Characteristic variety of the Gauss-Manin differential equations of a generic parallelly translated arrangement

We consider a weighted family of $n$ generic parallelly translated hyperplanes in $\C^k$ 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 Plucker coordinates of the associated point in the Grassmannian Gr(k,n). The Laurent polynomials are in involution.

• KZ equations, κ ∂I ∂z i (z) = K i (z)I(z), z = (z 1 , . . . , z n ), i = 1, . . . , n. Here κ is a parameter, I(z) a V -valued function, where V is a vector space from representation theory, K i (z) : V → V are linear operators, depending on z. The connection is flat for all κ, see for example [EFK,V2].
• Quantum differential equations, κ ∂I ∂z i (z) = p i * z I(z), z = (z 1 , . . . , z n ), i = 1, . . . , n. Here p 1 , . . . , p n are generators of some commutative algebra H with quantum multiplication * z depending on z. The connection is flat for all κ. These equations are part of the Frobenius structure on the quantum cohomology of a variety, see [D, M].
If κ ∂I ∂z i (z) = K i (z)I(z), i = 1, . . . , n, is a system of V -valued differential equations of one of these types, then its characteristic variety is It is known that the characteristic varieties of the first two types of differential equation are interesting. For example, the characteristic variety of the quantum differential equation of the flag variety is the zero set of the Hamiltonians of the classical Toda lattice, according to [G, GK], and the characteristic variety of the gl N KZ equations with values in the tensor power of the vector representation is the zero set of the Hamiltonians of the classical Calogero-Moser system, according to [MTV].
In this paper we describe the characteristic variety of the Gauss-Manin differential equations for hypergeometric integrals associated with a weighted family of n generic parallelly translated hyperplanes in C k . The characteristic variety is given as the zero set of Laurent polynomials, whose coefficients are determined by weights and the Plücker coordinates of the associated point in the Grassmannian Gr(k, n). The Laurent polynomials are in involution.
It is known that the KZ differential equations can be identified with Gauss-Manin differential equations of certain weighted families of parallelly translated hyperplanes, see [SV], as well as some quantum differential equations can be identified with Gauss-Manin differential equations of certain weighted families of parallelly translated hyperplanes, see [TV]. Therefore, the results in this paper on the characteristic variety of the Gauss-Manin differential equations associated with a family of generic parallelly translated hyperplanes can be considered as a first step to studying characteristic varieties of more general KZ and quantum differential equations, which admit integral hypergeometric representations.
The Laurent polynomials, defining our characteristic variety, are regular functions of the Plücker coordinates of the associated point in Gr(k, n). Therefore they can be used to study the characteristic varieties of more general Gauss-Manin differential equations for multidimensional hypergeometric integrals.
Our description of the characteristic variety is based on the fact, proved in [V4], that the characteristic variety of the Gauss-Manin differential equations is generated by the master function of the corresponding hypergeometric integrals, that is, the characteristic variety coincides with the Lagrangian variety of the master function. That fact is a generalization of Theorem 5.5 in [MTV], proved with the help of the Bethe ansatz, that the local algebra of a critical point of the master function associated with a gl N KZ equation can be identified with a suitable local Bethe algebra of the corresponding gl N module.
In Section 2, we consider the algebra of functions on the critical set of the master function and describe it by generators and relations.
In Section 3, we show that these relations give us equations defining the Lagrangian variety of the master function. We show that the corresponding functions are in involution. We define coordinate systems (z I , pĪ) on the Lagrange variety and for each of them a function Φ(z I , pĪ) also generating the Lagrangian variety. We describe the Hessian of the master function lifted to the Lagrangian variety and relate it to the Jacobian of the projection of the Langrangian variety to the base of the family.
In Section 4, we remind the identification from [V4] of the Lagrangian variety of the master function and the characteristic variety of the Gauss-Manin differential equations.
2. Algebra of functions on the critical set 2.1. An arrangement in C n × C k . Let n > k be positive integers. Denote J = {1, . . . , n}. Consider C k with coordinates t 1 , . . . , t k , C n with coordinates z 1 , . . . , z n . Fix n linear functions . We assume that all the numbers d i 1 ,...,i k are nonzero if i 1 , . . . , i k are distinct. In other words, we assume that the collection of functions g j , j ∈ J, is generic. We define n linear functions on C n × C k , f j = z j + g j , j ∈ J. We define the arrangement of hyperplanesC = {H j | j ∈ J} in C n × C k , whereH j is the zero set of f j . Denote by U(C) = C n × C k − ∪ j∈JHj the complement.
For every z = (z 1 , . . . , z n ) ∈ C n , the arrangementC induces an arrangement C(z) in the fiber over z of the projection π : C n × C k → C n . We identify every fiber with C k . Then C(z) consists of hyperplanes H j (z), j ∈ J, defined in C k by the equations f j = 0. Denote by U(C(z)) = C k − ∪ j∈J H j (z) the complement.
The arrangement C(z) is with normal crossings if and only if z ∈ C n − ∆, where H i 1 ,...,i k+1 is the hyperplane in C n defined by the equation f i 1 ,...,i k+1 (z) = 0, We have the following identify Proof. The dimension of S equals the codimension in C n of X 1 = {z ∈ C n | f I (z) = 0 for all I}. The subspace X 1 is the image of the subspace X 2 = {(z, t) ∈ C n × C k | f j (z, t) = 0 for all j ∈ J} under the projection π : C n × C k → C n . Clearly the subspace X 2 is kdimensional and the projection π| X 2 : X 2 → X 1 is an isomorphism. Hence dim X 1 = k and dim S = n − k.
2.2. Plücker coordinates. The matrix (b m j ) is an n × m-matrix of rank k. The matrix defines a point in the Grassmannian Gr(k, n) of k-planes in C n . The numbers d i 1 ,...,i k are Plücker coordinates of this point. Most of objects in this paper is determined in terms of these Plücker coordinates. We will use the following Plücker relation.

Algebra
The critical point equations are Denote by I(z) ⊂ O(U(C(z))) the ideal generated by the functions ∂Φ/∂t j , j ∈ J. The algebra of functions on the critical set is For a function g ∈ O(U(C(z))), denote by [g] its projection to A Φ (z). Denote We introduce the following polynomials in z 1 , . . . , z n , p 1 , . . . , p n . For every subset I = {i 1 , . . . , i k−1 } of distinct elements in J, we set For every subset I = {i 1 , . . . , i k+1 } of distinct elements in J, we set F I (z 1 , . . . , z n , p 1 , . . . , p n ) = (2.10) The following lemma collects properties of the elements p 1 , . . . , p n .
Relation (2.11) will be called the I-relation of first kind.
(iii) For every subset I = {i 1 , . . . , i k+1 } of distinct elements in J, we have Relation (2.12) will be called the I-relation of second kind.
Note that the polynomials F I in (2.11) and (2.12) are homogeneous if we put Then the projection to A Φ (z) of the left hand side of equation (2.3) can be written as and the functions p j are nonzero at every point of the critical set of the master function.
Example. If k = 1 and f j = t 1 + z j , then the ideal I(z) is generated by the function j∈J a j /(t 1 + z j ), while the idealĨ(z) is generated by the functions p 1 + · · · + p n , or by the functions 2.6. Proof of Theorem 2.4.
Lemma 2.5. Let I = {i 1 , . . . , i k } be a subset of distinct elements. Then inÃ(z), we have Proof. The statement easily follows from (2.11), that is, from relations of first kind. For example, if k = 2 and I = {1, 2}, then the two relations of first kind p 1 = 1 Lemma 2.6. InÃ(z), we have 1 = 1 |a| j∈J z j p j . Proof. We have where the first equality follows from Lemma 2.5, the second equality follows from the relations of second kind, the third equality follows from the relations of first kind. Denote by C(z) ⊂ (C × ) n the zero set of the idealĨ(z). Then the function p 1 . . . p k is nonvanishing on C(z).
The previous calculation shows that the multiplication of the invertible function p 1 . . . p k by 1 |a| j∈J z j p j does not change the invertible function. This gives the lemma.
Lemma 2.7. Let s k be a natural number and M = j∈J p s j j , j∈J s j = s, a monomial of degree s. Let J k−s+1 = {j 1 , . . . , j k−s+1 } be any subset in J with distinct elements. Then by using the relations of first kind only, the monomial M can be represented as a C-linear combination of monomials p i 1 . . . p is with 1 i 1 < · · · < i s n and {i 1 , . . . , i s } ∩ J k−s+1 = ∅.
Lemma 2.8. Let s k be a natural number and M = j∈J p s j j a monomial of degree s. Fix and element j 1 ∈ J. Then by using the relations of first kind and the relation 1 = 1 |a| j∈J z j p j only, the monomial M can be represented as a linear combination of monomials p i 1 . . . p i k with 1 i 1 < · · · < i k n and j 1 / ∈ {i 1 , . . . , i s }, where the coefficients of the linear combination are homogeneous polynomials in z of degree s − k.
Recall the deg z j = −1 for all j ∈ J.
Lemma 2.9. Let s > k be a natural number and M = j∈J p s j j a monomial of degree s. Then by using the relations of first kind and second kinds, the monomial M can be represented as a linear combination of monomials p i 1 . . . p i k of degree k, where the coefficients of the linear combination are rational functions in z, regular on C n − ∆ and homogeneous of degree s − k.
Let us finish the proof of Theorem 2.4. Let P (p 1 , . . . , p n ) be a polynomial. Fix j 1 ∈ J. By using the relations of first and second kinds only, the polynomial can be represented as a linear combinationP of monomials p i 1 . . . p i k with 1 i 1 < · · · < i k n and j 1 / ∈ {i 1 , . . . , i s }, see Lemmas 2.7 -2.9. Assume that P (p 1 , . . . , p n ) projects to zero in A Φ (z), then all coefficients of that linear combinationP must be zero, see part (v) of Lemma 2.3. This means that P lies in the idealĨ(z). Theorem 2.4 is proved.

Lagrangian variety of the master function
3.1. Critical set. Recall the projection π : C n × C k → C n . For any z ∈ C n − ∆, the arrangement C(z) in π −1 (z) has normal crossings. Recall the complement U(C) ⊂ C n × C k to the arrangementÃ in C n × C k . Denote Consider the master function Φ(z, t), defined in (2.5), as a function on U 0 . Denote by C Φ the critical set of Φ with respect to variables t, Lemma 3.1. The set C Φ is a smooth n-dimensional subvariety of U 0 .
Theorem 3.2. The natural homomorphismÃ → A Φ , p j → [a j /f j ], is an isomorphism.
The proof is the same as the proof of Theorem 2.4.

Lagrangian variety.
Consider the cotangent bundle T * (C n −∆) with dual coordinates z 1 , . . . , z n , p 1 , . . . , p n with respect to the standard symplectic form ω = n j=1 dp j ∧ dz j . Consider the open subset (C n − ∆) × (C × ) n ⊂ T * (C n − ∆) of all points with nonzero coordinates p 1 , . . . , p n . Consider the map Denote by Λ the image ϕ(C Φ ) of the critical set. The set Λ is invariant with respect to the action of C × which multiplies all coordinates p j and divides all coordinates z j by the same number. Denote byÎ ⊂ O((C n − ∆) × (C × ) n ) the ideal of functions that equal zero on Λ.
Proof. It is clear thatĨ ⊂Î. The proof of the inclusionÎ ⊂Ĩ is basically the same as the proof of Theorem 2.4. This gives the first statement of the theorem. It is clear that dim Λ = n. To prove that Λ is smooth, it is enough to show that at any point of Λ, the span of the differentials of the functions F I (p), |I| = k − 1, and G I (z, p), |I| = k + 1 is at least n-dimensional. By Lemma 2.1, the span of the z-parts of the differentials of the functions G I (z, p), I = |I| = k + 1, is n − k-dimensional. It is easy to see that the span of the differentials of the functions F I (p), I = |I| = k + 1, is at least k-dimensional, c.f. the example in the proof of Lemma 2.5. Hence Λ is smooth.
By the definition of ϕ, the set Λ is isotropic. Hence Λ is Lagrangian.
3.3. Generating functions. Consider the function Ψ = j∈J a j ln p j − i∈I z i p i (3.6) of n + k variables z j , j ∈ I, p j , j ∈ J. Express in Ψ the variables p i , i ∈ I, according to (3.5). Denote by Ψ(z I , pĪ) the resulting function of variables z I , pĪ.
Theorem 3.5. The function Ψ(z I , pĪ) is a generating function of the Lagrangian variety Λ. Namely, Λ lies the image of the map Proof. The proof that these formulas give (3.5) is by straightforward verification.

Integrals in involution.
Consider the standard Poisson bracket on T * (C n ), Recall the function G j (z j , p j ) in (2.15). It is clear that {G j , G j ′ } = 0 for all j, j ′ ∈ J. Now {G I , G I ′ } = 0 for all I, I ′ with |I| = |I ′ | = k + 1, since G I , G I ′ are linear combination of G j with constant coefficients.
Remark. An interesting property of the Hamiltonians F I , G I is that they are regular with respect the Plücker coordinates d i 1 ,...,i k . Hence, they can be used to study the Langrange varieties of the arrangements in C n × C k associated with not necessarily generic matrices (b i j ). 3.5. Hessian as a function on the Lagrange variety. Let z ∈ C n − ∆ and t 0 a critical point of the master function Φ(z, · ). An important characteristic of the critical point is the Hessian see, for example, [AGV,MV,V2,V3]. For a subset I = {i 1 , . . . , i k } ⊂ J, we denote by d 2 I the number (d i 1 ,...,i k ) 2 . Lemma 3.7. We have Proof. In [V3], the formula Hess Φ = (−1) k 1 i 1 <···<i k n d 2 i 1 ,...,i k k m=1 a im /f 2 im is given, which is the right hand side of (3.8). The formula itself is obvious.
3.6. Hessian and Jacobian. Let M = {m 1 , . . . , m k } ⊂ J be a k-element subset and z M , pM the corresponding ordered coordinate system on Λ. The functions z 1 , . . . , z n form an ordered coordinate system on C n − ∆. Consider the projection Λ → C n − ∆, (z, p) → z, and the Jacobian Jac I (z M , pM ) of the projection with respect to the chosen coordinate systems.  (3.10) 4. Characteristic variety of the Gauss-Manin differential equations 4.1. Space Sing V . Consider the complex vector space V generated by vectors v i 1 ,...,i k with i 1 , . . . , i k ∈ J subject to the relations v i σ(1) ,...,i σ(k) = (−1) σ v i 1 ,...,i k for any i 1 , . . . , i k ∈ J and σ ∈ S k . The vectors v i 1 ,...,i k with 1 i 1 < · · · < i k n form a basis of V . If v = 1 i 1 <···<i k n c i 1 ,...,i k v i 1 ,...,i k is a vector of V , we introduce the numbers c i 1 ,...,i k for all i 1 , . . . , i k ∈ J by the rule: c i σ(1) ,...,i σ(k) = (−1) σ c i 1 ,...,i k . We introduce the subspace Sing V ⊂ V of singular vectors by the formula The symmetric bilinear contravariant form on V is defined by the formulas: . . , i k are distinct. Denote by s ⊥ : V → Sing V the orthogonal projection with respect to the contravariant form.
4.2. Differential equations. Consider the master function Φ(z, t) as a function on U 0 ⊂ C n × C k . Let κ be a nonzero complex number. The function e Φ(z,t)/κ defines a rank one local system L κ on U 0 whose horizontal sections over open subsets ofŨ are univalued branches of e Φ(z,t)/κ multiplied by complex numbers. The vector bundle ∪ z∈C n −∆ H k (U(C(z)), L κ | U (C(z)) ) → C n − ∆ has the canonical flat Gauss-Manin connection. For a horizontal section γ(z) ∈ H k (U(C(z)), L κ | U (C(z)) ), consider the V -valued function I γ (z) = 1 i 1 <···<i k n γ(z) e Φ(z,t)/κ d ln f i 1 ∧ · · · ∧ d ln f i k v i 1 ,...,i k .
For any horizontal section γ(z), the function I γ (z) takes values in Sing V and satisfies the Gauss-Manin differential equations κ ∂I γ ∂z j = K j (z)I γ , j ∈ J, (4.1) where K j (z) ∈ End(Sing V ) are suitable linear operators independent of κ and γ. Formulas for K j (z) see, for example, in [V4,Formula (5.3)].
For z ∈ C n −∆, the subalgebra B(z) ⊂ End(Sing V ) generated by the identity operator and the operators K j (z), j ∈ J, is called the Bethe algebra at z of the Gauss-Manin differential equations. The Bethe algebra is a maximal commutative subalgebra of End(Sing V ), see [V4,Section 8].
We define the characteristic variety of the κ-dependent D-module associated with the Gauss-Manin differential equations (4.1) as Spec = {(z, p) ∈ T * (C n − ∆) | ∃v ∈ Sing V with K j (z)v = p j v, j ∈ J}.
Theorem 4.1 ([V5, Corollary 6.16]). The linear map µ does not depend on j 1 and is an isomorphism of complex vector spaces. For any j ∈ J, the isomorphism µ identifies the operator of multiplication by p j on A Φ (z) and the operator K j (z) on Sing V .
Corollary 4.2. The characteristic variety Spec of the Gauss-Manin differential equations coincides with the Lagrangian variety of the master function.