On the better behaved version of the GKZ hypergeometric system.

*(English)*Zbl 1284.33012Generalizations of the classical hypergeometric function as solutions of certain systems of partial differential equations defined by combinatorial data have been introduced in [I. M. Gel’fand et al., Funct. Anal. Appl. 23, No. 2, 94–106 (1989); translation from Funkts. Anal. Prilozh. 23, No. 2, 12–26 (1989; Zbl 0721.33006)]. These functions, denoted by GKZ in short, appear in the study of mirror symmetry of hypersurfaces and complete intersection in toric varieties.

In the paper under review the authors deal with better behaved GKZ which are defined as follows. Let us consider a system of partial differential equations on sequences of functions of \(n\) variables \(\Phi_{c} (x_{1}, \dots, x_{n})\) with \(c\) in the preimage \(K,\) in a finitely generated abelian group \(N,\) of the cone of a convex hull \(\Delta\) of a finite set \(\{v_{k}\}_{k=1}^{n}\) in \(N\bigotimes\mathbb R\) under the natural map \(\pi: N \rightarrow N\bigotimes\mathbb R\). Then

\(\partial_{j} \Phi_{c} = \Phi_{c + v_{j}}, j=1,2, \dots, n, \) for all \(c\in K\)

\(\sum_{j=1}^{n} \mu (v_{j}) x_{k} \partial_{j} \Phi_{c} = \mu (\beta -c) \Phi_{c}\), for all \(\mu\in M\), \(c\in K\).

Here, \(M\) denotes the free abelian group \(\operatorname{Hom}(N, \mathbb Z)\) and \(\beta\in N \bigotimes\mathbb C\) is a fixed parameter.

The space of the solutions is related to the logarithmic Jacobian rings and the authors prove that their dimension is the product of the normalized volume of the polytope \(\Delta\) and the torsion order of the abelian group \(N\). The effects of torsion in \(N\) and repetitions among the elements \(v_{k}\) are studied. The restriction map from the solution space of GKZ for the cone to that for its interior is analyzed. These results are intimately related to the work V. V. Batyrev [Duke Math. J. 69, No. 2, 349–409 (1993; Zbl 0812.14035)], but now the treatment is more algebraic and self-contained.

Partial semigroups version of the better behaved GKZ hypergeometric system can also be considered in this more general framework.

Some open problems are stated in terms of an appropriately defined category of better behaved GKZ systems. In particular, the better behaved GKZ system lends itself to a process of categorification which is expected to provide a non-commutative categorical resolution of a Gorenstein toric singularity. On the other hand, an interesting question is to study an appropriately defined category of better behaved GKZ systems and its functorial properties, part of which would mirror the properties of the category of toric Deligne-Mumford stacks.

In the paper under review the authors deal with better behaved GKZ which are defined as follows. Let us consider a system of partial differential equations on sequences of functions of \(n\) variables \(\Phi_{c} (x_{1}, \dots, x_{n})\) with \(c\) in the preimage \(K,\) in a finitely generated abelian group \(N,\) of the cone of a convex hull \(\Delta\) of a finite set \(\{v_{k}\}_{k=1}^{n}\) in \(N\bigotimes\mathbb R\) under the natural map \(\pi: N \rightarrow N\bigotimes\mathbb R\). Then

\(\partial_{j} \Phi_{c} = \Phi_{c + v_{j}}, j=1,2, \dots, n, \) for all \(c\in K\)

\(\sum_{j=1}^{n} \mu (v_{j}) x_{k} \partial_{j} \Phi_{c} = \mu (\beta -c) \Phi_{c}\), for all \(\mu\in M\), \(c\in K\).

Here, \(M\) denotes the free abelian group \(\operatorname{Hom}(N, \mathbb Z)\) and \(\beta\in N \bigotimes\mathbb C\) is a fixed parameter.

The space of the solutions is related to the logarithmic Jacobian rings and the authors prove that their dimension is the product of the normalized volume of the polytope \(\Delta\) and the torsion order of the abelian group \(N\). The effects of torsion in \(N\) and repetitions among the elements \(v_{k}\) are studied. The restriction map from the solution space of GKZ for the cone to that for its interior is analyzed. These results are intimately related to the work V. V. Batyrev [Duke Math. J. 69, No. 2, 349–409 (1993; Zbl 0812.14035)], but now the treatment is more algebraic and self-contained.

Partial semigroups version of the better behaved GKZ hypergeometric system can also be considered in this more general framework.

Some open problems are stated in terms of an appropriately defined category of better behaved GKZ systems. In particular, the better behaved GKZ system lends itself to a process of categorification which is expected to provide a non-commutative categorical resolution of a Gorenstein toric singularity. On the other hand, an interesting question is to study an appropriately defined category of better behaved GKZ systems and its functorial properties, part of which would mirror the properties of the category of toric Deligne-Mumford stacks.

Reviewer: Francisco Marcellán (Leganes)

##### MSC:

33C70 | Other hypergeometric functions and integrals in several variables |

##### Keywords:

generalized hypergeometric system; finitely generated abelian groups; logarithmic Jacobian ring; mixed Hodge structures; dimension
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\textit{L. A. Borisov} and \textit{R. Paul Horja}, Math. Ann. 357, No. 2, 585--603 (2013; Zbl 1284.33012)

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