Arrangements of hyperplanes arising from sections of rigid local systems

Yoshishige Haraoka Department of Mathematics, Kumamoto University Arrangements of hyperplanes in the thoery of differential equations

Arrangements of hyperplanes appear:

- as a singular locus of a completely integrable system

[figure: x = 0; ∞; y = 0; x = y; y = 1; x = 1]

Singular locus of the system for Appell’s F_{1}(α, β, β′, γ; x, y)

- as a set of branching points for an integral representation of solutions of a linear differential equation

[figure: s_{2} = 0; s_{2} = x; s_{1} = s_{2}; s_{1} = 0; s_{1} = 1]

Branching points for the integral representation of the generalized hypergeometric function

3F_{2} ( \begin{matrix}α_{1} α_{2} α_{3} \\ β_{1} β_{2} \end{matrix} ; x) = C ∫∫_{Δ} s_{1}^{α_{2}−β_{1}}(s_{1}−1)^{β_{1}−α_{1}−1}s_{2}^{α_{3}−β_{2}}(s_{2}−s_{1})^{β_{2}−α_{2}−1}(s_{2}−x)^{−α_{3}} ds_{1} ds_{2}

Problem: Describe the relation between the asymptotic behaviors and the cycles

\begin{cases}x = 0 & x = 1 & x = ∞ \\ 0 & 0 & α_{1} \\ 1 − β_{1}^{(∗1)} & 1 & α_{2} \\ 1 − β_{2}^{(∗2)} & ρ^{(∗3)} & α_{3} \end{cases}\} (ρ = ∑ β_{i} − ∑ α_{i})

[figure: s_{2} = 0; s_{2} = x; s_{1} = s_{2}; s_{1} = 0; s_{1} = 1; ∗1; ∗2; ∗3]

Remark: Hypersurfaces may appear

[figure: x = 0; L_{∞}; y = 0; C]

Singular locus of the system for Appell’s F_{4}(α, β, γ, γ′; x, y)

Rigid Fuchsian systems

[figure: Fuchsian ODE; rigid]

rigid = free of accessory parameters Theorem (N. M. Katz) Any irreducible rigid Fuchsian equation is connected to rank 1 Fuchsian equation by \underline{additions} and \underline{middle convolutions}.

We consider a Fuchsian system

\frac{dU}{dx} = (∑_{j=1}^{p} \frac{A_{j}}{x − a_{j}} ) U

A_{j}: n × n-constant matrix singularities: a_{0} := ∞, a_{1}, …, a_{p} We denote the system by (A_{1}, …, A_{p}) Addition: α = (α_{1}, …, α_{p}) ∈ ℂ^{p}

a_{α} : (A_{1}, …, A_{p}) ↦ (A_{1} + α_{1}, …, A_{p} + α_{p})

analytically:

U(x) ↦ ∏_{j=1}^{p}(x − a_{j})^{α_{j}} · U(x)

Middle convolution: μ ∈ ℂ We first define convolution c_{μ} with parameter μ

c_{μ} : (A_{1}, …, A_{p}) ↦ (G_{1}, ….G_{p})

where

G_{j} = (\begin{matrix}O & ⋯ & O & ⋯ & O \\ ⋮ & & ⋮ & & ⋮ \\ O & ⋯ & O & ⋯ & O \\ A_{1} & ⋯ & A_{j} + μ & ⋯ & A_{p} \\ O & ⋯ & O & ⋯ & O \\ ⋮ & & ⋮ & & ⋮ \\ O & ⋯ & O & ⋯ & O \end{matrix}) (j

We can find in an explicit way an invariant subspace W ⊂ ℂ^{pn} for (G_{1}, …, G_{p}).

Ĝ_{j} : action of G_{j} on ℂ^{pn}/W

Middle convolution mc_{μ} with parameter μ is defined by

mc_{μ} : (A_{1}, …, A_{p}) ↦ (Ĝ_{1}, …, Ĝ_{p}) analytically (Dettwiler-Reiter):

U(x) ↦ V (x) := (\frac{U(x)}{x−a_{1}} ; \frac{U(x)}{x−a_{2}} ; ⋮ ; \frac{U(x)}{x−a_{p}} ) ↦ Ṽ (x) := ∫_{Δ} V (s)(s − x)^{μ} ds

Then Ṽ (x) satisfies the system (G_{1}, …, G_{p}). Take P such that P^{−1}G_{j}P = ( \begin{matrix}∗ & ∗ \\ O & Ĝ_{j} \end{matrix} ) , and define Z̃(x) by Ṽ (x) = P Z̃(x). Set Z̃(x) =: (\begin{matrix}Z_{1}(x) \\ Z(x) \end{matrix}) . Then Z(x) satisfies

\frac{dZ}{dx} = (∑_{j=1}^{p} \frac{Ĝ_{j}}{x − a_{j}} ) Z.

Thus

mc_{μ} : U(x) ↦ Z(x)

Example: We start from a rank 1 equation

\frac{du}{dx} = (\frac{α_{1}}{x} + \frac{β_{1}}{x − 1}) u,

which has a solution u(x) = x^{α_{1}}(x − 1)^{β_{1}}.

u(x) ↦ (\begin{matrix}x^{α_{1}−1}(x − 1)^{β_{1}} \\ x^{α_{1}}(x − 1)^{β_{1}−1} \end{matrix})

↦ (\begin{matrix}∫_{δ} s^{α_{1}−1}(s − 1)^{β_{1}}(s − x)^{γ_{1}} ds \\ ∫_{δ} s^{α_{1}}(s − 1)^{β_{1}−1}(s − x)^{γ_{1}} ds \end{matrix})

↦ x^{α_{2}}(x − 1)^{β_{2}} (\begin{matrix}∫_{δ} s^{α_{1}−1}(s − 1)^{β_{1}}(s − x)^{γ_{1}} ds \\ ∫_{δ} s^{α_{1}}(s − 1)^{β_{1}−1}(s − x)^{γ_{1}} ds \end{matrix})

↦ (\begin{matrix}∫∫_{Δ} s_{1}^{α_{1}−1}(s_{1} − 1)^{β_{1}}(s_{1} − s_{2})^{γ_{1}}s_{2}^{α_{2}−1}(s_{2} − 1)^{β_{2}}(s_{2} − x)^{γ_{2}} ds_{1} ds_{2} \\ ∫∫_{Δ} s_{1}^{α_{1}}(s_{1} − 1)^{β_{1}−1}(s_{1} − s_{2})^{γ_{1}}s_{2}^{α_{2}−1}(s_{2} − 1)^{β_{2}}(s_{2} − x)^{γ_{2}} ds_{1} ds_{2} \\ ∫∫_{Δ} s_{1}^{α_{1}−1}(s_{1} − 1)^{β_{1}}(s_{1} − s_{2})^{γ_{1}}s_{2}^{α_{2}}(s_{2} − 1)^{β_{2}−1}(s_{2} − x)^{γ_{2}} ds_{1} ds_{2} \\ ∫∫_{Δ} s_{1}^{α_{1}}(s_{1} − 1)^{β_{1}−1}(s_{1} − s_{2})^{γ_{1}}s_{2}^{α_{2}}(s_{2} − 1)^{β_{2}−1}(s_{2} − x)^{γ_{2}} ds_{1} ds_{2} \end{matrix})

mc_{μ_{1}} → a_{α^{(1)}} → mc_{μ_{2}} → ⋯ → a_{α^{(ℓ−1)}} → mc_{μ_{ℓ}}

mc_{μ_{j}} = L_{j} ∘ c_{μ_{j}}

c_{μ_{1}} → a_{α^{(1)}} → c_{μ_{2}} → ⋯ → a_{α^{(ℓ−1)}} → c_{μ_{ℓ}} → L Starting from a rank 1 equation

\frac{du}{dx} = (∑_{j=1}^{p} \frac{α^{(1)}}{x − a_{j}} ) u

Solution: u(x) = ∏_{j=1}^{p}(x − a_{j})^{α_{j}^{(1)}}

⟶^{mc_{μ_{1}}}  ⟶^{a_{α^{(2)}}}  ⟶^{mc_{μ_{2}}}  ⋯  ⟶^{a_{α^{(ℓ)}}}  ⟶^{mc_{μ_{ℓ}}}
⟶^{c_{μ_{1}}}  ⟶^{a_{α^{(2)}}}  ⟶^{c_{μ_{2}}}  ⋯  ⟶^{a_{α^{(ℓ)}}}  ⟶^{c_{μ_{ℓ}}}  ⟶^{L}

Thus we have a general form of solutions of rigid Fuchsian equations:

U(x) = L ∫_{Δ} ∏_{i=1}^{ℓ} ∏_{j=1}^{p}(s_{i} − a_{j})^{α_{j}^{(i)}} ∏_{i=1}^{ℓ−1}(s_{i} − s_{i+1})^{μ_{i}}(s_{ℓ} − x)^{μ_{ℓ}} (\frac{ds_{1} ∧ ⋯ ∧ ds_{ℓ}}{∏_{i=1}^{ℓ}(s_{i} − a_{j_{i}})} )_{1≤j_{1},...,j_{ℓ}≤p}

Generating system (of rank p^{ℓ})

\frac{dU}{dx} = (∑_{j=1}^{p} \frac{G_{j}}{x − a_{j}} ) U

where

(G_{1}, …, G_{p}) = c_{μ_{ℓ}} ∘ a_{α^{(ℓ)}} ∘ ⋯ ∘ a_{α^{(2)}} ∘ c_{μ_{1}}(α_{1}^{(1)}, …, α_{p}^{(1)})

eigenvalue

multiplicity

0  p^{ℓ} − p^{ℓ−1}
α_{j}^{(ℓ)} + μ_{ℓ}  p^{ℓ−1} − p^{ℓ−2}
(α_{j}^{(ℓ−1)} + μ_{ℓ−1}) + (α_{j}^{(ℓ)} + μ_{ℓ})  p^{ℓ−2} − p^{ℓ−3}
⋮ ⋮
(α_{j}^{(q)} + μ_{q}) + ⋯ + (α_{j}^{(ℓ)} + μ_{ℓ})  p^{q−1} − p^{q−2}
⋮ ⋮
(α_{j}^{(1)} + μ_{1}) + (α_{j}^{(2)} + μ_{2}) + ⋯ + (α_{j}^{(ℓ)} + μ_{ℓ})  1

Table Eigenvalues of G_{j} Solution:

U(x) = ∫_{Δ} ∏_{i=1}^{ℓ} ∏_{j=1}^{p}(s_{i} − a_{j})^{α_{j}^{(i)}} ∏_{i=1}^{ℓ−1}(s_{i} − s_{i+1})^{μ_{i}}(s_{ℓ} − x)^{μ_{ℓ}} (\frac{ds_{1} ∧ ⋯ ∧ ds_{ℓ}}{∏_{i=1}^{ℓ}(s_{i} − a_{j_{i}})} )_{1≤j_{1},...,j_{ℓ}≤p}

In the following we take the arrnagement of hyperplanes

D = ⋃_{{1≤i≤ℓ \atop 1≤j≤p}}\{s_{i} = a_{j}\} ∪ ⋃_{1≤i≤ℓ−1}\{s_{i} = s_{i+1}\} ∪ \{s_{ℓ} = x\}

in ℂ^{ℓ}, and set X = ℂ^{ℓ} \ D. We consider the multi-valued function

Φ(s) = ∏_{i=1}^{ℓ} ∏_{j=1}^{p}(s_{i} − a_{j})^{α_{j}^{(i)}} ∏_{i=1}^{ℓ−1}(s_{i} − s_{i+1})^{μ_{i}}(s_{ℓ} − x)^{μ_{ℓ}}

on X.

Remark: The solution of a generating system is a special kind of Selberg integral. A general one contains ∏_{i<j}(s_{i} − s_{j})^{μ_{ij}} instead of ∏_{i=1}^{ℓ−1}(s_{i} − s_{i+1})^{μ_{i}}.

Φ(s) = ∏_{i=1}^{ℓ} ∏_{j=1}^{p}(s_{i} − a_{j})^{α_{j}^{(i)}} \underline{\overline{∏_{i=1}^{ℓ−1}(s_{i} − s_{i+1})^{μ_{i}}}} (s_{ℓ} − x)^{μ_{ℓ}}

Topological Theory

L: the local system on X of local solutions of

∇z := dz + d log Φ(s) ∧ z = 0.

where we have defined

Φ(s) := ∏_{i=1}^{ℓ} ∏_{j=1}^{p}(s_{i} − a_{j})^{α_{j}^{(i)}} ∏_{i=1}^{ℓ−1}(s_{i} − s_{i+1})^{μ_{i}}(s_{ℓ} − x)^{μ_{ℓ}}.

We shall study

• cohomology group H^{i}(X, L) and contiguity relations • homology group H_{i}(X, L^{∨}) and asymptotic behaviros Twisted Cohomology Theorem 1. For generic values of parmaeters,

\begin{aligned}H^{i}(X, L) & = 0 (i ≠ ℓ) \\ H^{ℓ}(X, L) & = ⟨\frac{ds_{1} ∧ ⋯ ∧ ds_{ℓ}}{∏_{i=1}^{ℓ}(s_{i} − a_{j_{i}})} | j_{1}, j_{2}, …, j_{ℓ} ∈ \{1, 2, …, p\}⟩ . \end{aligned}

In the proof we use [Esnault-Schechtman-Vieweg], [Schechtman-Terao-Varchenko], and χ(X) = (−p)^{ℓ}.

Contiguity Relations

U_{j_{1}...j_{ℓ}}(x) ↔ \frac{ds_{1} ∧ ⋯ ∧ ds_{ℓ}}{∏_{i=1}^{ℓ}(s_{i} − a_{j_{i}})}

Let Ū_{j_{1}...j_{ℓ}}(x) be obtained by increasing α_{k}^{(m)} by 1.

\frac{s_{m} − a_{k}}{∏_{i=1}^{ℓ}(s_{i} − a_{j_{i}})} = \frac{s_{m} − a_{j_{m}}}{∏_{i=1}^{ℓ}(s_{i} − a_{j_{i}})} + \frac{a_{j_{m}} − a_{k}}{∏_{i=1}^{ℓ}(s_{i} − a_{j_{i}})}

= \frac{1}{∏_{i≠m}(s_{i} − a_{j_{i}})} + \frac{a_{j_{m}} − a_{k}}{∏_{i=1}^{ℓ}(s_{i} − a_{j_{i}})}

≡ ∑_{j′_{1},...,j′_{ℓ}} c_{j_{1},...,j_{ℓ}}^{j′_{1},...,j′_{ℓ}} \frac{1}{∏_{i=1}^{ℓ}(s_{i} − a_{j′_{i}})} + \frac{a_{j_{m}} − a_{k}}{∏_{i=1}^{ℓ}(s_{i} − a_{j_{i}})}

From which we obtain

Ū_{j_{1}...j_{ℓ}}(x) = ∑_{j′_{1},...,j′_{ℓ}} c_{j_{1},...,j_{ℓ}}^{j′_{1},...,j′_{ℓ}}U_{j′_{1}...j′_{ℓ}}(x) + (a_{j_{m}} − a_{k})U_{j_{1}...j_{ℓ}}(x)

Twisted Homology and Asymptotic Behavior We assume a_{1}, …, a_{p} ∈ ℝ and moreover

a_{1} < a_{2} < ⋯ < a_{p}

We are interested in the asymptotic behavior as x → a_{j}, and then we assume a_{j} < x < a_{j+1}

a^{1} a^{2}  a^{j}  a^{j+1}  a^{p−1} a^{p} x

Recall:

eigenvalues of G_{j} multiplicity

0  p^{ℓ} − p^{ℓ−1}
α_{j}^{(ℓ)} + μ_{ℓ}  p^{ℓ−1} − p^{ℓ−2}
⋮ ⋮
(α_{j}^{(q)} + μ_{q}) + ⋯ + (α_{j}^{(ℓ)} + μ_{ℓ})  p^{q−1} − p^{q−2}
⋮ ⋮
(α_{j}^{(1)} + μ_{1}) + (α_{j}^{(2)} + μ_{2}) + ⋯ + (α_{j}^{(ℓ)} + μ_{ℓ})  1

at x = a_{j}, the generating system has the exponent

ρ_{jq} := ∑_{k=q}^{ℓ}(α_{j}^{(k)} + μ_{k})

of multiplicity p^{q−1} − p^{q−2}.

Theorem 2. The exponent ρ_{jq} is realized by the following twisted cycles:

s_{i} : (1 ≤ i ≤ q − 2) s_{q−1} : s_{q}, …, s_{ℓ} :  a_{1} a_{1} a_{1} a_{2} a_{2} a_{2}  a_{j} a_{j} a_{j}  a_{j+1} a_{j+1} a_{j+1}  a_{p−1} a_{p−1} a_{p−1} a_{p} a_{p} a_{p} a_{j−1}  a_{j+2} s_{q} s_{q} ⋯ s_{ℓ−1}s_{ℓ} x  s_{i+1} ◯ ◯  × ×  ×  ◯  ◯ ◯ ◯  ◯  ◯
# of cycles = (p − 1) × p^{q−2} = p^{q−1} − p^{q−2}

We call them asymptotic cycles. cf.

Φ(s) = ∏_{i=1}^{ℓ} ∏_{j=1}^{p}(s_{i} − a_{j})^{α_{j}^{(i)}} ∏_{i=1}^{ℓ−1}(s_{i} − s_{i+1})^{μ_{i}}(s_{ℓ} − x)^{μ_{ℓ}}

These cycles are independent:

∫_{Δ} Φ(s)\frac{ds_{1} ∧ ⋯ ∧ ds_{ℓ}}{∏_{i=1}^{ℓ}(s_{i} − a_{j_{i}})} ≈ C(x − a_{j})^{ρ_{jq}} ∫_{Δ′} Φ′(s′)\frac{ds_{1} ∧ ⋯ ∧ ds_{q−1}}{∏_{i=1}^{q−1}(s_{i} − a_{j_{i}})} ,

where

\begin{matrix}π : ℂ^{ℓ} & → ℂ^{q−1}, \\ (s_{1}, …, s_{ℓ}) & ↦ (s_{1}, …, s_{q−1}) \end{matrix} Δ′ = π(Δ),

Φ′(s′) := ∏_{i=1}^{q−1} ∏_{k=1}^{p}(s_{i} − a_{k})^{α_{k}^{(i)}} ∏_{i=1}^{q−2}(s_{i} − s_{i+1})^{μ_{i}}(s_{q−1} − a_{j})^{μ_{q−1}}.

We can show that (∫_{Δ′} Φ′(s′)\frac{ds_{1} ∧ ⋯ ∧ ds_{q−1}}{∏_{i=1}^{q−1}(s_{i} − a_{j_{i}})} ) is of full rank.

Problems.

• Specify the asymptotic cycles when the parameters are resonant. (cf. K. Mimachi’s talk) Can we characterize such cycles using the terminology of arrangements of hyperplanes?

• Middle convolution may be defined also for completely integrable systems. We are interested in the change of the singular locus under middle convolution,

or

arrangements of hyperplanes which are invariant under an appropriate middle convolution.

dU = [∑_{i=1}^{q} ∑_{j=1}^{p} A_{ij} \frac{dx_{i}}{x_{i} − a_{j}} + ∑_{i≠i′} C_{ii′} \frac{d(x_{i} − x_{i′})}{x_{i} − x_{i′}}] U

We fix i. Then we can define a convolution w.r.t. x_{i} by

i
V_{j} = ∫_{Δ} \frac{1}{s − a_{j}} U(x_{1}, …,s, …, x_{q})(s − x_{i})^{μ} ds (1 ≤ j ≤ p)
V_{p+i′} = ∫_{Δ} \frac{1}{s − x_{i′}} U(x_{1}, …,s, …, x_{q})(s − x_{i})^{μ} ds (i′ ≠ i)
V = ^{t}(V_{1}, …, V_{p+q})

Then we can define the middle convolution in a similar way as ODE case, and get a completely integrable system with the same singular locus.