Invariant of the hypergeometric group associated to the quantum cohomology of the projective space.

Susumu TANABÉ

Abstract - We present a simple method to calculate the Stokes matrix for the quantum cohomology of the projective spaces CP^{k−1} in terms of certain hypergeometric group. We present also an algebraic variety whose fibre integrals are solutions to the given hypergeometric equation.

1. Generalized hypergeometric function

We begin with a short review on the motivation of our problem making reference to the works [5], [9] where one can find precise definitions of the notions below.

At first, we consider a k− dimensional Frobenius manifold F with flat coordinates (t_{1}, ⋯, t_{k})∈ F where the coordinate t_{i} corresponds to coefficients of the basis Δ_{i} of the quantum cohomology H^{∗}(CP^{k−1}). On H^{∗}(CP^{k−1}) one can define so called quantum multiplication

Δ_{α} • Δ_{β} = C_{α,β}^{γ}Δ_{γ},

or

\frac{∂}{∂t_{α}} · \frac{∂}{∂t_{β}} = C_{α,β}^{γ} \frac{∂}{∂t_{γ}} ,

on the level of vector fields on F. The Frobenius manifold is furnished with the Frobenius algebra on the tangent space T_{t}F depending analytically on t ∈ F, T_{t}F = (A_{t}, < , >_{t}) where A_{t} is a commutative associative C algebra and < , >_{t}: A_{t} × A_{t} → C a symmetric non-degenerate bilinear form. The bilinear form < , >_{t} defines a metric on F and the Levi-Civita connexion ∇ for this metric can be considered. Dubrovin introduces a deformed flat connexion ˜∇ on F by the formula ˜∇_{u}v := ∇_{u}v + xu · v with x ∈ C the deformation parameter. Further he extends ˜∇ to F × C. Especially we have ˜∇_{\frac{∂}{∂x}} = \frac{∂}{∂x} − E(t) − \frac{μ}{x}, where E(t) corresponds to the multiplication by the Euler vector field E(t) = ∑_{1≤j≠2≤k−1}(2 − j)t_{j} \frac{∂}{∂t_{j}} + kt_{2} \frac{∂}{∂t_{2}} .

After [5], [9] the quantum cohomology u⃗(x) = (u_{1}(x), ⋯, u_{k}(x)) for the projective space CP^{k−1} at a semisimple point (0, t_{2}, 0, …, 0) (i.e. the algebra (A_{t}, < , >_{t}) is semisimple there) satisfies the following system of differential equation:

(1.1)  ∂_{x}u⃗(x) = (kC_{2}(t) + \frac{μ}{x})u⃗(x).

where

C_{2}(0, t_{2}, 0, …, 0) = (C_{2,β}^{γ})_{1≤β,γ≤k} = ( \begin{matrix}0 & 0 & ⋯ & 0 & e^{t_{2}} \\ 1 & 0 & ... & 0 & 0 \\ 0 & 1 & ... & 0 & 0 \\ ⋮ & ... & ... & ⋮ & ⋮ \\ 0 & 0 & ⋯ & 1 & 0 \end{matrix} ) .

The matrix μ denotes a diagonal matrix with rational entries:

μ = diag\{−\frac{k − 1}{2}, −\frac{k − 3}{2}, ⋯, \frac{k − 3}{2}, \frac{k − 1}{2}\}.

The last component u^{k}(z) (after the change of variables z := kxe^{\frac{t_{2}}{k}}) of the above system for the quantum cohomology satisfies a differential equation as follows [9]:

(1.2)  [(ϑ_{z})^{k} − z^{k}]z^{\frac{−k+1}{2}} u^{k}(z) = 0,

with ϑ_{z} = z \frac{∂}{∂z} . After the Fourier-Laplace transformation

ũ(λ) = ∫ e^{λz}z^{\frac{−k+1}{2}} u^{k}(z)dz,

we obtain an equation as follows:

[(ϑ_{λ} + 1)^{k} − (\frac{∂}{∂λ})^{k}]ũ(λ) = 0.

Here the notation ϑ_{λ} stands for λ\frac{∂}{∂λ}. After multiplying λ^{k} from the left, we obtain

[λ^{k}(ϑ_{λ} + 1)^{k} − ϑ_{λ}(ϑ_{λ} − 1)(ϑ_{λ} − 2) ⋯ (ϑ_{λ} − (k − 1))]ũ(λ) = 0.

The equation for λũ(λ), the Fourier-Laplace transform of \frac{∂}{∂z} z^{\frac{−k+1}{2}} u^{k}(z) should be

(1.3)  [λ^{k}(ϑ_{λ})^{k} − (ϑ_{λ} − 1)(ϑ_{λ} − 2) ⋯ (ϑ_{λ} − k)](λũ(λ)) = 0.

It is evident that the Stokes matrix for \frac{∂}{∂z} z^{\frac{−1+k}{2}} u^{k}(z) is identical with that of the original solution u^{k}(z).

Before proceding further, we remind the following theorem that gives connexion between the Stokes matrix of the system (1.1) with the monodromy of the equation (1.3). Let us consider the Fourier-Laplace transform of the system (1.1):

(1.1)′  (ϑ_{λ} + id_{k})ũ(λ) = (kC_{2}(t)∂_{λ} − μ)ũ(λ).

It is evident that the system (1.1)’ yields the equation (1.3) for its solution component after corresponding modifications.

In a slightly more general setting, let us observe a system with regular singularities:

(1.4)  (Λ − λ · id_{k})∂_{λ}ũ(λ) = (id_{k} + A_{1}(λ))ũ(λ)

with Λ ∈ GL(k, C) whose eigenvalues (λ_{1}, ⋯, λ_{k}) are all distinct, A_{1}(λ) ∈ End(C^{k}) ⊗ O_{C} with A_{1}(0) = diag(ρ_{1}, ⋯, ρ_{k}) where none of the ρ_{j}’s is an integer. Theorem 1.1. ( [2], [5]) Under the assumption that the eigenvalues of the matrix A_{1}(0) are distinct, the Stokes matrix S for the solution component of (1.1) expresses the symmetric Gram matrix G of the solution component of (1.1)’ as follows:

tS + S = 2G.

As for the definition of the Stokes matrix S for the system (1.4) we refer to [5], [9]. The main theorem of this article is the following:

Theorem 1.2. The i, j component S_{ij}, 1 ≤ i, j ≤ k of the Stokes matrix to the system (1.1) has the following expression:

S = \{ \begin{matrix}(−1)^{i−j} _{k}C_{i−j} & i ≥ j \\ 0 & i < j \end{matrix}

This theorem has already been shown by D.Guzzetti [9] by means of a detailed study of braid group actions etc on the set of solutions to (1.2). We present here another approach to understand the structure of the Stokes matrix.

Remark 1. In this article we observe the convention of the matrix multiplicatin as follows:

A · x = (a_{ij})_{0≤i,j≤k−1}(x_{i})_{0≤i≤k−1} = ⟨∑_{i=0}^{k−1} a_{ij}x_{i}⟩_{0≤j≤k−1}.

The matrix operatoes on the vector from left, in contrast to the convetion used in [5], [9].

On the other hand it has been known since [4] that a collection of coherent sheaves O(−i) 0 ≤ i ≤ k − 1 on CP^{k−1} satisfies the following relation

Hom(O(−i), O(−j)) = S^{i−j}(C^{k}), 0 ≤ i, j ≤ k − 1
Ext^{ℓ}(O(−i), O(−j)) = 0, 0 ≤ i, j ≤ k − 1, ℓ > 0

These relation entails immediately the equality

χ(O(−i), O(−j)) := ∑_{ℓ=0}(−1)^{ℓ}Ext^{ℓ}(O(−i), O(−j)) = \{ \begin{matrix}_{k}C_{i−j} & i ≥ j \\ 0 & i < j. \end{matrix}

The conjugate of the matrix χ := (χ(O(−i), O(−j)) 0 ≤ i, j ≤ k − 1 with the diagonal matrix diag(1, −1, ⋯, (−1)^{k−1}) gives us our Stokes matrix. Therefore the Theorem 1.2 is a confirmation of an hypothesis [6] that the matrix for certain exceptional collection of coherent sheaves on a good Fano variety Y must coincide with the Stokes matrix for the quantum cohomology of Y.

Our strategy to prove theorem 1.2 consists in the study of system (1.1)’, instead of (1.1) itself.

Further we consider so called the Kummer covering (naming after N.Katz) of the projective space CP^{1} by ζ = λ^{k} to deduce an hypergeometric equation:

(1.4)  [ζ(ϑ_{ζ})^{k} − (ϑ_{ζ} − \frac{1}{k})(ϑ_{ζ} − \frac{2}{k}) ⋯ (ϑ_{ζ} − 1)]v(ζ) = 0,

for v(λ^{k}) = λũ(λ). We remind of us here a famous theorem due to A.H.M.Levelt that allows us to express the (global) monodromy group of the solution to (1.4) in quite a simple way. For the hypergeometric equation in general,

(1.5)  [∏_{ℓ=1}^{k}(ϑ_{ζ} − α_{ℓ}) − ζ ∏_{ℓ=1}^{k}(ϑ_{ζ} − β_{ℓ})]v(ζ) = 0,

we define two vectors (A_{1}, ⋯, A_{k}) and (B_{1}, ⋯, B_{k}) in the following way:

∏_{ℓ=1}^{k}(t − e^{2πα_{ℓ}i}) = t^{k} + A_{1}t^{k−1} + A_{2}t^{k−2} + ⋯ + A_{k},

∏_{ℓ=1}^{k}(t − e^{2πβ_{ℓ}i}) = t^{k} + B_{1}t^{k−1} + B_{2}t^{k−2} + ⋯ + B_{k}.

Definition 1.3. A linear map L ∈ GL(k, C) is called pseudo-reflexion if it satisfies the condition rank(id_{k} − L) = 1. A pseudo-reflexion R satisfiying an additional condition R^{2} = id_{k} is called a reflexion.

Proposition 1.4. ( [1], [11]) For the solutions to (1.5), the monodromy action on them at the points ζ = 0, ∞, 1 has the follwing expressions:

(1.6)  h_{0} = ( \begin{matrix}0 & 0 & ⋯ & 0 & −A_{k} \\ 1 & 0 & ... & 0 & −A_{k−1} \\ 0 & 1 & ... & 0 & −A_{k−2} \\ ⋮ & ... & ... & ⋮ & ⋮ \\ 0 & 0 & ⋯ & 1 & −A_{1} \end{matrix} ) ,

(h_{∞})^{−1} = ( \begin{matrix}0 & 0 & ⋯ & 0 & −B_{k} \\ 1 & 0 & ... & 0 & −B_{k−1} \\ 0 & 1 & ... & 0 & −B_{k−2} \\ ⋮ & ... & ... & ⋮ & ⋮ \\ 0 & 0 & ⋯ & 1 & −B_{1} \end{matrix} ) ,

whereas h_{1} = (h_{0}h_{∞})^{−1} is a pseudo-reflexion.

It is worthy to notice that the above proposition does not precise for which bases of solution to (1.5) the monodromy is calculated. As a corollary to the Proposition 1.4, however, we see that the monodromy action on the solutions to our equation (1.4) can be written down with respect to a certain basis as follows:

(1.7)  h_{0} = ( \begin{matrix}0 & 0 & ⋯ & 0 & 1 \\ 1 & 0 & ... & 0 & 0 \\ 0 & 1 & ... & 0 & 0 \\ ⋮ & ... & ... & ⋮ & ⋮ \\ 0 & 0 & ⋯ & 1 & 0 \end{matrix} ) ,

h_{∞} = ( \begin{matrix}_{k}C_{1} & 1 & 0 & ⋯ & 0 & 0 \\ −_{k}C_{2} & 0 & 1 & ... & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ \\ (−1)^{k−1} _{k}C_{k−2} & 0 & 0 & ... & 1 & 0 \\ (−1)^{k} _{k}C_{k−1} & 0 & 0 & ⋯ & 0 & 1 \\ −(−1)^{k} & 0 & 0 & ⋯ & 0 & 0 \end{matrix} ) .

In other words,

(1.8)  det(t − h_{0}) = t^{k} − 1, det(t − h_{∞}) = (t − 1)^{k}.

Furthermore we have,

(1.9)  h_{1} = ( \begin{matrix}(−1)^{k−1} & 0 & 0 & ⋯ & 0 & 0 \\ (−1)^{k−2} _{k}C_{k−1} & 1 & 0 & ... & 0 & 0 \\ (−1)^{k−3} _{k}C_{k−2} & 0 & 1 & ... & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ \\ _{k}C_{1} & 0 & 0 & ⋯ & 0 & 1 \end{matrix} ) .

In the next sectin we will see that the theory of Levelt supplies us with necessary data to calculate further the Stokes matrix of the solutions to (1.1).

2. Invariants of the Hypergeometric group

Let us begin with a detailed description of the generators of the hypergeometric group defined for the solutions to the equation (1.3). Proposition 2.1. (cf. [8] I, 8.5) The generators of the hypergeometric group H of the equation(1.3) are expressed in terms of the matrices introduced in the Proposition 1.4 as follows:

(2.1)  M_{0} = h_{0}^{k} = 1, M_{1} = h_{1} = (h_{0}h_{∞})^{−1}, M_{∞} = h_{∞}^{k}, M_{ω^{i}} = h_{∞}^{−i}h_{1}h_{∞}^{i}(i = 1, 2, ⋯, k − 1),

where M_{t} denotes the monodromy action around the point t ∈ CP_{λ}^{1}. These generators naturally satisfy the Riemann-Fuchs relation:

(2.2)  M_{∞}M_{ω^{k−1}}M_{ω^{k−2}} ⋯ M_{ω}M_{1} = id_{k}.

proof Let us think of a k−leaf covering C̃P_{λ}^{1} of CP_{ζ}^{1} that corresponds to the Kummer covering ζ^{k} = λ. In

lifting up the path around ζ = 1 the first leaf of C̃P_{λ}^{1}, the monodromy h_{1} is sent to the conjugation with a path around λ = ∞. That is to say we have M_{ω} = h_{∞}^{−1}h_{1}h_{∞}. For other leaves the argument is similar. Q.E.D.

Let us denote by X^{K} a k × k matrix that satisfies the relation

(2.3)  ḡX^{K} ^{t}g = X^{K},

for every element g of a group K ⊂ GL(k, C). From the definition, the set of all X^{K} for a group K reperesents a C vector space in general. We will call a matrix of this space the quadratic invariant of the group K. In the special case in which we are interested, the following statement holds.

Lemma 2.2. For the hypergeometric group H generated by the pseudo-reflexions as in (2.1), for every X^{H} there exists a non zero k × k matrix X̃^{H} such that X^{H} = λX̃^{H} for some λ ∈ C \ {0}.

proof The relation

(2.4)  h_{1}X ^{t}h_{1} = X

gives rise to equations on x_{0j} and x_{j0}. That is to say, the first row of (2.4) corresponds to

(−1)^{i} _{k}C_{i}x_{00} − (−1)^{k−1}x_{0i} = x_{0i}, 1 ≤ i ≤ k − 1,

while

(−1)^{i} _{k}C_{i}x_{00} − (−1)^{k−1}x_{i0} = x_{i0}, 1 ≤ i ≤ k − 1.

Thus we obtained 2(k − 1) linearly independent equations. Further by concrete calculus one can easily see that

M_{ω^{ℓ}} = id_{k} + T_{ℓ},

where

T_{ℓ} = ( \begin{matrix}t_{0}^{(ℓ)}τ_{0} & t_{1}^{(ℓ)}τ_{0} & ⋯ & t_{ℓ}^{(ℓ)}τ_{0} & 0 & ⋯ & 0 \\ t_{0}^{(ℓ)}τ_{1} & t_{1}^{(ℓ)}τ_{1} & ⋯ & t_{ℓ}^{(ℓ)}τ_{1} & 0 & ⋯ & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ t_{0}^{(ℓ)}τ_{k−1} & t_{1}^{(ℓ)}τ_{k−1} & ⋯ & t_{ℓ}^{(ℓ)}τ_{k−1} & 0 & ⋯ & 0 \end{matrix} ) ,

with (k − ℓ)− zero columns from the right. The remaining columns are generated from T_{1} after simple linear recurrent relations by an inductive way. The relation M_{ω}X ^{t}M_{ω} = X gives rise to new equations

(1 + t_{1}^{(1)}τ_{1})^{2}x_{11} + linear functions in (x_{0i}, x_{i0} ) = x_{11} ,

with (1 + t_{1}^{(1)}τ_{1}) = −1 + (_{k}C_{1})^{2} ≠ 1 and

(1 + t_{1}^{(1)}τ_{1})x_{1i} + linear functions in (x_{0i}, x_{i0} , x_{11} ) = x_{1i},
(1 + t_{1}^{(1)}τ_{1})x_{i1} + linear functions in (x_{0i}, x_{i0} ) = x_{i1} .

Thus we get 2k − 3 new linearly independent equations. In general for (ℓ, ℓ) term, we get from the relation M_{ω^{ℓ}}X ^{t}M_{ω^{ℓ}} = X, 1 ≤ ℓ ≤ k − 1

(1 + t_{ℓ}^{(ℓ)}τ_{ℓ})^{2}x_{ℓℓ} + linear functions in (x_{νi}, x_{iν}, 0 ≤ ν ≤ ℓ − 1 ) = x_{ℓℓ},

with 1 + t_{ℓ}^{(ℓ)}τ_{ℓ} = −1 + (_{k}C_{ℓ})^{2} ≠ 1. For x_{iℓ}

(1 + t_{ℓ}^{(ℓ)}τ_{ℓ})x_{iℓ} + linear functions in (x_{νi}, x_{iν}, 0 ≤ ν ≤ ℓ − 1 , x_{ℓℓ}) = x_{iℓ}.

In this way we get a set of 2(k − 1) + ∑_{ℓ=1}^{k−1}(2(k − ℓ) − 1) = k^{2} − 1 independent linear equations with respect to the elements of X. Q.E.D.

From the above Lemma 2.2, we can calculate the quadratic invariant X^{H} after the following relations,

(2.5)  h̄_{0}X^{H} ^{t}h_{0} = X^{H}, h̄_{∞}X^{H} ^{t}h_{∞} = X^{H}.

From [1] we know that the inverse to X^{H_{0}} = X^{H} for H_{0} = {h_{0}, h_{∞}}, if it exists, must be a Toeplitz matrix i.e.:

(X^{H_{0}})^{−1} = ( \begin{matrix}x_{0} & x_{1} & x_{2} & ⋯ & x_{k−1} \\ x_{−1} & x_{0} & x_{1} & ⋯ & x_{k−2} \\ x_{−2} & x_{−1} & x_{0} & ⋯ & x_{k−3} \\ ⋮ & ... & ... & ⋮ & ⋮ \\ x_{−(k−1)} & x_{−(k−2)} & x_{−(k−3)} & ⋯ & x_{0} \end{matrix} ) .

Making use of this circumstances, it is possible to show that the system of equations that arises from the relations

th_{∞}(X^{H_{0}})^{−1} h̄_{∞} = (X^{H_{0}})^{−1},^{t} h_{0}(X^{H_{0}})^{−1} h̄_{0} = (X^{H_{0}})^{−1},

for (X^{H_{0}})^{−1} consists of 2(k − 1) equations.

(2.6)′  x_{k−1−i} = x_{−i−1},
(2.6)′′  (−1)^{k+1}x_{k−1−i} + (−1)^{k} _{k}C_{k−1}x_{k−2−i} + ⋯ + _{k}C_{3}x_{2−i} − _{k}C_{2}x_{1−i} + kx_{−i} = x_{−1−i}

This calculates the matrix X^{H} for the case k− odd.

As for the case k− even, our matrix X^{H} has the following form

X^{H} = ( \begin{matrix}0 & y_{1} & y_{2} & ⋯ & y_{k−1} \\ y_{−1} & 0 & y_{1} & ⋯ & y_{k−2} \\ y_{−2} & y_{−1} & 0 & ⋯ & y_{k−3} \\ ⋮ & ... & ... & ⋮ & ⋮ \\ y_{−(k−1)} & y_{−(k−2)} & y_{−(k−3)} & ⋯ & 0 \end{matrix} ) ,

where y_{−(k−1)}, ⋯ y_{k−1} satisfy 2(k − 1) equations for some constant y_{0},

(2.6)′′′  y_{i} + y_{−i} = 0, y_{i} − y_{−i} = 2(−1)^{i} _{k}C_{i}y_{0} 1 ≤ i ≤ k − 1,

which are derived from (2.5). Thus the matrix X^{H} for the case k−even is obtained.

We remember here a classical theorem on the pseudo-reflexions.

Theorem 2.3. (cf. Bourbaki Groupe et Algèbre de Lie Chapitre V, §6, Exercise 3) Let E be a vector space with basis (e_{1}, ⋯, e_{d}), and their dual basis (f_{1}, ⋯, f_{d})∈ E^{∗}. Let us set a_{ij} = f_{i}(e_{j}). The pseudo-reflexion s_{i} with respec to the basis f_{i} is defined as

s_{i}(e_{j}) = e_{j} − f_{i}(e_{j})e_{i} = e_{j} − a_{ij}e_{i}.

Set

(2.7)  V = ( \begin{matrix}a_{11} & a_{21} & a_{31} & ⋯ & a_{d1} \\ 0 & a_{22} & a_{32} & ⋯ & a_{d2} \\ 0 & 0 & a_{33} & ⋯ & a_{d3} \\ ⋮ & ... & ... & ⋮ & ⋮ \\ 0 & 0 & ⋯ & 0 & a_{dd} \end{matrix} ) , U = ( \begin{matrix}0 & 0 & 0 & ⋯ & 0 \\ a_{12} & 0 & 0 & ⋯ & 0 \\ a_{13} & a_{23} & 0 & ⋯ & 0 \\ ⋮ & ... & ... & ⋮ & ⋮ \\ a_{1d} & a_{2d} & ⋯ & a_{d−1,d} & 0 \end{matrix} )

Under these notations, the composition of all possible reflexions s_{d}s_{d−1} ⋯ s_{1} (a Coxeter element) with respect to the basis (e_{1}, ⋯, e_{d}) is expressed as follows:

(2.8)  s_{d}s_{d−1} ⋯ s_{1} = (id_{d} − V)(id_{d} + U)^{−1}.

proof For 1 ≤ i, k ≤ d we define

y_{i} = s_{i−1} ⋯ s_{1}(e_{i}).

It is possible to see that

e_{i} = y_{i} + ∑_{k<i≤d} a_{ki}y_{k}, s_{d} ⋯ s_{1}(e_{i}) = y_{i} − ∑_{i≤k≤d} a_{ki}y_{k}.

The statement follows immediately from these relations. Q.E.D.

To establish a relationship between the invariant X^{H} and the Gram matrix necessary for calculus of the Stokes matrix, we investigate a situation where the generators of the hypergeomeric group have special forms. Namely consider a hypergeometric group Γ of rank k generated by pseudo-reflexions R_{0}, ⋯ R_{k−1} where

(2.9)  R_{j} = id_{k} − Q_{j},

with

(2.10)  Q_{j} = ( \begin{matrix}0 & ⋯ & 0 & t_{j0} & 0 & ⋯ & 0 \\ 0 & ⋯ & 0 & t_{j1} & 0 & ⋯ & 0 \\ 0 & ⋯ & 0 & t_{j2} & 0 & ⋯ & 0 \\ ⋮ & ⋯ & ⋮ & ⋮ & ⋮ & ⋯ & ⋮ \\ 0 & ⋯ & 0 & t_{j,k−1} & 0 & ⋯ & 0 \end{matrix} ) , 0 ≤ j ≤ k − 1,

all zero components except for the j−th column. Let us define the Gram matrix G associated to the above collection of pseudo-reflexions:

(2.11)  G = ( \begin{matrix}t_{00} & t_{10} & ⋯ & t_{k−1,0} \\ t_{01} & t_{11} & ⋯ & t_{k−1,1} \\ t_{02} & t_{12} & ⋯ & t_{k−1,2} \\ ⋮ & ⋮ & ⋯ & ⋮ \\ t_{0,k−1} & t_{1,k−1} & ⋯ & t_{k−1,k−1} \end{matrix} ) .

We shall treat the cases where G is either symmetric or anti-symmetric. Let us introduce an upper triangle matrix S satisfying

G = S +^{t} S (resp. G = S −^{t} S),

for a symmetric (anti-symmetric) matrix G. In the anti-symmetric case, we shall use a convention so that the diagonal part of S is a scalar multiplication on the unit matrix. It is easy to see that for the symmetric (resp. anti-symemtric) G the diagonal element t_{jj} = 2 (resp. t_{jj} = 0).

Proposition 2.4. For an hypergeometric group Γ defined over R, the following statements hold.

1) Suppose that the space of quadratic invariant matrices X^{Γ} is 1 dimensional. Then X^{Γ} coincides with the Gram matrix G (2.11) up to scalar multiplication.

2)The composition of all generators R_{0}, ⋯, R_{k−1} gives us the Seifert form:

(2.12)  R_{k−1} ⋯ R_{0} = ∓^{t}S · S^{−1},

where to the minus sign corresponds symmetric G and to the plus sign anti-symmetric G.

proof 1) It is enough to prove that the Gram matrix is a quadratic invariant. We calculate

R_{j}G^{t}R_{j} = (id_{k} − Q_{j})G(id_{k} − ^{t}Q_{j}).

It is esy to compute

Q_{j}G = (t_{aj}t_{jb})_{0≤a,b≤k−1}, G ^{t}Q_{j} = (t_{ja}t_{jb})_{0≤a,b≤k−1},
Q_{j}G^{t}Q_{j} = t_{jj}G ^{t}Q.

It yields the following equality,

G^{t}Q_{j} + Q_{j}G − Q_{j}G^{t}Q_{j} = (t_{jb}((1 − t_{jj})t_{ja} + t_{aj}))_{0≤a,b≤k−1},

that vanishes for G symmetric with t_{jj} = 2 and for G anti-symmetric with t_{jj} = 0.

2) It is possible to apply directly our situation to that of Theorem 2.3. In the symmetric case, t_{ii} = 2 and

V = ( \begin{matrix}2 & t_{10} & t_{20} & ⋯ & t_{k−1,0} \\ 0 & 2 & t_{21} & ⋯ & t_{k−1,1} \\ 0 & 0 & 2 & ⋯ & t_{k−1,2} \\ ⋮ & ... & ... & ⋮ & ⋮ \\ 0 & 0 & ⋯ & 0 & 2 \end{matrix} ) , U = ( \begin{matrix}0 & 0 & 0 & ⋯ & 0 \\ t_{10} & 0 & 0 & ⋯ & 0 \\ t_{20} & t_{21} & 0 & ⋯ & 0 \\ ⋮ & ... & ... & ⋮ & ⋮ \\ t_{k−1,0} & t_{k−1,1} & ⋯ & t_{k−1,k−2} & 0 \end{matrix} ) ,

in accordance with the notation (2.7). The formula (2.8) means (2.12) with minus sign. In the anti-symmetric case t_{ii} = 0, 0 ≤ i ≤ k − 1 and (2.7) yields (2.11) with plus sign. Q.E.D.

Corollary 2.5. We can determine the Stokes matrix S by the following relation

(2.13)  S = (id_{k} − R_{k−1} ⋯ R_{0})^{−1}G,

with the aid of the Gram matrix and pseudo-reflexions.

In some sense, a converse to the Proposition 2.4 holds. To show this, we remember a proposition from [12].

Definition 2.6. The fundamental set (u_{0}(λ), ⋯ u_{k−1}(λ)) of the system (1.4) is a set of solutions satisfying the following asymptotic expansion:

u_{j}(λ) = (λ − λ_{j})^{ρ_{j}} ∑_{r=0}^{∞} g_{r}^{(j)}(λ − λ_{j})^{r},

where (λ_{0}, ⋯ λ_{k−1}) are eigenvalues of the matrix Λ. The exponenets ρ_{j} are diagonal elements of the matrix A_{1}(0).

Proposition 2.7. ( [12]) Every generator of an hypergeometric group Γ over R defined for the system of type (1.4) (without logarithmic solution) is a product of pseudo-reflexions of the follwing form expressed with respect to its fundamental set:

(2.14)  M_{j} = id_{k} − ( \begin{matrix}0 & ⋯ & 0 & s_{j0} & 0 & ⋯ & 0 \\ 0 & ⋯ & 0 & s_{j1} & 0 & ⋯ & 0 \\ ⋮ & ⋯ & ⋮ & ⋮ & ⋮ & ⋯ & ⋮ \\ 0 & ⋯ & 0 & s_{j,k−1} & 0 & ⋯ & 0 \end{matrix} ) .

where s_{jj} = 2 or 0.

We get the following corollary to the above proposition 2.7.

Corollary 2.8. Assume that the hypergeometric group Γ is generated by pseudo-reflexions T_{0}, ⋯, T_{k−1} such that rank(T_{i} − id_{k}) = 1 for 0 ≤ i ≤ k − 1. Then it is possible to choose a suitable set of pseudo-reflexions generators R_{j} like (2.9), (2.10), up to constant multiplication on Q_{j}, so that they determine the quadratic invariant Gram matrix like (2.11).

proof The proposition 2.7 implies that every generator T_{i} is a product of pseudo-reflexions M_{j} with s_{jk} possibly different from t_{jk}. From the condition on the quadratic invariant X^{Γ} and the proposition 2.4, s_{jk} must coincide with t_{jk}. That is to say Γ must be generated by M_{0}, ⋯, M_{k−1} with \frac{s_{ja}}{t_{ja}} = \frac{s_{jb}}{t_{jb}} for all a, b, j ∈ \{0, ⋯, k−1\}. This means that Γ has as its generators the pseudo-reflexions R_{0}, ⋯, R_{k−1} of (2.12) up to constant multiplication on Q_{j}. Q.E.D. proof of Theorem 1.2 First we remark that solutions to (1.3) have no logarithmic asymptotic behaviour around any of their singular points except for the infinity.

In the case with k odd for X^{H_{0}}, there exists a ≠ 0 such that the vector v⃗_{0} :=^{t} (1 + (−1)^{k−1}, −k, _{k}C_{2}, ⋯, (−1)^{k−2} _{k}C_{k−2}, (−1)^{k−1} _{k}C_{k−1}) ∈ R^{k} satisfies the relation:

X^{H_{0}} v⃗_{0} =^{t} (a, 0, 0, ⋯, 0).

Actually this fact can be proven almost without calculation in the follwing way. First we introduce a series of vetors

w⃗_{ℓ} = (x_{−ℓ}, x_{−ℓ+1}, ⋯, x_{k−1−ℓ}), ℓ = 0, 1, ⋯, k − 1.

Then the equation (2.6)′′ can be rewritten in terms of w⃗_{ℓ}:

w⃗_{ℓ} · v⃗_{0} = ∑_{i=0}^{k−1}(−1)^{i} _{k}C_{i} · x_{i−ℓ} = 0 for 1 ≤ ℓ ≤ k − 1.

On the other hand, the vector w⃗_{0} is linearly independent of the vectors w⃗_{1}, … w⃗_{k−1} by virtue of the construction of the matrix X. Therefore w⃗_{0} · v⃗_{0} ≠ 0 as v⃗_{0} ≠ 0. This means the existence of the non zero constant a as above.

This relation with the corollary 2.8 gives immediately the expression below for the pseudo-reflexions

(2.15)  R_{j} = id_{k} − ( \begin{matrix}0 & ⋯ & 0 & (−1)^{j+k−1} _{k}C_{j} · r & 0 & ⋯ & 0 \\ ⋮ & ⋯ & ⋮ & ⋮ & ⋮ & ⋯ & ⋮ \\ 0 & ⋯ & 0 & −(−1)^{k−1}k · r & 0 & ⋯ & 0 \\ 0 & ⋯ & 0 & (1 + (−1)^{k−1}) · r & 0 & ⋯ & 0 \\ 0 & ⋯ & 0 & −k · r & 0 & ⋯ & 0 \\ 0 & ⋯ & 0 & _{k}C_{2} · r & 0 & ⋯ & 0 \\ ⋮ & ⋯ & ⋮ & ⋮ & ⋮ & ⋯ & ⋮ \\ 0 & ⋯ & 0 & (−1)^{k−1−j} _{k}C_{k−j−1} · r & 0 & ⋯ & 0 \end{matrix} ) , 0 ≤ j ≤ k − 1,

whose Gram matrix is equal to

\begin{matrix}G_{ij} = (−1)^{i−j+k−1} _{k}C_{i−j} · r & i > j \\ G_{ii} = (1 + (−1)^{k−1}) · r & i = j \\ G_{ij} = (−1)^{i−j} _{k}C_{j−i} · r & j > i \end{matrix}

with some constant r. As for the case k− even, the equations (2.6)′′′ and the Corollary 2.8 gives us the expression (2.15) for the pseudo-reflexion generators.

Taking into account the theorem 1.1 for the symmetric Gram matrix, we obtain the desired statement for the case k− odd, as it is required G_{ii} = 2 = 2r.

For the case k−even, we remember a statement on the Stokes matrix from [2] which claims that if the matrix μ of (1.1) has integer eigenvalues, the equality det(S + ^{t}S) = 0 must hold. The Corllary 2.5 gives us the relation

S = (id_{k} + (id_{k} − V)(id_{k} + U)^{−1})^{−1}G = (id_{k} + U)G^{−1}G = id_{k} + U,

with

U_{ij} = (−1)^{i−j+k−1} _{k}C_{i−j} · r i > j.

We shall choose the constant r = 1 so that S + ^{t}S = 2id_{k} + U +^{t} U posesses an eigenvector (1, −1, ⋯, 1, −1) with zero eigenvalue. Q.E.D.

Remark 2. The Gram matrix (2.13) that has been calculated for the fundamental set (Definition 2.6) of the equation (1.3) gives directly a suitable Stokes matrix we expected. For other Fano varieties, however, the Gram matrix calculated with respect to the fundamental set does not necessarily give a desirable form, as it is seen from the case of odd dimensional quadrics. This situation makes us to be careful in the choice of the base of solutions for which we calculate the Gram matrix.

3. Geometric interpretation of the hypergeometric equation

In this section we show that the equation (1.4) arises from the differential operator that annihilates the fibre integral associated to the family of variety defined as a complete intersection

(3.1)  X_{s} := {(x_{0}, ⋯ x_{k}) ∈ C^{k+1}; f_{1}(x) + s = 0, f_{2}(x) + 1 = 0}.

where

f_{1}(x) = x_{0}x_{1} ⋯ x_{k}, f_{2}(x) = x_{0} + x_{1} + ⋯ x_{k}.

This result has been already announced by [7], [8] and [3]. Our main theorem of this section is the following Theorem 3.1. Let us assume that ℜ(f_{1}(x) + s)|_{Γ} < 0, ℜ(f_{2}(x) + s)|_{Γ} < 0, out of a compact set for a Leray coboundary cycle Γ ∈ H^{k+1}(C^{k+1} \ X_{s}) avoiding the hypersurfacess f_{1}(x) + s = 0 and f_{2}(x) + 1 = 0. For such a cycle we consider the following residue integral:

(3.2)  I_{x^{i},Γ}^{(v_{1},v_{2})}(s) = ∫_{Γ} x^{i}(f_{1}(x) + s)^{−v_{1}}(f_{2}(x) + 1)^{−v_{2}}dx,

for the monomial x^{i} := x^{i_{0}} ⋯ x^{i_{k−1}}. Then the integral I_{1,Γ}^{(1,1)}(s) satisfies the following hypergeometric differential equation

(3.3)  [ϑ_{s}^{k} − k^{k}s(ϑ_{s} + \frac{1}{k})(ϑ_{s} + \frac{2}{k}) ⋯ (ϑ_{s} + \frac{k}{k})]I_{1,Γ}^{(1,1)}(s) = 0

which has unique holomorphic solution at s = 0,

(3.4)  I_{0}(s) = ∑_{m≥0} \frac{(km)!}{(m!)^{k}}s^{m}.

We shall put ζ = \frac{k^{k}}{s} , to get (1.4) from (3.3). Our calculus is essentially based on the Cayley trick method developed in [13].

Proof of Theorem 3.1 Let us consider the Mellin transform of the fibre integral (3.2)

(3.5)  M_{i,Γ}^{(v_{1},v_{2})}(z) := ∫_{Π} s^{z}I_{x^{i}}^{(v_{1},v_{2})}(s)\frac{ds}{s}

For the Mellin transform (3.5), we have the following

(3.5)′  M_{i,Γ}^{(v_{1},v_{2})}(z) = g(z) ∏_{ℓ=0}^{k−1} Γ(z + i_{ℓ} + 1 − v_{2})Γ(− ∑_{ℓ=0}^{k−1}(i_{ℓ} + 1) − kz + v_{1} + kv_{2})Γ(−z + v_{2})Γ(z),

with g(z) a rational function in e^{πiz}. The formula (3.5)’ shall be proven below. In substituting i = 0, v_{1} = v_{2} = 1, we see that

I_{1,Γ}^{(1,1)}(s) = ∫Π̌ s^{−z}g(z)\frac{Γ(z)^{k}}{Γ(kz)}dz,

where Π̌ denotes the path (−i∞, +i∞) avoidoing the poles of Γ(z) = 0, −1, −2, · · · . From this integral representation, the equation (3.3) immediately follows in taking account the fact that the factor g(z) plays no role in establishment of the differential equation. Q.E.D.

Proof of (3.5)’ In making use of the Cayley trick, we transform the integral (3.5) into the following form

(3.6)  M_{i,Γ}^{(v_{1},v_{2})}(z) = ∫_{Π×Γ×R_{+}^{2}} x^{i}e^{y_{1}(f_{1}(x)+s)+y_{2}(f_{2}(x)+1)}y_{1}^{v_{1}}y_{2}^{v_{2}}s^{z} \frac{dx}{x^{1}} \frac{dy}{y^{1}} \frac{ds}{s^{1}}.

Here we introduce new variables T_{0}, ⋯ T_{k+2},

(3.7)  T_{i} = y_{1}x_{i}, 0 ≤ i ≤ k − 1, T_{k} = y_{1}s, T_{k+1} = y_{2}x_{0}x_{1} ⋯ x_{k−1}, T_{k+2} = y_{2}

in such a way that the phase function of the right hand side of (3.6) becomes

y_{1}(f_{1}(x) + s) + y_{2}(f_{2}(x) + 1) = T_{0} + T_{1} + ⋯ + T_{k+2}.

If we set

Log T :=^{t} (log T_{0}, ⋯, log T_{k+2})
Ξ :=^{t} (x_{0}, ⋯, x_{k−1}, s, y_{1}, y_{2})
Log Ξ :=^{t} (log x_{0}, ⋯, log x_{k−1}, log s, , log y_{1}, log y_{2}).

Gram matrix 11

Then the above relationship (3.7) can be written down as

(3.8)  Log T = L · Log Ξ,

where

L = [ \begin{matrix}1 & 0 & 0 & ⋯ & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & ⋯ & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & ⋯ & 0 & 0 & 1 & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & ⋯ & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & ⋯ & 0 & 0 & 1 & 0 \\ 1 & 1 & 1 & ⋯ & 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & ⋯ & 0 & 1 & 0 & 1 \end{matrix} ] .

This yields immediately

Log Ξ = L^{−1} · Log T,

with

L^{−1} = [ \begin{matrix}1 & 0 & 0 & ⋯ & 0 & −1 & 0 & 0 \\ 0 & 1 & 0 & ⋯ & 0 & −1 & 0 & 0 \\ 0 & 0 & 1 & ⋯ & 0 & −1 & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & ⋯ & 1 & −1 & 0 & 0 \\ 1 & 1 & 1 & ⋯ & 1 & −k & −1 & 1 \\ 0 & 0 & 0 & ⋯ & 0 & 1 & 0 & 0 \\ −1 & −1 & −1 & ⋯ & −1 & k & 1 & 0 \end{matrix} ] .

If we set

(3.9)  (i_{0}, ⋯ i_{k−1}, z, v_{1}, v_{2}) · L^{−1} = (L_{0}(i, z, v_{1}, v_{2}), ⋯, L_{k+2}(i, z, v_{1}, v_{2})).

then we can see that

M_{i,Γ}^{(v_{1},v_{2})}(z) = ∫_{Π×Γ×R_{−}^{2}} x^{i}e^{T_{0}+···+T_{k+2}}y_{1}^{v_{1}}y_{2}^{v_{2}}s^{z} \frac{dx}{x^{1}} \frac{dy}{y^{1}} \frac{ds}{s^{1}}

= ∫_{T_{0}+···+T_{k+2}≠0} e^{T_{0}+···+T_{k+2}} ∏_{0≤i≤k+2} T_{i}^{L_{k+2}(i,z,v_{1},v_{2})} ⋀_{0≤i≤k+2} \frac{dT_{i}}{T_{i}} .

In view of the choice of the cycle Γ, we can apply the formula to calculate Γ function to our situation:

∫_{C} e^{−T} T^{σ} \frac{dT}{T} = (1 − e^{2πiσ})Γ(σ),

for the unique nontrivial cycle C turning around T = 0 that begins and returns to ℜT → +∞. The direct calculation of (3.9) shows that

L_{ℓ}(i, z, v_{1}, v_{2}) = z + i_{ℓ} + 1 − v_{2}, 0 ≤ ℓ ≤ k − 1,

L_{k}(i, z, v_{1}, v_{2}) = − ∑_{ℓ=0}^{k−1}(i_{ℓ} + 1) + v_{1} + kv_{2}), L_{k+1}(i, z, v_{1}, v_{2}) = −z + v_{2}, L_{k+2}(i, z, v_{1}, v_{2}) = z.

This shows the formula (3.5)’. Q.E.D.

Remark 3. In combining Theorems 1.2, 3.1, we can state that we found out a deformation of an algebraic variety X_{λ} = \{(\frac{λ}{k})^{k}(x_{0}x_{1} ⋯ x_{k}) + 1 = 0, x_{0} + x_{1} + ⋯ + x_{k} = 1 = 0\}. such that its variation gives rise to the equation (1.3). It means that we establish a connexion between an exceptional collection of CP^{k−1} and a set of vanishing cycles for its mirror counter part X_{λ}. Thus our theorems give an affirmative answer to the hypothesis stating the existence of such relationship between two mirror symmetric varieties (so called Bondal-Kontsevich hypothesis) in a special case. See [8] and [10] in this respect for the detail.

References

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[3] S.Barannikov, Semi-infinite Hodge structures and mirror symmetry for projective spaces , math.AG/0010157.

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[5] B.Dubrovin, Painlevé transcendents in two dimensional topological field theory ,The Painlevé property, CRM Ser Math. Physics, Springer (1999), pp. 487-412 math.AG/9803107.

[6] B.Dubrovin, Geometry and analytic theory of Frobenius manifolds ,,Proceedings ICM Berlin Vol.II,(1998), math.AG/9807034.

[7] A.Givental’, Equivariant Gromov-Witten invariants, Intern.Math. Res.Notices 13 (1996), pp.613-663.

[8] V.V.Golyshev, Riemann-Roch Variations , Izvestia Math. 65 (2001), no. 5, pp.853-887.

[9] D.Guzzetti, Stokes matrices and monodromy of the quantum cohomology of projective spaces , Comm. in Math. Physics 207 (1999), no.2, pp.341-383.

[10] M.Kontsevich, Homological algebra of mirror symmetry , Proceedings of ICM Zürich 1994, (1994) alg-geom/9411018.

[11] A.H.M.Levelt, Hypergeometric functions, Indagationes Math.23(1961), pp.361-403.

[12] K.Okubo, On the group of Fuchsian equations , Tokyo Metropolitan University, 1987.

Connection problems for systms of linear differential equations, Springer LNM, 243 (1971), pp.238-248.

[13] S.Tanabé Transformée de Mellin des intégrales- fibres associée aux singularités isolées d’intersection complète II, in preparation.

Independent University of Moscow Bol’shoj Vlasijevskij Pereulok 11, MOSCOW, 121002, Russia E-mails: tanabe@mccme.ru,