$A$-hypergeometric systems that come from geometry

We establish some connections between nonresonant $A$-hypergeometric systems and de Rham-type complexes. This allows us to determine which of these $A$-hypergeometric systems"come from geometry."


Introduction
Let A = {a (1) , . . . , a (N ) } ⊆ Z n with a (j) = (a (j) 1 , . . . , a (j) n ). We shall assume throughout this paper that these lattice points generate Z n as abelian group. Let L be the corresponding lattice of relations, L = (l 1 , . . . , l N ) ∈ Z N | N j=1 l j a (j) = 0 , and let α = (α 1 , . . . , α n ) ∈ C n . The A-hypergeometric system is the system of partial differential equations in the variables λ 1 , . . . , λ N consisting of the operators (we write ∂ j for ∂/∂λ j ) for all l ∈ L and the operators for i = 1, . . . , n. We denote by D = C λ 1 , . . . , λ N , ∂ 1 , . . . , ∂ N the ring of differential operators in the λ j . The associated hypergeometric D-module is Let C[λ] = C[λ 1 , . . . , λ N ] be the polynomial ring in N variables and let X be a smooth variety over C [λ]. Let G be a finite group acting on X/C [λ]. Then G acts on the relative de Rham cohomology groups H i DR (X/C[λ]). For an irreducible representation χ of G, let H i DR (X/C[λ]) χ denote the χ-isotypic component of H i DR (X/C[λ]), i.e., H i DR (X/C[λ]) χ is the sum of all G-submodules of H i DR (X/C[λ]) that are isomorphic to χ. The H i DR (X/C[λ]) χ are D-modules via the Gauss-Manin connection. We say that a D-module M comes from geometry if it is isomorphic to H i DR (X/C[λ]) χ for some (X/C[λ], G, χ). The main idea of this paper is to show that the M α are isomorphic as Dmodules to certain cohomology groups that arise in algebraic geometry.
, the coordinate ring of the n-torus T n over C[λ]. Put In terms of the above basis, we have for ξ ∈ R ′ Formally, so for any derivation ∂ ∈ Der C (C[λ]) the operator commutes with ∇ α (where f ∂ denotes the polynomial obtained from f by applying ∂ to its coefficients). This defines an action of Der C (C[λ]) on , ∇ α ). In particular, ∂ j acts as D j = ∂ j + x a (j) . This action extends to an action of D on (Ω , ∇ α ) into D-modules. Let C(A) ⊆ R n be the real cone generated by A and let ℓ 1 , . . . , ℓ s ∈ Z[u 1 , . . . , u n ] be homogeneous linear forms defining the codimension-one faces of C(A), normalized so that the coefficients of each ℓ i are relatively prime and so that ℓ i ≥ 0 on C(A) for each i. We say that α is nonresonant for A if ℓ i (α) ∈ Z for all i. Note that α is nonresonant for A if and only if α + u is nonresonant for A for all u ∈ Z n . (If α ∈ R n , this is equivalent to saying that no proper face of C(A) contains a point of α + Z n .) Remark. It is straightforward to check that for u ∈ Z n multiplication by x u defines an isomorphism of complexes of D-modules (its inverse is multiplication by x −u ). Thus when α is nonresonant the M α+u for u ∈ Z n are all isomorphic as D-modules.
Consider the special case with coordinates x 1 , . . . , x n−1 . Let U ⊆ T n−1 be the open set where g is nonvanishing and let Ω k U/C[λ] be the module of relative k-forms over the ring of regular functions on U . The map ∇ α : we get an action of Der C (C[λ]) on this complex. The action extends to an action of D, making (Ω • U/C[λ] , ∇ α ) into a complex of D-modules. Note that for u ∈ Z n , multiplication by x u1 1 · · · x un−1 n−1 /g un defines an isomorphism of complexes of D-modules x u1 1 · · · x un−1 n−1 g un , ∇ α ) come from geometry. (We sketch the proof of this fact in Section 4.) From Theorems 1.4 and 1.5, we then get the following result.
. If in addition α ∈ Q n , then M α comes from geometry.
The proofs of Theorems 1.4 and 1.5 are based on ideas from [1] and [3]. Those papers in turn are related to earlier work of Dwork, Dwork-Loeser, and N. Katz. We refer the reader to the introductions of [1] and [3] for more details on the connections with that earlier work.

Proof of Theorem 1.4
It is straightforward to check that the Z i,α commute with one another and that . It follows that right multiplication by Z i,α maps the left ideal l∈L D l into itself. If we put P = D/ l∈L D l , then right multiplication by the Z i,α is a family of commuting endomorphisms of P as left D-module. Let C • be the cohomological Koszul complex on P defined by the Z i,α . Concretely, where the e i are formal symbols satisfying e i ∧ e j = −e j ∧ e i and the boundary operator δ α : C k → C k+1 is defined by additivity and the formula (for σ ∈ P) One obtains a complex of left D-modules (C • , δ α ) for which .
is an isomorphism of D-modules by [1,Theorem 4.4]. It extends to a map φ : C k → Ω k R log by additivity and the formula By [2, Corollary 2.4], this is an isomorphism of complexes of D-modules: . By (2.1) and (2.2), Theorem 1.4 is a consequence of the following result.
We state and prove a generalization of Proposition 2.3. Let U ⊆ Z n be a nonempty subset satisfying the condition: be the subset 3 is the special case U = U 0 of the following more general result.
For ( is a quasi-isomorphism of complexes of D-modules. Since W (v, T ) ⊆ Z n for all v, T , Lemma 2.6 implies that the inclusion , ∇ α ) is a quasi-isomorphism for all v, T . The quasi-isomorphisms (2.7) and (2.8) imply Proposition 2.5.
Proof of Lemma 2.6. For T ′ ⊆ T , let is a quasi-isomorphism. Let Q • be the quotient complex We show that (2.9) is a quasi-isomorphism by showing that H k (Q • ) = 0 for all k.
We define a filtration {F p } p≤vt on the complex Ω log be the C[λ]-submodule spanned by differential forms By (1.2) and (1.3), ∇ α respects this filtration. We also denote by F p the induced filtrations on the subcomplex Ω • M U (v,T ′′ ) log and the quotient complex Q • . Note that since To show that H k (Q • ) = 0 for all k, it suffices to show that H k (gr p Q • ) = 0 for all k and p, where gr p Q • denotes the p-th graded piece of the associated graded of the filtration F p . Write It is straighforward to check that and a calculation using (1.1) and (1.3) shows that Suppose that the form (2.10) lies in F p \ F p+1 . Then ℓ t (α + u) = ℓ t (α) + p and if ℓ t (a (j) ) = 0, then ℓ t (u + a (j) ) > p. It follows that the second term on the right-hand side of (2.11) lies in F p+1 , so on the associated graded complex the induced map is just multiplication by ℓ t (α) + p. Since α is nonresonant for A, ℓ t (α) + p = 0. Thus multiplication by a nonzero constant is homotopic to the zero map, which implies that H k (gr p Q • ) = 0 for all k.

Proof of Theorem 1.5
In this section we assume that f = x n g(x 1 , . . . , x n−1 ). By the Remark following Theorem 1.4, we may assume that if α n ∈ Z, then α n ≥ 1.
, ∇ α ) is a quasi-isomorphism of complexes of D-modules. Theorem 1.5 is then a consequence of the following result. Proof. We regard Ω • R+ log as the total complex associated to a certain two-row double complex: let where d ′ is exterior differentiation relative to the variables x 1 , . . . , x n−1 , and let ∂ v : Ω k,0 → Ω k,1 be the map where d ′′ is exterior differentiation relative to the variable x n . Since ∂ v is injective, the canonical projection Ω k R+ log → Ω k−1,1 induces a quasi-isomorphism To complete the proof of Proposition 3.1, we define a quasi-isomorphism of complexes between (Ω •,1 /∂ v Ω •,0 , ∂ h ) and (Ω • U/C[λ] , ∇ α ). Define γ : Ω k,1 → Ω k U/C[λ] to be the C[λ]-module homomorphism satisfying where (α n ) un = α n (α n + 1) · · · (α n + u n − 1). It is straightforward to check that γ commutes with boundary operators, hence defines a homomorphism of complexes from Ω •,1 to Ω • U/C [λ] . The hypothesis that α n ∈ Z ≤0 implies that γ is surjective, and it is straightforward to check that γ(∂ v Ω k,0 ) = 0. It remains only to show that ker(γ : Ω k, where ξ i ∈ Ω k,0 is in the C[λ]-span of the forms We prove by induction on M − m that ξ ∈ ∂ v Ω k,0 . We have Since (α n ) M = 0, this equation implies ξ M = gη, where η is in the C[λ]-span of the forms (3.3). It follows that By induction we are reduced to the case ξ = x m i ξ m dxn xn . But in this case 0 = γ(ξ) = (−1) m (α n ) m ξ m /g m , so ξ m = 0.
If α n ∈ Z, there is no need to introduce the complex Ω • R+ log .
The proof then proceeds unchanged. (See [3, Lemma 2.5] for the details in a similar situation.)

Application of N. Katz's results
We begin by sketching the proof that the H i (Ω • U/C[λ] , ∇ α ) come from geometry.
is a free module with basis {dx i } n−1 i=1 , D acts on global i-forms by acting on their coefficients relative to exterior powers of this basis. The group (µ D ) n acts on X and its relative de Rham complex (Ω • X/C[λ] , d). The irreducible representations χ of (µ D ) n can be indexed by n-tuples (a 1 , . . . , a n ), 0 ≤ a i < D, so that if χ corresponds to (a 1 , . . . , a n ), then there is an isomorphism of complexes of , ∇ α ), where α = (a 1 /D, . . . , a n /D).) The first assertion of Corollary 1.6 then implies that ) χ , which establishes the second assertion of Corollary 1.6. Now suppose that C = Spec(A) is a smooth connected curve over C and φ : ) is a morphism. Let X ′ be the pullback of X to a variety over C, i.e., X ′ = A ⊗ C[λ] X. Then X ′ is the hypersurface in T n A defined by the equation is the Laurent polynomial obtained from g by applying φ to its coefficients (by abuse of notation, we also denote by φ the homomorphism C[λ] → A corresponding to φ : C → A N ). The varieties X and X ′ are smooth affine schemes whose de Rham cohomology can be computed as the cohomology of the complex of global sections of the de Rham complex. By the right-exactness of tensor products, one has It follows from [9, Section 14] that H n−1 DR (X ′ /A) has regular singular points and quasi-unipotent local monodromy at infinity (i.e., at all points of the quotient field of A). Therefore H n−1 DR (X/C[λ]) has regular singular points and quasi-unipotent local monodromy at infinity (in the sense of [11, Section VIII]). Equation (4.1) then implies that M α has regular singular points and quasi-unipotent local monodromy at infinity.
To apply the results of [10], we observe that the results of this paper are valid when one replaces C[λ] by C(λ). LetD denote the ring of differential operators with coefficients in C(λ) and definē
Explicitly, lettingX ⊆ T n C(λ) be the hypersurface x D n g(x D 1 , . . . , x D n−1 ) − 1 = 0, we have (corresponding to Equation (4.1)) (4.6)M α ∼ = H n−1 DR (X/C(λ)) χ . By [10, Theorem 5.7] H n−1 DR (X/C(λ)) χ has a full set of polynomial solutions modulo p for almost all primes p if and only if it has a full set of algebraic solutions. Note that the solution sets of M α andM α in the algebraic closure of C(λ) are identical. From Equation (4.6), we then get the following result.