutative rings of differential operators and their modules
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References:

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