Hypergeometric D-modules and
twisted Gauß–Manin systems
The Euler–Koszul complex is the fundamental tool in the homological study of -hypergeometric differential systems and functions. We compare Euler–Koszul homology with D-module direct images from the torus to the base space through orbits in the corresponding toric variety. Our approach generalizes a result by Gel’fand et al. [GKZ90, Thm. 4.6] and yields a simpler, more algebraic proof.
In the process we extend the Euler–Koszul functor to a category of infinite toric modules and describe multigraded localizations of Euler–Koszul homology.
Key words and phrases:toric ring, hypergeometric system, Euler–Koszul homology, D-module, direct image, Gauß-Manin system
1991 Mathematics Subject Classification:13N10,14M25
1.1. Definition of GKZ-systems
Let and denote the free -modules with bases and respectively. Let be an integer -matrix with columns . We consider both as a map with respect to the bases above and as the finite subset of consisting of the images of the . We assume that and that is a positive semigroup which means that is the only unit in . To this type of data, Gel’fand, Graev, Kapranov and Zelevinskiĭ [GGZ87, GKZ89] associated in the 1980’s a class of -modules today called GKZ- or -hypergeometric systems and defined as follows.
Let be the coordinate system on corresponding to , and let be the corresponding partial derivative operators on the sheaf of regular functions on or its ring of global sections . Then the Weyl algebra
is the ring of algebraic differential operators on and is the ring of global sections of the sheaf of algebraic differential operators on . With and , write for where here and elsewhere we freely use multi-index notation. The toric relations of are then
while the Euler vector fields to are
Finally, for , the -hypergeometric system is the -module
The structure of the solutions to the (always holonomic) modules is tightly interwoven with the combinatorics of the pair , and -hypergeometric structures are nearly ubiquitous. Indeed, research of the past two decades revealed that toric residues, generating functions for intersection numbers on moduli spaces, and special functions (Gauß, Bessel, Airy, etc.) may all be viewed as solutions to GKZ-systems. In other directions, varying Hodge structures on families of Calabi–Yau toric hypersurfaces as well as the space of roots of univariate polynomials with undetermined coefficients have -hypergeometric structure.
1.2. Torus action
Consider the algebraic -torus with coordinate functions corresponding to . One can view the columns of , as characters on , and the parameter vector as a character on its Lie algebra via . A natural tool for investigating is the torus action of on the cotangent space of at given by
The coordinate ring of is which contains the toric ideal generated by the toric relations . For , is the ideal of the closure of the orbit of whose coordinate ring is the toric ring
The contragredient action of on given by
for , defines a -grading on and on the coordinate ring of by
Note that for -homogeneous the commutator equals where is the -th component of . As , (1.2) also defines a -grading on the sheaves of differential operators and under which and are homogeneous.
Note that and naturally define an algebraic -module
-isomorphic to but equipped with a twisted -module structure expressed symbolically as
on which acts via the product rule.
1.3. Questions, results, techniques
In our algebraic setting, the ring of global sections of is identified with via the Fourier transform. Under this correspondence a natural question is the following:
Study the relationship between (the Fourier transform of) and the direct image of under the orbit map
An important result in this direction was given in [GKZ90, Thm. 4.6]: for non-resonant the two modules are isomorphic. Here, a parameter is non-resonant if it is not contained in the locally finite subspace arrangement of resonant parameters
the union being taken over all linear subspaces that form a boundary component of the rational polyhedral cone .
A powerful way of studying is to consider it as a -th homology of a Koszul type complex of on . The idea of such Euler–Koszul complex is already visible in [GKZ89] and was significantly enhanced in [MMW05]. Results from [MMW05] show that is a resolution of if and only if is not in the -exceptional locus , a well-understood (finite) subspace arrangement of .
In [Ado94, §4] a variation of this complex can be found, whose is reminiscent of that of the Gauß–Manin system of the map
by factorization through its graph, see for example [Pha79]. This suggests that might be suitable for representing the direct image , an observation (inspired by a talk by Adolphson) that became the catalyst for this article.
The main results in this article, contained in Section 3.2, make the relationship between , and precise. We determine in Corollary 3.7 the exact set of parameters for which the first and last of these agree. In view of [MMW05], this provides a considerable sharpening of [GKZ90, Thm. 4.6] expressing GKZ-systems in terms of twisted Gauß–Manin systems, stated in Corollary 3.7. In [GKZ90, Thm. 2.11], this latter result is used in the homogeneous case to show that the generic monodromy representation on the (solution) space of -hypergeometric functions is irreducible for non-resonant .
The parameters identified in Corollary 3.7 are precisely those for which left-multiplication by induces a quasi-isomorphism on for each . We show in Corollary 3.9 that, given , has this property for . By Remark 3.6, left-multiplication by is a quasi-isomorphism on if and only if the contiguity operator is a quasi-isomorphism for all (which holds in general for ). In this way, arises as the direct limit of the Euler–Koszul complexes induced by the contiguity operators.
realized by a suitable (product of) contiguity operator(s). Example 4.1 suggests that there might be a filtration on the cone of (1.4) whose graded pieces consist of direct images of under -orbit maps to border tori of , that is, tori forming . In essence this would ask the following.
Is there a relation between local cohomology of the Euler–Koszul complex with -invariant support and direct images of -modules under -orbit maps to border tori of ?
In the special case where is Cohen–Macaulay and hence is a resolution of , the local cohomology of hypergeometric systems was studied by Okuyama [Oku06a]. His main result [Oku06a, Thm. 3.12] shows indeed some similarity with Theorem LABEL:32 in Section 4 where we explicitly describe the direct images in Problem 1.2. Problem 1.2 also motivates to study general -graded localizations of Euler–Koszul complexes. This is the subject of Section LABEL:39, where we generalize study the Euler–Koszul functor on direct limits of toric modules generalizing ideas from [MMW05] and [Oku06a].
2. Direct image via torus action
In this section we determine the direct image (complex) of under . The orbit map identifies the coordinate ring of with the coordinate ring of where . The inclusion of the closure corresponds to the canonical projection .
Put where is as in (1.1). As is -graded, each , and hence as well, acts by right multiplication on representatives of classes in . The total complex induced by these operators is the right Koszul complex induced by .
The Fourier transform of the direct image is represented by the right Koszul complex of on which is acyclic except in degree .
We use the abbreviation . Then the right Koszul complex of on is a free -resolution of . Since is -affine it suffices to check this on global sections. But grading the global section complex with respect to the order filtration yields the Koszul complex of on the Laurent polynomial ring in the variables and which is clearly exact.
In order to compute the (Fourier transformed) direct image of this complex we factorize into the closed embedding
and the open embedding
The direct image is represented by the right Koszul complex of on where is the transfer -bimodule [Bor87, VI.5.1]. Since is a closed embedding one can identify and as left -modules [Bor87, VI.7.3] and we have to verify that the right-action of on this module translates into that of under this identification. The transpose of is the -bimodule
whose left structure is given by the chain rule of differentiation [Bor87, VI.4.1]. The transpose of acts from the left on by
where is considered as partial derivative with respect to via the Fourier transform. Under the identification
this becomes whose transpose is . Therefore the right action of on coincides with that of . As is affine, the direct image functor is exact [Bor87, Prop. VI.8.1]. Thus the direct image is represented by the right Koszul complex of on which is acyclic except in degree .
The direct image functor for the open embedding is the exact functor [Bor87, VI.5.2]. As is affine and since , is represented by the acyclic right Koszul complex of on as claimed. ∎
We shall see an alternative proof for the acyclicity statement in Proposition 2.1 in Remark LABEL:37.(LABEL:37a).
3. Euler–Koszul homology on localizations
In [MMW05] a generalization of the right Koszul complex from Proposition 2.1 is developed as follows. Interpret right multiplication of on as the effect of the left -linear endomorphism that sends a -homogeneous to
and extending -linearly. The advantage of this point of view is that the definition extends verbatim to any left -module with -grading (1.2).
For a -graded -module , the Euler–Koszul complex of with parameter is the Koszul complex of these (obviously commuting) endomorphisms on . The homology of this complex in the category of left -modules is the Euler–Koszul homology, a generalization of the -hypergeometric system . In this section we determine when is a representative for . By Proposition 2.1, this amounts to describing when and are quasi-isomorphic.
Throughout, we identify a submatrix of columns of with the corresponding set of column indices. Denoting and , is -graded if so is . With this setup, the right action of on in Proposition 2.1 coincides with the left action (3.1). For the present section, (3.1) is more convenient.
By the left and right Ore property of , the left and right localizations of coincide. In particular there is for any -module an isomorphism
of -modules where acts on the left module by . This induces an isomorphism and, by exactness of localization, a corresponding isomorphism in homology.
The complex represents if and only if left-multiplication by is invertible on for .∎
Gel’fand et al. [GKZ90, Thm. 4.6] show that if is homogeneous and non-resonant for then the -hypergeometric system represents . The non-resonance condition means that, for each proper face of the cone ,
By [MMW05, Prop. 5.3], non-resonance implies that the Euler-Koszul complex is a resolution. Homogeneity of is equivalent to being in the row span of .
We shall describe the set of parameters for which represents without any homogeneity assumption. At the same time we weaken the non-resonance condition. To do so we give a description of the parameter set in for which left-multiplication by is invertible on .
Left-multiplication by is injective on for .
Consider the weight vector where and . Since is -homogeneous, it suffices to check the statement after grading with respect to . But . Thus, is a domain and multiplication by injective. ∎
In [MMW05, Def. 5.2], the following notion was introduced.
For a finitely generated -graded -module , the set of quasi-degrees is the Zariski closure of the set of all for which .
We can now formulate an important class of parameters.
we let be the strongly resonant parameters of .
For the following conditions are equivalent:
Left-multiplication by is a quasi-isomorphism on .
Without loss of generality we may assume . By nature of the Euler–Koszul complex and Lemma 3.2, left-multiplication by defines a chain map
We prove that its cokernel , and thus by Lemma 3.2 its cone, is exact precisely if is not strongly resonant for .
Let be obtained from by deleting the last column , set and note that is a toric -module. Each is a direct sum of copies of the -module
Note that the element lies in degree as is -homogeneous of degree zero. In order to compute the induced differential in , pick -homogeneous and . Then for one computes
It follows that decomposes as the sum of -complexes
By [MMW05, Prop. 5.3], exactness of the -th summand is equivalent to and the equivalence follows. ∎
Consider the cokernel of the (injective) chain map
induced by acting by right-multiplication on . The modules of , considered as -modules, have a direct sum decomposition equal to those of the complex that appears in the proof of Theorem 3.5 as cokernel of left-multiplication by on . However, the differentials in and are not the same: the differential in is -linear, while that of is only -linear. It follows from [MMW05] that (3.4) is a quasi-isomorphism if and only if .
The following corollary is the promised sharpening of [GKZ90, Thm. 4.6]; it determines when the hypergeometric module is isomorphic to .
The following are equivalent:
is naturally isomorphic to .
If (2) holds then left-multiplication by is a quasi-isomorphism on by Corollary 3.1. Since higher Euler–Koszul homology is -torsion it must vanish in this case. It follows that when (2) holds then is quasi-isomorphic to and hence to . Thus, (2) implies (3).
Conversely, if (3) holds, then is a resolution of . Since left-multiplication by is always a quasi-isomorphism on by Proposition 2.1, is invertible on . For the cokernel of (3.3) we have then . Since decomposes into a sum of Euler–Koszul complexes over , vanishing of is equivalent to vanishing of by [MMW05, Prop. 5.3]. It follows that (3.3) is a quasi-isomorphism and by Theorem 3.5 we conclude that (1) holds. ∎
It is natural to ask whether there are any parameters that satisfy the hypothesis of Corollary 3.7. To answer this question, we denote
for any . Note that multiplication by the invertible function on defines an isomorphism of -modules shifting the degree by .
For fixed and ,
Obviously, elements satisfy the condition of Corollary 3.7 and, for any , we have for . ∎
For the contiguity operator induces a quasi-isomorphism between the complexes and by Remark 3.6. Thus,
for . Hence, for , .
For fixed and ,
4. Direct images through border tori
For any and there is a natural localization map
which for most is an isomorphism according to Corollary 3.7. One wonders what the cone of this map is when it is not an isomorphism. Its Fourier transform must be supported in since both complexes in question are supported in and agree on .
Consider the case with . According to our results above, is in the set of strongly resonant parameters sketched below and for . A calculation with Macaulay2 [M2] shows that the cone over the localization map has homology .
Consider the projection ; corresponding direct images are computed by the left Koszul complex induced by . Combine this projection with the embedding sending to induced by the face of . The direct image of under the composition has the same cohomology as . Note that also “causes” to be in .
Example 4.1 suggests the investigation of direct images of under orbit maps factoring through tori in , as well as partial -graded localizations of Euler–Koszul homology of . In the final two sections we follow this line of thought: we determine the structure of these direct images in Theorem LABEL:32, and we give a result that mirrors Corollary 3.9 in Corollary LABEL:12.
By abuse of language, we call a submatrix of columns of a face of if is a face of the cone . The toric variety is the closure of the orbit through . The complement is a union of other orbits,
the union being taken over the faces of . Here, is the orbit of where for and for . As is well-known, the closure of corresponds to the -graded prime of . Note that for no .
Denote by the -equivariant map induced by sending to . Denote , and . Then has a natural factorization as follows.