A simple counterexample to the Fuchsian version of the Hilbert 21st problem.

1

D.V.Anosov

1. Standing notations, definitions and several basic facts. Content of this section is completely covered by the book [1]. Perhaps this especially concerns notations, because essential facts can be found in other places as well (but in [1] they are collected together and presented in the form appropriate to our present goals). We begin from some of the notations.

S := \overline{ℂ} \ \{a_{1}, …, a_{n}\} is the punctured Riemann sphere, π : S̃ → S — its universal covering ^{2}, Δ — the group of deck transformations. Points of S̃ are denoted with “tilde”: x̃.

Assume we are given a linear system of ordinary differential equations which in the standard vector-matrix notations looks as

\frac{dy}{dx} = A(x)y,  y ∈ ℂ^{p},  (1)

where the matrix A(x) is holomorphic in S. Generally solutions of (1) are branching, but they are single-valued holomorphic functions on S̃. As a linear combination of solutions (with constant complex coefficients) is again a solution, the set 𝔙 of all solutions ^{3} is a complex linear vector space (which is p-dimensional). Given (1), one can associate to it a matrix ODE

\frac{dY}{dx} = A(x)Y (where Y is a (p × p) − matrix).  (2)

One deals mainly with those solutions of (2) for which the determinant det Y ≠ 0. (If it is ≠ 0 at some point, it is ≠ 0 everywhere.) Columns of such Y constitute a fundamental system of solutions of (1), i.e. a basis in 𝔙.

The system (1) is called Fuchsian if the entries of A(x) have only simple poles at all a_{i}. It is called regular if its solutions grow or decrease no faster than some power of |x − a_{i}| when x = π(x̃) → a_{i} provided that x remains inside some sector having the vertex in a_{i}. (The latter proviso is necessary, because the solution can branch logarithmically, and if in such case x → a_{i} along an appropriate spiral, one can get arbitrarily big growth of |y(x̃)|.) Fuchsian systems are regular, the converse is not true.

To any σ ∈ Δ one associates the map

σ^{∗} : 𝔙 → 𝔙 σ^{∗}y := y ∘ σ^{−1}  (3)

1This work was partially supported by the Göran Gustaffson foundation, Royal Inst. of Technology (Stokholm), and by the RFBR – DFG grant 436 RUS 113/73/3. Author expresses his gratitude to the staff of the Royal Inst. of Technology and of the Max Planck Inst. for Math (Bonn) for hospitality.

Extended version of this paper is supposed to appear in the ”Ulmer Seminare über Funktionalanalysis und Differentialgleichungen”.

2In [1] the map π was denoted by p, but p is also the number of equations in the system which we shall consider. Of course π has also the different meaning, namely \frac{22}{7}, but here the danger of confusion is less.

3In [1] gothic letters were not used, but it was for a technical reason only.

(in other words, (σ^{∗}y)(x̃) = y(σx̃) or (σ^{∗}y)(σx̃) = y(x̃)). The map

Δ → GL(𝔙) σ ↦ σ^{∗}

turns out to be a group representation. It is called the (abstract) monodromy representation for the system (1). We call it “abstract” because the space 𝔙 is a little bit abstract — a space of solutions; it is not so “concrete” as ℂ^{p} (although they are isomorphic). If we choose a basis Y = (y_{1}, …, y_{p}) in 𝔙 (speaking more classically, a fundamental system of solutions of (1)), the monodromy representation takes a more concrete form: we get a representation

χ : Δ → GL(p, ℂ) σ^{∗}Y; = (σ^{∗}y_{1}, …, σ^{∗}y_{p}) = Y χ(σ),  (4)

where the (p × p)-matrix χ(σ) acts from the right in a standard way on a row (y_{1}, …, y_{p}); this simply means that χ(σ) is the coefficient matrix of the linear transformation σ^{∗} in terms of coordinates defined by the basis (y_{1}, …, y_{p}). If we replace this basis by another one, the “concrete” monodromy representation χ is replaced by a conjugate representation. We say that χ arises from the system (1) or is realized by this system.

The 21st Hilbert problem (or the Riemann-Hilbert problem) has two versions: the regular and the Fuchsian one. (See [1] on the historical discussion.) One inquires whether there exists a regular (Fuchsian) system realizing the given monodromy representation χ. (As we speak about S, we tacitly assume that singularities a_{i} are also prescribed.) J.Plemelj proved many years ago that the answer to the regular version is always affirmative. Bolibruch discovered that the answer to the Fuchsian version can be negative. The goal of this note is to describe one of his examples which seems to be the simplest one. (As it often happens, the first example was somewhat more complicated.) This description is somewhat simplified comparatively to his original exposition.

The above general information was given in order to explain the general goal of this paper. In order to describe the example mentioned and to prove that it is really a counterexample, we need more background.

First of all, we need some local theory, i.e. theory concerning the behavior of solutions of (1) near singular points a_{i}. Denote by U_{i} a small open disk centered at a_{i} that does not contain other a_{j}; on occasion we can demand a little bit more. (If a_{i} = ∞, “small disk” means “\overline{ℂ} \ D where D is a big closed disk containing all other a_{i}”.) Denote by U_{i}^{∗} the punctured neighborhood U_{i} \ {a_{i}} and by Ũ_{i}^{∗} its universal covering surface. The group Δ_{i} of deck transformations of the covering Ũ_{i}^{∗} → U_{i}^{∗} is the infinite cyclic group generated by the deck transformation σ_{i} corresponding to one turn around a_{i} counterclockwise (if a_{i} = ∞, we take a clockwise turn). Denote by 𝔙_{i} the space of solutions of our system (1) considered only over U_{i}^{∗}, so that they are holomorphic functions on Ũ_{i}^{∗}. Analogously to what was said before, Δ_{i} acts on 𝔙_{i}, but in this case all reduces to the action of σ_{i} (and its iterates). If Y is a non-degenerate solution of (2), i.e. a fundamental system of solutions of (1), then

σ_{i}^{∗}Y = Y G_{i}^{−1} with some G_{i} ∈ GL(p, ℂ),  (5)

i.e. G_{i} = χ(σ_{i}^{∗−1}). It may seem more natural to take G_{i} = χ(σ_{i}^{∗}), but the definition (5) leads to the formula Y (σ_{i}x̃) = Y G_{i} which is classical. Define E_{i} := \frac{1}{2πi} ln G_{i}, where the choice of ln is normalized by demanding that if λ_{i}^{j} are eigenvalues of G_{i} and μ_{i}^{j} — of E_{i}, then

μ_{i}^{j} = \frac{1}{2πi} ln λ_{i} and 0 ≤ ℜμ_{i}^{j} < 1.

Besides this, for a regular system Levelt defines some integer numbers φ_{i}^{j}; i = 1, …, n; j = 1, …, p. Numbers φ_{i}^{1}, …, φ_{i}^{p} somehow characterize growth (or decreasing) of solutions near a_{i} or, to be more precise, the part of this growth which is described in terms of holomorphic functions while there can remain some growth related to fractional powers or logarithms which is not taken into consideration (so for x̃^{3/2} the corresponding φ is 1). For a fixed i numbers φ_{i}^{j} are numbered in such a way that φ_{i}^{1} ≥ … ≥ φ_{i}^{p} We shall refer to these φ_{i}^{j}, so to say, twice: they figure in I) below; numbers β_{i}^{j} := φ_{i}^{J} + μ_{i}^{j} which are defined in the case of a regular system and are called its exponents figure in II).

We shall use two theorems proved by Levelt (in the Fuchsian case which we only need first of them essentially was somewhat earlier proved by Gantmacher). One can find their proofs in [1].

I). For each a_{i} there exists such a basis Y_{i} in 𝔙 (i.e. such a non-degenerate solution of (2)) that in Ũ_{i}^{∗}

Y_{i}(x̃) = V_{i}(x)(x − a_{i})^{Φ_{i}}(x − a_{i}^{})^{E′_{i}},  (6)

where V_{i}(x) is holomorphic in the (non-punctured!) neighborhood U_{i} of a_{i}, Φ_{i} is the diagonal matrix with the diagonal elements φ_{i}^{1}, …, φ_{i}^{p} and E′_{i} is an upper-triangular matrix similar to E_{i} (thus E′_{i} also is the coefficient matrices of the linear transformation \frac{1}{2πi} ln σ_{i}^{∗}, but in another basis than E_{i}, — in the basis Y_{i}). ((x − a_{i}^{})^{E′_{i}} is interpreted in a natural way; see [1] where we wrote (x̃ − a_{i})^{E_{i}}.) Further, it turns out that Fuchsian case is just the case when det V_{i}(a_{i}) ≠ 0. We call Y_{i} a Levelt basis for (near) a_{i}.

For our current purposes we can understand I) as the statement about the existence of Y_{i}, V_{i}, Φ_{i} and E′_{i} with the properties indicated. From this point of view the only thing which is stated about Φ_{i} is that it is a diagonal matrix with integer entries ordered in such a way that φ_{i}^{j} decrease when j increases and that Φ_{i} is defined in a unique way (while there can be some freedom in the choice of Y_{i}, V_{i} and E′_{i}). Then we consider φ_{i}^{j} as being by definition diagonal elements of Φ_{i} which existence is asserted in I). The fact that they coincide with some numbers defined in another way can be considered as a separate theorem (which we do not need).

II). For any regular system

∑_{i,j} β_{i}^{j} ≤ 0 and ∑_{i,j} β_{i}^{j} = 0 ⇔ the system is Fuchsian.  (7)

The latter theorem is not of the purely local character — it involves β_{i}^{j} for various a_{i}, —- although its proof does not require the serious discussion of connections between local and global theories. However, for some other purposes this discussion is necessary, and we shall turn to it.

One has to tie Ũ_{i}^{∗} and S̃. If n > 2, π^{−1}U_{i}^{∗} (which is a subset of S̃) has infinitely many connected components which are all isomorphic (as a spaces covering U_{i}^{∗}) to Ũ_{i}^{∗}. After we identify one of these components with Ũ_{i}^{∗}, we can consider for y ∈ 𝔙 their restrictions on Ũ_{i}^{∗} and in this way we get a map

𝔙 → 𝔙_{i} y ↦ y|Ũ_{i}^{∗}

which, of course, is an isomorphism of two p-dimensional complex vector spaces. (This linear map is non-degenerate because the only holomorphic function on S̃ which is identically zero on Ũ_{i}^{∗} is 0 everywhere.) After this we can consider σ_{i}^{∗} as acting on 𝔙. Once more: this interpretation of σ_{i}^{∗} as elements of GL(𝔙) depends upon the choice of the identification of Ũ_{i}^{∗} with one of the connected components of π^{−1}(U_{i}^{∗}).

In order to describe various identifications in more detail ^{4}, it is useful to appeal to the continuous paths in S. The product αβ of two such paths is defined in a well-known way, if the end of α coincides with the beginning of β. (We simply pass first α, then β.) If α and β vary continuously, but their endpoints remain fixed, αβ also varies continuously with the fixed endpoints, thus the multiplication is correctly defined for some classes each of which consists of paths which are mutually homotopy equivalent as paths with fixed endpoints; one can speak about the product [α][β] of such classes if the end of the paths from [α] (this end is the same for all paths from this class, that is why we say “end”, not “ends”) coincides with the beginning of the paths from [β]. Fix a base point b ∈ S and consider classes of all paths beginning at b. It is well known that one can introduce a topology in this set so that this set together with its evident projection on S (the projection maps every class simply to the end of its paths) becomes the universal covering space S̃. In these terms the fundamental group π_{1}(S, b) acts on the S̃ from the left as

([α], [β]) ↦ [αβ].

It turns out that in this way one gets all the deck transformations thus establishing an isomorphism π_{1}(S, b) ≈ Δ. Identification of S̃ with the space of classes of paths to some extent depends on the choice of b and for this reason generally one does not get a unique isomorphism π_{1}(S, b) = Δ which would be the same for all b. It is well known that various isomorphisms defined by this way differ by the conjugation: if via our construction we get isomorphisms

j_{1} : π_{1}(S, b_{1}) → Δ, j_{2} : π_{1}(S, b_{2}) → Δ,

then there exists such path α from b_{1} to b_{2} that for all [β] ∈ π_{1}(S, b_{1})

j_{2}([β]) = j_{1}([α][β][α]^{−1}).

Analogously, if we fix a base point b_{i} ∈ U_{i}^{∗}, then we get an isomorphism π_{1}(U_{i}, b_{i}) = Δ_{i}. Because this group is commutative, now the isomorphism does not depend on various choices made in the process of construction. Now, if we choose a class of paths α_{i} beginning in b and ending at b_{i}, we get an inclusion

Ũ_{i}^{∗} → S̃ [β] ↦ [α_{i}][β].

Note that σ_{i} corresponds to a path β_{i} in U_{i}^{∗} from b_{i} to b_{i} going around a_{i} and making one turn in positive direction (counterclockwise for a_{i} ≠ ∞ and clockwise for a_{i} = ∞) or, to be more precise, to the class [β_{i}]. Namely, for [γ] ∈ Ũ_{i}^{∗} (which means that γ lies in U_{i}^{∗} and begins in b_{i})

4See also [1].

σ_{i}[γ] = [β_{i}][γ].

Now the natural way to make σ_{i} acting on S̃ is to pass to the element [α_{i}][β_{i}][α_{i}]^{−1} ∈ π_{1}(S, b) and to consider its action on S̃. In any case, for elements of Ũ_{i}^{∗} considered as being inclosed to S̃ (with the aid of the same [α_{i}], of course) this leads to the same result as before. After we replaced σ_{i} by elements of Δ which we shall still denote by σ_{i}, we can take usual σ_{i}^{∗}. They turn out to be the same σ_{i}^{∗} as considered earlier.

Various inclusions Ũ_{i}^{∗} ↪ S̃ constructed in this way (with various [α_{i}]) can be obtained from a fixed inclusion by deck transformations. Namely, if the previous inclusion (obtained via using [α_{i}]) is j and another inclusion (obtained via using [α′_{i}]) is j′, then for all x̃ ∈ Ũ_{i}^{∗}

j′(x̃) = δj(x̃), where δ = [α′_{i}][α_{i}]^{−1}

(expression for δ is written having in mind the previous identification of Δ with π_{1}(S, b)). If we use a third path α′′_{i} = δ′α_{i} and get the inclusion j′′, then

j′Ũ_{i}^{∗} = j′′Ũ_{i}^{∗} ⇔ δ′ ∈ δΔ_{i}

(having in mind the previous Δ_{i} ↪ Δ), so the connected components of π^{−1}U_{i}^{∗} can be “numbered” by the elements of Δ/Δ_{i}. Various inclusions Δ_{i} ↪ Δ constructed in this way can be obtained from each other by conjugations in the group Δ.

Geometrically a rather pictorial description of Δ and π_{1}(S, b) is the following. Let first all a_{i} ≠ ∞. Take b ∈ ℂ outside of all direct lines a_{i}a_{j}. Then one can number our a_{i} in such a way that if we take a ray with the end in b and passing through b_{1} and rotate it counterclockwise making one turn around a_{i}, then it meets points a_{2}, …,a_{n} in the same order in which they are numbered. Take b_{i} on this ray when it passes through a_{i} between b and a_{i}, and take α_{i} to be the path from b to a_{i} along the interval ba_{i}. If one of a_{i} is ∞, this is modified as follows. We can assume that ∞ is a_{n}. Choice of b, the numbering of remaining points a_{1}, …, a_{n−1}, points b_{1}, …, b_{n−1} and paths α_{1}, …, α_{n−1} are as before with n − 1 instead of n. Take a ray with the vertex at b lying in the angle which is bounded by the rays ba_{1} and ba_{n−1} and which does not contain other points a_{i}. Take a point b_{n} lying on this ray sufficiently far (outside the disk D figuring in U_{i}‘n^{∗} = \overline{ℂ} \ D) and take α_{n} to be a path from b to b_{n} along bb_{n}. Under such choice of the objects figuring in our construction, π_{1}(S, b) can be described as the group generated by σ_{1},…,σ_{n} which satisfy the only relation

σ_{1} … σ_{n} = 1

(thus Δ is the free group with n − 1 generators). In order to define its representation in GL(p, ℂ), one has to choose n non-degenerate matrices G_{1}, …, G_{n} satisfying

G_{n} … G_{1} = I  (8)

(where I is the identity matrix; the identity transformation will be also denoted by I) and let χ(σ_{i}) := G_{i}^{−1}.

A representation is called a B-representation (in honor of Bolibruch; this was suggested by Yu.S.Il’yashenko), if

a) χ is reducible;

b) χ(σ_{i}) are similar to one Jordan block. (Condition b) does not depend upon a concrete choice of inclusion Δ_{i} ↪ Δ: different inclusions are conjugate to each other and so matrices χ(σ_{i}) for two different inclusions are similar, thus they have the same Jordan normal form.) In this case we can write λ_{i} instead of λ_{i}^{j}.

A regular system is called B-system if its monodromy representation is a B-representation. We shall prove

Theorem. For a Fuchsian B-system

λ_{1} … λ_{n} = 1.  (9)

This theorem is due to A.A.Bolibruch [2]. The proof given below is essentially the same, but its exposition seems to be somewhat simplified.

Then in section 3 we shall give an example of a B-representation for which this product is ≠ 1. Such representation cannot be realized by a Fuchsian system, independently of concrete choice of a_{i}.

Remarks. 1). There exist other examples where some representation can be realized by a Fuchsian system for one choice of a_{i} and cannot for another.

2). This theorem gives a necessary condition for a B-representation to be realizable as a monodromy representation for a Fuchsian system. This condition is not sufficient. If p = 3, then any B-representation satisfies this condition, while there exist non-realizable B-representations with p = 3. Indeed, such was the first counterexample discovered.

3). In the case p = 3, all the representations Δ → GL(p, ℂ) which cannot be realizable as monodromy representations of Fuchsian systems are B-representations. In this case necessary and sufficient conditions for realizability are known [1].

4). There exist representations (with p > 3) which are non-realizable and are not Brepresentations. For p = 4 such example was given in [3]. It can be easily extended to any bigger p.

2. Proof of the theorem. Assume we have a B-representation χ which is realized by a Fuchsian system (1) (under some choice of the fundamental system of solutions Y of the latter). Then in the space 𝔙 we have the equivalent representation σ ↦ σ^{∗} (see (3) and (4)).

a) means that in the space 𝔙 there exists a subspace 𝔜 which is invariant with respect to the representation σ ↦ σ^{∗}. Let l := dim 𝔜. Let y_{1}, …, y_{p} be a basis in 𝔙 such that y_{1}, …, y_{l} ∈ 𝔜. Y^{1} := (y_{1}, …, y_{l}) is a row consisting of l elements which are vectors from ℂ^{p}, i.e. columns consisting of p elements (the latter being some functions of x̃; Y^{1}(x̃) = (y_{1}(x̃), …, y_{l}(x̃)) is a p × l-matrix consisting of numbers). In this basis χ has the form

χ(σ) = ( \begin{matrix}χ_{1}(σ) & ∗ \\ 0 & ∗ \end{matrix} )

where χ_{1} is also a representation of the deck transformations group — this times a representation by (p × p)-matrices. Then

Y^{1}(σx̃) = Y^{1}(x̃)χ_{1}(σ^{−1}).  (10)

b) implies that for any i there exists a filtration in 𝔙 invariant with respect to σ_{i}^{∗}:

{0} = 𝔙_{i}^{0} ⊂ 𝔙_{i}^{1} ⊂ … ⊂ 𝔙_{i}^{p} = 𝔙

such that dim 𝔙_{i} = i. Namely,

𝔙_{i}^{j} = Ker (σ_{i}^{∗} − λ_{i}I)^{j}.

The following statement is a general statement for linear operators A : 𝔙 → 𝔙 which are similar to one Jordan block. Let ℨ be a k-dimensional invariant subspace for A. Then ℨ = Ker (A − αI)^{k}, where α the the eigenvalue of A. Indeed, (λ − α)^{k} is the characteristic polynomial of A|ℨ. (It must divide the characteristic polynomial (λ − α)^{p} of A and be of degree k.) Thus

(A − αI)^{k}ℨ = 0 and so ℨ ⊂ Ker (A − αI)^{k}.

But the latter spaces are both k-dimensional.

In particular, 𝔜 = 𝔙_{i}^{l}, — both are l-dimensional subspaces invariant for σ_{i}^{∗}.

Let y_{i1}, …, y_{ip} be a Levelt basis for a_{i}. Then ∑_{j=1}^{l} ⊕ℂy_{ij} is an l-dimensional invariant subspace, so

∑_{j=1}^{l} ⊕ℂy_{ij} = 𝔙_{i}^{l} = 𝔜 = ∑_{j=1}^{l} ⊕ℂy_{j},

and for all i

(y_{1}, …, y_{l}) = (y_{i1}, …, y_{il})C_{i} with some constant matrix C_{i}.

For a polyvector w ∈ Λ^{l}(ℂ^{p}) denote by w^{i_{1}...i_{l}} its coordinates in the basis

{e_{j_{1}} ∧ … ∧ e_{j_{l}}}, 1 ≤ j_{1} < … < j_{l} ≤ p

of the space Λ^{l}(ℂ^{p}); here e_{1}, …, e_{p} is the standard basis of ℂ^{p}. If the polyvector is a simple (decomposable) one, say w = w_{1} ∧ … ∧ w_{l}, then the coordinates

w^{i_{1}...i_{l}} = det (w_{j}^{i_{h}})j, h = 1, …, l

(where w_{i}^{j} are the coordinates of w_{i}) are the Plücker coordinates of the space ∑ ⊕ℂw_{i} (spanned by w_{1}, …, w_{l}). For any x̃ ∈ S̃ we have already introduces some Y^{1}(x̃) = (y_{1}(x̃), …, y_{l}(x̃)); introduce now the polyvector

w(x̃) := y_{1}(x̃) ∧ … ∧ y_{l}(x̃)

with coordinates w^{i_{1}...i_{l}}(x̃). Note that

w^{j_{1}...j_{l}}(σx̃) = w^{j_{1}...j_{l}}(x̃) det χ_{1}(σ^{−1}).

Let w^{j_{1}...j_{l}} ≢ 0. If w^{j_{1}...j_{l}}(x̃) = 0, then all w^{j_{1}...j_{l}}(σx̃) = 0. Thus singular points and zeroes for w^{j_{1}...j_{l}} are: possibly a_{1}, …, a_{n} and perhaps some other points

\underbrace{b_{1}, …, b_{m}}_{only zeroes} .

(We write b_{j}, not b̃_{j}, because zeroes are in all points of S̃ covering b_{j}.) Now

d ln w^{j_{1}...j_{l}}(σx̃) = d ln w^{j_{1}...j_{l}}(x̃) + \underbrace{d const}_{=0},

and we see that the form

ω := d ln w^{j_{1}...j_{l}}(x̃) = 1-form on S′ := S \ {b_{1}, …, b_{m}}.

ω is a form with only finite number of singularities (all of them being poles). Sum of all residues of such form equals to zero:

∑ res_{a_{i}}ω + ∑ res_{b_{j}}ω = 0.

Let k_{j} := res_{b_{j}}ω; this number equals to the order of zero of w^{j_{1}...j_{l}} at the point b_{j}. All k_{j} ≥ 0. Now consider res_{a_{i}}ω. Since some fundamental system of solutions (the system which is Levelt system at a_{i}) has the form (6) with E′_{i} having all diagonal elements equal to μ_{i}, we have

(y_{i1}(x̃), …, y_{il}(x̃)) = (v_{i1}(x), …, v_{il}(x))T,

where T is an upper-triangular matrix with diagonal elements t_{jj} = (x − a_{i}^{})^{φ_{i}^{j}+μ_{i}}. Then

y_{i1} ∧ … ∧ y_{il} = t_{11} … t_{ll}v_{i1} ∧ … ∧ v_{il}.

Consequently, in Ũ^{∗} y_{1} ∧ … ∧ y_{l} = cy_{i1} ∧ … ∧ y_{il} =

= c(x − a_{i}^{})^{φ_{i}^{1}+...+φ_{i}^{l}+lμ_{i}}v_{i1} ∧ … ∧ v_{il}

(with c = const.)

Assume (v_{i1} ∧ … ∧ v_{il})^{j_{1}...j_{l}} has a zero of order l_{j} at a_{i}. Then

d ln(y_{i1} ∧ … ∧ y_{il})^{j_{1}...j_{l}} = (l_{i} + φ_{i}^{1} + … + φ_{i}^{l} + lμ_{i})\frac{dx}{x} + …,

res_{a_{i}}ω = l_{i} + φ_{i}^{1} + … + φ_{i}^{l} + lμ_{i},  (11)

If at least one of the numbers k_{i} or l_{j} is > 0, then

0 > ∑(φ_{i}^{1} + … + φ_{i}^{l} + lμ_{i}).

But in the Fuchsian case ∑_{i,j} β_{i}^{j} = 0, and as β_{i}^{j} = φ_{j} + μ_{i}, it follows that

\frac{1}{p} ∑(φ_{i}^{1} + … + φ_{i}^{p} + pμ_{i}) = 0.  (12)

Left hand side of (12) is ≤ \frac{l}{p} ∑(φ_{i}^{1} + … + φ_{i}^{l} + lμ_{i}), as φ_{i}^{j} decrease as j increases. Here strict inequality holds, if there exists such i that φ_{i}^{1} ≠ φ_{i}^{p}. Indeed, in order to prove that

φ_{i}^{1} ≠ φ_{i}^{p} ⇒ left hand side < 0,

write the equivalent formulas

p(φ_{i}^{1} + … + φ_{i}^{l}) > l(φ_{i}^{1} + … + φ_{i}^{p}),  (?)
(p − l)(φ_{i}^{1} + … + φ_{i}^{l}) > l(φ_{i}^{l+1} + … + φ_{i}^{p}),  (?)
∑_{j=1}^{l}[(p − l)φ_{i}^{j} − φ_{i}^{l+1} − … − φ_{i}^{p}] > 0,  (?)

All summands are ≥ 0 and the first summand […] is > 0.

So, if either one of the numbers k_{i} or l_{j} is > 0 or one of φ_{i}^{1} ≠ φ_{i}^{p}, we arrive at the contradiction. We have proved that in the Fuchsian case all k_{i} = 0 (i.e. ω has no zeroes outside of the set of singular points), all l_{i} = 0 (i.e.

(u_{i1} ∧ … ∧ u_{il})^{j_{1}...j_{l}} ≠ 0 at a_{i})

and all φ_{i}^{j} depend only on i. We shall denote the latter numbers simply by φ_{i}. We already know that μ_{i}^{j} = μ_{i} depend only on i, thus the same is correct for β_{i}^{j} and we can write simply β_{i}.

Now (7) reduces to ∑ p(φ_{i} + μ_{i}) = 0, i.e. ∑(φ_{i} + μ_{i}) = 0. As φ_{i} are integers, ∑ μ_{i} ∈ ℤ. But λ_{i} = e^{2πiμ_{i}}, and we arrive at (9).

Remark. Considering the Plücker coordinates w^{j_{1}...j_{l}}(z̃) of the linear subspace

𝔜(x̃) := ∑_{i=1}^{l} ⊕ℂy_{i}(x̃)

one can also prove that in our case 𝔜(x̃) = const (does not depend on x̃). This is not needed here, but this is an interesting general property of Fuchsian B-systems, and it is used in the proofs of the non-realizability of some B-representations.

If we are going to prove that 𝔜(x̃) = const, it is natural to use the Plücker coordinates of this space and (considering them as functions of x̃) to pay attention to their zeroes and singularities. In the proof of our theorem it does not seem to be so natural. It seems as if the proof arised as a by-product of the proof of the statement 𝔜(x̃) = const (although actually the history was more complicated). However, if we take the statement 𝔜(x̃) = const as granted, we can prove our theorem in a somewhat different way, just appealing to the ”⇐” part of (7) which is applied to our original system (Fuchsian system realizing χ) and also to the l-th order subsystem which after an appropriate change of coordinates in ℂ^{p} allowing to consider 𝔜(x̃) as the space y_{l+1} = … = y_{p} = 0 splits off from this system and which realizes the representation χ_{1}. But in the proof of the ”⇐” part of (7) one considers the differential 1-form d ln det Y (x̃) which for the new (smaller) system is just d ln w^{1...l}(x̃); so actually in this argument we also use our d ln w^{i_{1}...i_{l}}(x̃), only this is hidden in the proof of (7) and only one special array of i_{1}, …, i_{l} is taken.

3. An example. Finally, we shall give an example of the B-representation violating (9). In this example n = 3, p = 4. χ is completely defined by giving three matrices G_{1}, G_{2}, G_{3} such that G_{3}G_{2}G_{1} = I. All three matrices will have the following block structure with (2 × 2)-blocks:

( \begin{matrix}A & ∗ \\ 0 & A \end{matrix} ) .

So representation will be reducible with 𝔜 = ℂe_{1} ⨁ ℂe_{2}. For G_{1} we write A, for G_{2} — B instead of A and for G_{3} — C. A, B, C will be similar to two-dimensional Jordan blocks with eigenvalue, correspondingly, -1, 1, 1. G_{i} will have the same eigenvalues, and (9) certainly will be violated. For “generic” blocks ∗ matrix G_{i} − λ_{i}I will have rank 3, so we have a good chance to find suitable blocks ∗. Finally, we must ascertain that

G_{3}G_{2}G_{1} = I.  (13)

We have

( \begin{matrix}C & ∗ \\ 0 & C \end{matrix} ) ( \begin{matrix}B & ∗ \\ 0 & B \end{matrix} ) ( \begin{matrix}A & D \\ 0 & A \end{matrix} ) = ( \begin{matrix}CBA & CBD + ∗ \\ 0 & CBA \end{matrix} ) ,

where we need not write explicitly ∗’s; we need only to observe that in CBD + ∗ the ∗ does not depend on F. Thus (13) is equivalent to

CBA = I  (14)

and

CBD + ∗ = 0.  (15)

If the first of them is satisfied, we can satisfy the second one taking D = −(CB)^{−1}∗. Of course one has also to check whether G_{i} will be equivalent to one Jordan block, but, as it was already said, other cases which can happen seem to be exceptional, so more or less any choice of ∗ in G_{1} and G_{2} has a good chance to be successful.

The problem of finding (2×2)-matrices A, B, C satisfying the above-mentioned conditions is an exercise on linear algebra which is perhaps different from the exercises usually given to the students and which in this sense is unusual, but which is not a very difficult one. And the choice of the upper right blocks of two matrices can be more or less accidental (for the third matrix the corresponding block will be then uniquely defined). I shall not dwell on it and simply write the matrices G_{i} which were suggested by Bolibruch:

G_{3} = ( \begin{matrix}1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{matrix} )

(so G_{3} is already a Jordan block with the eigenvalue 1),

G_{2} = ( \begin{matrix}3 & 1 & 15 & 4 \\ −4 & −1 & −19 & −5 \\ 0 & 0 & 3 & 1 \\ 0 & 0 & −4 & −1 \end{matrix} ) ,

G_{1} = ( \begin{matrix}−1 & 0 & 0 & 1 \\ 4 & −1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 4 & −1 \end{matrix} ) .

Thus we take

A = ( \begin{matrix}−1 & 0 \\ 4 & −1 \end{matrix} ) , B = ( \begin{matrix}3 & 1 \\ −4 & −1 \end{matrix} ) , C = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) .

It is almost evident that A, B, C have the desired Jordan normal form; it is easy to check that (14) holds; checking of (15) needs more calculations, but of course they are made automatically. Finally, it is easy to see that all Rk (G_{i} − λ_{i}I) = 3.

References

[1] D.V.Anosov, A.A.Bolibruch, The Riemann-Hilbert Problem. — A Publication from the Steklov Institute of Mathematics. Vieweg, Braunschweig; Wiesbaden, 1994, pp.1-190.

[2] A.A.Bolibruch, Holomorphic Bundles, Associated with Linear Differential Equations and the Riemann–Hilbert Problem, in : B.L.J.Braaksma, G.K.Immink, M.van der Put (editors), The Stokes Phenomenon and Hilbert’s 16th Problem, Groningen, The Netherlands 31 May–3 June 1995. World Scientific, Singapore-New Jersey-London-Hong Kong, p. 51–70.

[3] A.A.Bolibruch, On the problem of Existence of Fuchsian Systems with Given Asymptotics, in: Proc. of the Steklov Inst. of Math., v. 216, 1996, p. 26–37.