Linear differential equations

Contents

1 Holomorphic systems of first order LDE on complex domains 21.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Basic existence and uniqueness theorem . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Wronskian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Fundamental solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Convergence of formal solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.6 Analytic continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Monodromy and Riemann-Hilbert correspondence 82.1 Monodromy of holomorphic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Holomorphic gauge transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Local systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Holomorphic vector bundles and connections . . . . . . . . . . . . . . . . . . . . . . 132.5 Integrable connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.6 Riemann-Hilbert correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Meromorphic systems, local theory 183.1 Some terminology for meromorphic systems . . . . . . . . . . . . . . . . . . . . . . . 183.2 Fuchsian singularities and their classification . . . . . . . . . . . . . . . . . . . . . . 203.3 Regular singularities of meromorphic systems . . . . . . . . . . . . . . . . . . . . . . 253.4 Regular singularities of differential equations . . . . . . . . . . . . . . . . . . . . . . 273.5 Cyclic vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4 Fuchsian systems over P1, global theory 304.1 Fuchsian systems over P1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2 Local and global monodromy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.3 The Riemann-Hilbert problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1 Holomorphic systems of first order LDE on complex domains

Main references for this lecture:

? Wasow, Asymptotic Expansions for ordinary differential equations, chapter 0 [16].? Mitschi/Sauzin, Divergent Series, Summability and Resurgence I, chapter 1 [10].

1.1 Definitions

Definition 1.1: A holomorphic system of linear differential equations is a differential equationof the form

Y ′ = AY ,

where A ∈Mr×r(O(U )) is a square matrix whose coefficients are holomorphic functions on somecomplex domain U ⊂C (open, non-empty).A solution of Y ′ = AY over some open set U ′ ⊂ U is a vector-valued holomorphic functionY ∈Mr×1(O(U ′)) such that

Y ′(x) = A(x)Y (x) ∀x ∈U ′ .

Two closely related types of ordinary differential equations are the following :

• Higher order linear differential equations of the formy(r) = a0y + a1y

′ + · · ·+ ar−1y(r−1) ,where the ai ’s are holomorphic functions on U , are equivalent to particular holomorphicsystems Y ′ = AY , where the matrix A is of the form

A =

0 1 . . . 0

. . .. . .

.... . . 1

a0 a1 . . . ar−1

.Indeed, Y = t(y1, . . . , yr ) is a local solution of the latter if and only if y′i = yi+1 for all i ∈{1, . . . , r − 1} and y = y1 is a local solution of the former.

• Riccati equations are non-linear differential equations of the formy′ = ay2 + by + c , (1)

where a,b,c are holomorphic functions on U . Riccati equations can be seen as projectivizedversions of holomorphic systems of rank r = 2. Indeed, if Y = t(y1, y2) is a local solution of

Y ′ =(b2 + d c−a b2 + d

)Y ,

(with arbitrary holomorphic d), then y := y1y2 is, where it is defined, a local solution of (1).

Remark. In general, for all of the above types of differential equations, the solutions cannot be computedexplicitely in terms of usual functions. However, when one solution ϕ of the Riccati equation

y′ = ay2 + by + c

is given, the others can be found by considering the change of variable y = 1ŷ +ϕ. Indeed, the resultingdifferential equation is

ŷ′ = −a− (2aϕ + b)ŷ .

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1.2 Basic existence and uniqueness theorem

Theorem 1.2: Let A ∈Mr×r(O(U )), let x0 ∈U and let Y0 ∈Mr×1(C).There exists a unique germ of local solution Y of Y ′ = AY near x0 such that Y (x0) = Y0.

Proof. Note that several arguments of this proof will be useful for our later discussions.

• First, we show that there exists a unique formal power series solution at x0 with constantterm Y0.

LetA =

∑k≥0

Ak(x − x0)k

be the power series expansion of A at x0. Let

Y =∑k≥0

Yk(x − x0)k

be a formal power series with Yk ∈ Mr×1(C). It is formally solution of Y ′ = AY if andonly if

Yk+1 =1

k + 1

k∑`=0

Ak−`Y` ∀k ∈N . (2)

• Next, we show for the above formal Y , that RC(Y ) > 0, i.e. that Y has positive radius ofconvergence.

– Here and thereafter, we choose, for any (n,m) ∈ N∗, a norm || · || on Mm×nC, such thatthis collection of norms is submultiplicative: we have

||M ·N || ≤ ||M || · ||N ||

for all matricesM,N for which the productMN is defined. We may choose for example

||(mij )i,j || =∑i,j

|mij |.

Now from (2), we have

||Yk+1|| ≤k∑`=0

||Ak−` || ||Y` || ∀k ∈N . (3)

– We introduce a control-function

ζ(x) :=||Y0||

1− (x − x0)α(x), where α(x) =

∞∑k=0

||Ak ||(x − x0)k .

Note that RC(α) = RC(A), where the radius of convergence of A is by definition thesmallest one among the radii of convergence of the coefficients of A. In particular, α is

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holomorphic in a neighborhood of x0. We deduce that ζ is holomorphic in a neighbor-hood of x0. Moreover, we can compute its power series expansion

ζ =∑k≥0

ζk(x − x0)k .

It must satisfy ζ0 = ||Y0|| and

ζk+1 =k∑`=0

||Ak−` ||ζ` ∀k ∈N . (4)

– By induction, from (3) and (4), one deduces that ||Yk || ≤ |ζk | for all k ∈N. We concludethat

RC(Y ) ≥ RC(ζ) > 0.

Remark. The much more general so-called Cauchy-Kowalewski theorem states a similar result forholomorphic partial differential equations (in several variables), that are not necessarily linear, and thatmay be of higher order. See for example [15, p. 445] or [3, Chap 5, §15] for a precise statement.

1.3 Wronskian

Proposition 1.3: Let Y ′ = AY be a holomorphic system of rank r over U and let φ1, . . . ,φr be rsolutions over U . Consider the matrix Ω := (φ1| . . . |φr ). Then the holomorphic function

ω := det(Ω)

is a solution of the first order holomorphic differential equation

ω′ = tr(A)ω.

Proof. Let us denote the coefficients, columns and lines of Ω as follows:

Ω = (ωij )1≤i,j≤r =(φ1 . . . φr

)=

ψ1...ψr

.Since for each φi , we have φ′i = Aφi , the matrix Ω satisfies Ω

′ = AΩ. In particular,ψ′1...ψ′r

=

∑ni=1 a1iψi...∑n

i=1 ariψi

.By definition of the determinant, we have

det(Ω) =∑σ∈Sr

sgn(σ )ω1,σ (1) · · ·ωr,σ (r).

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For ω = det(Ω) we therefore obtain

ω′ =∑σ∈Sr sgn(σ )ω

′1,σ (1) · · ·ωr,σ (r) + . . .+

∑σ∈Sr sgn(σ )ω1,σ (1) · · ·ω

′r,σ (r)

= det

ψ′1...ψr

+ . . .+ detψ1...ψ′r

= ∑ni=1 aii detψ1...ψr

= tr(A)ω.

1.4 Fundamental solutions

Definition 1.4: Let Y ′ = AY be a holomorphic system of rank r over U . Let U ′ ⊂ U be open.A fundamental solution of Y ′ = AY over U ′ is a matrix-valued holomorphic function Φ ∈Mr×r(O(U ′)) such that

Φ ′(x) = A(x)Φ(x) ∀x ∈U ′ ,

and moreoverdet(Φ)(x) , 0 ∀x ∈U ′ .

Exercise 1. Assume there is a fundamental solution Φ of Y ′ = AY over U ′ and assume that U ′ isconnected. Show that

{solutions over U ′} = { Φ ·C | C ∈Mr×1C }

{fundamental solutions over U ′} = { Φ ·M |M ∈GLrC } .

Proposition 1.5 (Existence of local fundamental solutions):Let Y ′ = AY be a holomorphic system of rank r over U . Let x0 ∈ U and let D be the largest disccentered at x0 that is contained in U .Then there exists a fundamental solution Φ of Y ′ = AY over D.

Proof.

• Existence on a small disc D′.By the basic existence and uniqueness theorem, there exist solutions φ1, . . . ,φr of Y ′ = AYnear x0 such that for Φ := (φ1| . . . |φr ) we have Φ(x0) = Ir . In particular, for some small discD′ around x0, we have Φ ∈ Mr×r(O(D′)), as well as Φ ′ = AΦ over D′. By the Wronskianproposition, det(Φ) is of the form ce

∫trA. Since moreover det(Φ)(x0) = det(Ir ) = 1 , 0, we

have det(Φ)(x) , 0 for all x ∈D′. Hence Φ is a fundamental solution on D′.

• Extension to D.Assume for a contradiction that the greatest possible disc D′ that can be obtained in the firstpart of the proof is not already equal to D. Then there exists a point x1 in the boundary ofD′ such that Φ cannot be analytically continued to a neighborhood of x1. But there existsa disc ∆ around x1 contained in U over which there is a fundamental solution Φ̃ . On theintersection ∆∩D′, we must have Φ = Φ̃ ·M for some M ∈GLrC. Since Φ̃ ·M is holomorphicon ∆, Φ extends holomorphically to D′ ∪∆. This yields a contradiction.

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1.5 Convergence of formal solutions

We have already seen that for holomorphic systems, formal solutions exist and converge. It turnsout that for mermomorphic systems with only simple poles, formal solutions, when they exist,also converge.

Let A ∈Mr×r(O(U )) and x0 ∈ U . Consider now the meromorphic system with a simple pole atx0 given by

Y ′ =A

x − x0Y .

Existence and uniqueness of local solutions with prescribed initial conditions can only be guaran-teed near points in U \ {x0}. However, we have the following very useful result.

Theorem 1.6: Let A =∑k≥0Ak(x − x0)k with Ak ∈Mr×r(C) be a matrix power series with radius

of convergence R > 0. LetY =

∑k≥0

Yk(x − x0)k

with Yk ∈ Mr×1(C) be a formal power series satisfying (x − x0)Y ′ = AY . Then the radius ofconvergence of Y is at least R.

Sketch of proof. Let D be the largest disc centered at x0 which is contained in U .

• If A =∑∞k=0Ak(x − x0)k for all x ∈D, and Y is a formal solution of (x − x0)Y ′ = AY , then one

must have

[(k + 1)Ir −A0]Yk+1 =k∑`=0

Ak+1−`Y`

for all k ≥ 0.

• There exists k0 ∈N such that for all k ≥ k0, the matrix (k + 1)Ir −A0 is invertible. Moreover,there exists a constant c > 0 such that for all k ≥ k0, we have∣∣∣∣∣∣((k + 1)Ir −A0)−1∣∣∣∣∣∣ ≤ c .It follows that for all k ≥ k0, one has

||Yk+1|| ≤ ck∑`=0

||Ak+1−` || ||Y` || .

• The function

α(x) :=∞∑k=1

||Ak ||(x − x0)k

is holomorphic on D and vanishes at x0. Moreover, the function1

1−cα is well-defined andholomorphic on some non-empty disc D′ centered at x0.

• The function

ζ(x) :=1

1− cα(x)

||Y0||+ k0∑s=1

||Ys+1|| − c s−1∑t=0

||As−t || ||Yt ||

(x − x0)s

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is well-defined and holomorphic on D′, and its power series expansion ζ(x) =∑∞k=0ζk(x−x0)k

satisfies {ζk = ||Yk || ∀ k ≤ k0ζk = c

∑k−1s=0 ||Ak+1−s||ζs ∀ k > k0 .

The reason why we chose this control-function ζ becomes quite clear when we multiply bothsides of the definition of ζ with 1− cα(x), and consider the power series expansion.

• One has ||ζk || ≤ ||Yk || for all k ∈ N. From the fact that the power series expansion of thecontrol-function ζ converges on D′, one deduces that the formal power series Y convergeson D′. In particular, Y has positive radius of convergence.

• One deduces, similarly the the second part of the proof of Proposition 1.5, that Y convergesat least on the disc D around x0 with radius R.

Remark.

1. Note that the statement analogous to this theorem fails for higher order poles:The singular second order differential equation

x3y′′ + x2y′ = y − xy′

admits a formal power series solution near 0 which is divergent.

2. Note also that even for simple poles, formal solutions might not exist: consider y′ = 1xy.

1.6 Analytic continuation

Let Y ′ = AY be a holomorphic system overU ⊂C. Let x0 ∈U and let γ : [0,1]→U be a path (i.e., acontinuous map) with γ(0) = x0. For each t ∈ [0,1], let us denote by Dt the maximal disc centeredat γ(t) which is contained in U .

LetΦ be a fundamental solution on D0 (i.e., near x0). Since γ([0,1]) is compact, we may choosea finite sequence

0 = t0 < · · · < tN = 1

of points in [0,1] such that for each i ∈ {0, . . . ,N − 1}, we have

γ([ti , ti+1]) ⊂Dti ∩Dti+1 .

We now define Φ0 := Φ on D0 and then for each i ∈ {0, . . . ,N −1} successively define Φti+1 to be theunique fundamental solution on Dti+1 which coincides with Φti on Dti ∩Dti+1 .

One may easily convince oneself that these fit into a family (Φt ,Dt)t∈[0,1] of fundamental solu-tions on maximal discs, such that

∀ t ∈ [0,1] ∃ε > 0 ∀ t′ ∈ [0,1] , |t − t′ | < ε ⇒ γ(t′) ∈Dt and Φt |Dt∩Dt′ = Φt′ |Dt∩Dt′ .

In other words, the family (Φt ,Dt)t∈[0,1] is an analytic continuation of (Φ ,D) = (Φ0,D0) along γ .

Definition 1.7: With respect to the above notation, we denote by Φγ the fundamental solutionnear γ(1) obtained by analytic continuation of Φ along γ , i.e., Φγ = Φ1.

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From the general theory of analytic continuations, we immediately obtain the following.

Proposition 1.8: With respect to the above notation,

• Φγ is well-defined, and

• if γ ′ : [0,1]→U is homotopic to γ relatively to the endpoints, then Φγ ′ = Φγ .

Corollary 1.9: Let Y ′ = AY be a holomorphic system over U ⊂ C. If a non-empty open subsetU ′ ⊂U is simply connected, then

• there exists a fundamental solution Φ over U ′, and

• { solutions over U ′} ' Cr .

2 Monodromy and Riemann-Hilbert correspondence

Main references for this lecture:

? Sabbah, Isomonodromic deformations and Frobenius varieties, chapter 0+1 [12].

? Deligne, Équations différentielles à points singuliers réguliers, for further reading [5].

2.1 Monodromy of holomorphic systems

Let Y ′ = AY be a holomorphic system of rank r over U ⊂ C and let Φ be a fundamental solutionon a maximal disc D around x0 ∈ U . Consider a closed path γ : [0,1]→ U with γ(0) = γ(1) = x0.Then Φγ is also a fundamental solution on D. Hence there exists a matrix Mγ ∈GLrC such that

Φγ = Φ ·Mγ .

Definition 2.1: With respect to the above notation, the map

ρΦ :{π1(U,x0) → GLr

γ 7→ Mγ = Φ−1Φγ

is called the monodromy representation of Y ′ = AY with respect to Φ and x0.

Proposition 2.2: The monodromy representation ρΦ is an anti-representation:For any γ1,γ2 ∈ π1(U,x0), we have

ρΦ (γ1.γ2) = ρΦ (γ2) · ρΦ (γ1) .

Here γ1.γ2 denotes the usual concatenation “first γ1, then γ2”.

Proof. We haveΦγ1.γ2 = (Φγ1)γ2 = (Φ ·Mγ1)

γ2 = Φγ2 ·Mγ1 = Φ ·Mγ2 ·Mγ1 .

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For M ∈GLrC, we have

ρΦ ·M(γ) =M−1 · ρΦ (γ) ·M ∀γ ∈ π1(U,x0) .

In particular, for any two fundamental solutionsΦ andΨ over D, the associated anti-representationsρΦ and ρΨ are conjugated by an element M ∈GLrC.

Definition 2.3: Let Y ′ = AY be a holomorphic system of rank r over U . Let ρ be its monodromyrepresentation with respect to some fundamental solution. The conjugacy class

[ρ]

of the anti-representation ρ : π1(U,x0)→ GLrC up to global conjugation by constant matricesM ∈GLrC is called the monodromy of Y ′ = AY .

• Note that the monodromy [ρ] of a holomorphic system as above does not depend on thechoice of a fundamental solution Φ near x0.

• Moreover, if U is connected, it does not depend on the choice of the base-point x0 in thefollowing sense:

If U ⊂C is (open and) connected, it is arc-connected, and for any x0,x1 ∈U we may choose a pathα : [0,1]→U from x0 to x1. Such a path induces an isomorphism

τα :{π1(U,x1) → π1(U,x0)

γ 7→ α.γ.α−1 .

If ρ = ρΦ is the monodromy representation of a holomorphic system Y ′ = AY over U with respectto a fundamental solution Φ and the base-point x0, then

ρΦ ◦ τα = ρΦα .

If α′ is another path in U from x0 to x1, then

[ρΦα′ ] = [ρΦα ] .

Example 1 (Monodromy of an Euler system). An Euler system is a meromorphic system defined overC of the form

Y ′ =AxY

where A ∈Mr×rC is a constant matrix. This system is of course in general not holomorphic over C, but itdefines a holomorphic system over C∗, for which we can compute the monodromy. A local fundamentalsolution is given by

Φ(x) = xA := exp(A ln(x)) .

If γ denotes a closed path in C∗ with endpoint ε, such that γ is turning counterclockwise around 0, then

Φγ (x) = exp(A(ln(x) + 2iπ)) = Φ(x) · exp(2iπA) .

HenceMγ = exp(2iπA). Sinc π1(C∗, ε) is generated by this γ , the monodromy of the above Euler systemamounts to the conjugacy class of Mγ .

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Lemma 2.4: Any conjugacy class [M] of a matrix M = Mγ ∈ GLrC can be realized as the mon-odromy of an Euler system.

Proof. One only needs to show that for any M ∈ GLrC there exists A ∈ Mr×rC such that M =exp(2iπA). This is an easy exercise. See for example [7, pp. 35-36] for a proof.

2.2 Holomorphic gauge transformations

Let Y ′ = AY be a homomorphic system of rank r over U . Let U ′ ⊂ U be open and nonempty. LetΦ ∈ GLr(O(U ′)), that is, Φ is an r × r-matrix with coefficients holomorphic over U ′ and nowherevanishing determinant. Consider the change of variable Y = ΦŶ . It leads to a new holomorphicsystem

Ŷ ′ = ÂŶ

over U ′, with = Φ−1AΦ −Φ−1Φ ′ .

Such a change of variable is called a holomorphic gauge transformation over U ′. A global gaugetransformation of a system Y ′ = AY defined over U is a gauge transformation defined over U .

Example 2. If Y ′ = AY admits a fundamental solution Φ over U ′, then the associated gauge transfor-mation yields the trivial system Ŷ ′ = 0 over U ′.

Definition 2.5: Let A ∈Mr×r(O(U )).We denote by

[A]

the equivalence class of A, seen as the matrix of the holomorphic system Y ′ = AY over U , modulothe action of global holomorphic gauge transformations.

Theorem 2.6: The monodromy mapHolomorphic systems

of rank r over Umodulo global hol. gauge

→

Anti-representationsρ : π1(U )→GLrCmodulo conjugation

[A] 7→ [ρA] .

is well-defined. Moreover, if U ⊂C is connected, the monodromy map is bijective.

Proof.

• The well-definedness, i.e. the fact that the monodromy [ρA] of a holomorphic system Y ′ =AY only depends on [A], is easily checked.

• Injectivity:Assume that two holomorphic systems Y ′ = AY and Z ′ = BZ over a connected open domainU give rise to the same monodromy [ρ]. Let ρ : π1(U,x0) → GLrC be a representative of

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[ρ]. One can easily check that there exist local fundamental solutions ΦA and ΦB of Y ′ = AYand Z ′ = BZ respectively near x0, such that the respective monodromies with respect to ΦAand ΦB are both given by ρ. Consider Φ := ΦA ·Φ−1B . This Φ , which is well-defined on themaximal disc around x0, can be analytically continued along any path in U with startingpoint x0. For any closed path γ with end point x0, one has Φγ = Φ . Therefore Φ extends,by analytic continuation, to a holomorphic and everywhere invertible matrix function on U .The global gauge transformation Y = ΦŶ applied to Y ′ = AY yields Ŷ ′ = BŶ .

• Surjectivity:The surjectivity will be a direct consequence of two theorems which we will encounter dur-ing the rest of this chapter. Namely, the Riemann-Hilbert correspondence (version 2), com-bined with the Grauert-Röhrl theorem.

Remark. More generally, whenX is a (connected) Riemann surface, one can define holomorphic systemsdY = ΩY on X, where Ω ∈Mr×r(Ω1X(X)), i.e., the coefficients of Ω are global holomorphic 1-forms onX. One then obtains a similar monodromy map [Ω] → [ρΩ]. One again has well-definedness andinjectivity. Surjectivity holds if X is non-compact.In the general case, in order to construct an inverse to the monodromy map, one needs to broaden thediscussion to a notion larger than the one of holomorphic systems. This will be done in the following.

2.3 Local systems

Let X be a topological space.

Definition 2.7: A local system of rank r over X is a sheaf L of finite dimensional C-vectorspaces over X such that L is locally isomorphic to the constant sheaf CrX .

Recall that sections of the constant sheaf CrX over ∆ ⊂ X are continuous maps ∆→ Cr , where

Cr is given the discrete topology. Such continuous maps are constant if ∆ is connected.

Example 3. If Y ′ = AY is a holomorphic system of rank r over U ⊂C, then the sheaf Sol of solutions ofY ′ = AY is a local system of rank r over U . Indeed, around any point x ∈ U we may choose a maximaldisc D on which there exists a fundamental solution Φ . Then for any connected ∆ ⊂ D (open andnon-empty), we have

Sol(∆) = {Φ |∆ ·C | C ∈Mr×1(C)} ' Cr .

Assume now that the topological space X is connected and locally arc-connected, so that wemay consider its fundamental group with respect to some base-point p ∈ X. We have definedthe monodromy representation for holomorphic systems, and one may define more generally themonodromy representation of a local system:

Let L be a local system over X and let γ : [0,1] → X be a closed path with endpoint p ∈X. Then γ∗L is a local system over [0,1]. As one can easily check, this local system istrivial: any germ of section extends uniquely to a section over [0,1]. We thereby obtain anisomorphism (γ∗L)0

∼→ (γ∗L)1. On the other hand, the stalks (γ∗L)0 and (γ∗L)1 are bothcanonically identified with the stalk Lp. The composition

Lp = (γ∗L)0∼→ (γ∗L)1 = Lp

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yields an element ϕγ ∈ GL(Lp), which is well-defined and only depends on the homotopyclass of γ relative to the endpoint p. The map

ρL :

π1(X,p) → GL(Lp)γ 7→ ϕγis an anti-morphism of groups and is called the monodromy representation of L with re-spect to p.

Example 4. For the sheaf of solutions Sol of a holomorphic system Y ′ = AY over U ⊂C, for any x0 ∈Uwe have

Solx0 = { values at x0 of local solutions around x0 } = Mr×1C .

Using the natural identification GL(Mr×1C) = GLrC, the monodromy representation of the local systemSol with respect to the base-point x0 is of the form

ρSol : π1(U,x0)→GLrC .

As one can easily check, ifΦ is the unique fundamental solution over a maximal disc around x0 satisfyingΦ(x0) = Ir , then

ρΦ = ρSol .

We denote byRep(X,p)

the category of anti-representations π1(X,p)→ GL(V ), where V is a finite dimensional C-vectorspace. Here morphisms of anti-representations are defined in a natural way from from morphismsV →W of finite dimensional C-vector spaces. We denote by

LocSys(X)

the category of local systems over X. Again morphisms are defined in the natural way.

Theorem 2.8: Let X be a connected and locally arc-connected topological space. Let p ∈ X.The monodromy functor

LocSys(X)→ Rep(X,p)

is an equivalence of categories.

Sketch of proof. For a detailed proof we refer to [12, Chap 0, §15d]. Let us just give a few indica-tions concerning the construction of a quasi-inverse functor; this construction is called suspen-sion.Let ρ : π1(X,p)→ GL(V ) be an anti-representation. Consider the universal cover u : X̃ → X. Wehave an action x̃ 7→ x̃ · γ of π1(X,p) on X̃. Consider the sheaf L̃ := VX̃ , i.e., the constant sheafover X̃ with stalk V . We construct a local system L over X as follows. For any ∆ ⊂ X open, weconsider ∆̃ := u−1(∆). We have an action of π1(X,p) on L̃(∆̃) defined by (x̃,v) 7→ (x̃ · γ,ρ(γ)(v)).We may now set L(∆) := L̃(∆̃)π1 to be the π1(X,p)-invariant part of L̃ on the inverse image of ∆.One can check that L is a local system with monodromy representation ρL = ρ with respect to thebase-point p.

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2.4 Holomorphic vector bundles and connections

Let X be complex manifold.

Definition 2.9: A holomorphic vector bundle of rank r over X is a holomorphic map

π : E→ X

of complex manifolds such that for any point x ∈ X, the fiber Ex := π−1({x}) is an r-dimensionalcomplex vector space, and such that there exists an atlas (Ui ,ϕi)i∈I of trivialization maps fittinginto the diagram

E|Ui := π−1(Ui)

ϕi∼

//

π

&&

Ui ×Cr

pr1{{

Ui .

Here (Ui)i∈I has to be a covering of X by open subsets Ui , and the trivialization maps ϕi have tobe biholomorphic and linear in restriction to each fiber over an element x ∈Ui .

Definition 2.10: Let π : E→ X be a holomorphic vector bundle of rank r.A holomorphic section of E over U ⊂ X is a holomorphic map e :U → E such that π◦σ = idU .We denote by E the sheaf of holomorphic sections of π : E→ X.A frame of E over U is a sequence (e1, . . . , er ) of elements of E(U ) such that for each x ∈ U , onehas

Vect(e1(x), . . . , er(x)) = Ex .

Remark. • Frames exist locally; they are actually equivalent to local trivialization maps via

ϕ(ei(x)) = (x,εi) ,

where ε is the canonical basis of Cr .

• From a locally free sheaf E of OX-modules of rank r over X, one can construct a vector bundle Ewhose sheaf of sections is naturally isomorphic to E, by setting Ex = Ex/mEx, where m is the idealof OX,x given by germs vanishing at x.

• A vector bundle E over X admits a global frame if and only if E is trivial, i.e., if E is isomorphic tothe vector bundle pr2 : X ×Cr → X.

We will need the following two classification results of vector bundles over certain Riemannsurfaces.

Theorem 2.11 (Grauert-Röhrl):If X is a non-compact Riemann surface, then any vector bundle over X is trivial.

Proof. A detailed proof can be found for example in [6, §30].

13

Theorem 2.12 (Birkhoff-Grothendieck): Let E→ X be a vector bundle of rank r over the Rie-mann sphere X = P1 = P1C. Then there exists a sequence (k1, . . . , kr ) ∈Z

r , unique up to permuta-tion, such that

E ' O(k1)⊕ · · · ⊕O(kr ) .

Proof. A detailed proof can be found for example in [12, Chap 1 §4].

Definition 2.13: A holomorphic connection on a holomorphic vector bundle π : E → X is aC-linear morphism

∇ : E → E ⊗Ω1X ,

where Ω1X denotes the sheaf of holomorphic 1-forms on X, such that ∇ satisfies the Leibniz rule:for any U ⊂ X open, any e ∈ E(U ) and any f ∈ OX(U ), we have

∇(f e) = f ∇(e) + e⊗df .

Let e = (e1, . . . , er ) be a frame of E over U . Then any holomorphic section e of E over U can bewritten uniquely as

e = (e1, . . . , er ) ·Y :=r∑i=1

eiyi

for some Y = t(y1, . . . , yr ) ∈ Mr×1(O(U )). Given a holomorphic connection ∇ on E, we can asso-ciate to each local frame e of E a connection matrix: the well-defined matrix Ω ∈ Mr×r(Ω1X(U ))satisfying

∇ ((e1, . . . , er ) ·Y ) = (e1, . . . , er )⊗ (dY +ΩY )

for each Y ∈Mr×1(O(U )).

Exercise 2. Let Φ ∈GLr(O(U )). If Ω denotes the connection matrix with respect to the frame e over U ,then the connection matrix with respect to the frame ê = e ·Φ is given by

Ω̂ = Φ−1ΩΦ +Φ−1dΦ .

Remark. Up to isomorphism, a holomorphic vector bundle with connection may be reconstructed froman associated collection of connection matrices Ωi (over open sets Ui ⊂ X over which there exist globalframes ei , such that X =

⋃iUi) and transition maps Φij (with ej = ei ·Φij over non-empty Ui ∩Uj).

Example 5 (Relation to solutions of Y ′ = AY ). Let (E,∇) be a holomorphic connection over X. AssumedimCX = 1. Let U ⊂ X be an open subset on which there exists a coordinate x : U → C. Then E|U istrivial by the Grauert-Röhrl theorem. In particular, E|U admits a global frame e. The connection matrixΩ with respect to this frame writes

Ω = −Adx

for some holomorphic matrix function A on U . Let Y ∈ Mr×1O(U ′) for some U ′ ⊂ U . Then Y is asolution over U ′ of Y ′ = AY if and only if ∇(e ·Y ) = 0. Moreover, a change of frame for E|U ′ correspondsto a holomorphic gauge transformation of Y ′ = AY over U ′.

14

2.5 Integrable connections

Given a holomorphic connection ∇ on E, we may associate a C-linear morphism defined by

∇ :{E ×Ω1X → E ×Ω

2X

e⊗ω 7→ e⊗dω+∇(e)∧ω.

Definition 2.14: Given a holomorphic connection ∇ on E, one defines the curvature of ∇ by themorphism

∇◦∇ : E → E ⊗Ω2X .

A holomorphic connection is called flat or integrable if its curvature is zero.

Exercise 3. The curvature ∇◦∇ of a holomorphic connection ∇ on E isOX-linear, and given with respectto frames e of E by

∇◦∇ : e ·Y 7→ e · (dΩ+Ω∧Ω) ·Y ,where Ω is the connection matrix of ∇ with respect to e.

Definition 2.15: A holomorphic connection ∇ on E is called flat or integrable if

∇◦∇ = 0 .

A local section e of E is said to be flat or horizontal (with respect to ∇) if ∇(e) = 0. A framee = (e1, . . . , er ) of E is called horizontal if each ei is flat, or, equivalently, if the connection matrixof ∇ with respect to this frame is 0.

Remark. If dimCX = 1, then any holomorphic connection on a holomorphic vector bundle over X isintegrable. Indeed, there are no non-zero local holomorphic two-forms on X in that case.

Theorem 2.16 (Existence of horizontal frames): Let (E,∇) be an integrable holomorphic con-nection of rank r over a complex manifold X. Then any point x ∈ X admits a neighborhood overwhich there exists a horizontal frame.

Proof. • Since the statement is local, it can be reduced to the following.Let Ω ∈ Mr×r(Ω1(D)), where D is a polydisc in Cm centered at 0, such that Ω satisfies theintegrability condition

dΩ+Ω∧Ω = 0 .Show that up to shrinking D, there exists Φ ∈GLr(O(D)) such that Φ−1ΩΦ +Φ−1dΦ = 0 .

• We may write−Ω = A1dx1 + · · ·+Amdxm ,

where the Ai ’s are holomorphic matrix functions on D. By the Cauchy-Kowalewsky theo-rem, for any sufficiently small polydisc D′ in Cm centered at 0, there exists a unique matrixfunction Φ1 ∈Mr×r(O(D′)) such that{ ∂

∂x1Φ1 = A1Φ1

Φ1(0,x2, . . . ,xm) = Ir .

15

Since det(Φ) is non-vanishing over {x1 = 0}, up to shrinking D′, we may assume that

Φ1 ∈GLr(O(D′)) .

• For Ω̂ := Φ−11 ΩΦ1 +Φ−11 dΦ1 , one readily checks that

1. Ω̂ satisfies the integrability condition.2. −Ω̂ is of the form Â2dx2 + · · ·+ Âmdxm where the Âi ’s are holomorphic on D′.

From these and the fact that Φ1|{x1=0} = Ir , one deduces that

Âi(x1, . . . ,xm) = Âi(0,x2, . . . ,xm) = Ai(0,x2, . . . ,xm) ∀i ∈ {2, . . . ,m} .

In other words, Ω̂ satisfies the same conditions as Ω, but depends only on m− 1 variables.

• We conclude by induction on the dimension m.More precisely, by induction on m, for sufficiently small D′, there exists Φ2 ∈ GLr(O(D′)),depending only on (x2, . . . ,xm), such that

Φ−12 Ω̂Φ2 +Φ−12 dΦ2 = 0 .

Then Φ := Φ1Φ2 satisfies the requirements.

Corollary 2.17: Let (E,∇) be an integrable holomorphic connection of rank r over complex man-ifold X. Then the sheaf

ker∇

of sections of E that are horizontal with respect to ∇ is a local system of rank r over X.

Proof. Locally, if e is a horizontal frame over U , then ∇ : e ·Y 7→ e ·dY . If U is connected, then

ker∇(U ) = {e ·C | C ∈Mr×1C} ' Cr .

In particular, to an integrable holomorphic connection ∇ over E, one can associate a mon-odromy representation

ρ∇ : π1(X,x0)→GL(Ex0) ,namely, the monodromy representation of the local system ker∇.Remark. LetD be an effective divisor onX. A meromorphic connection∇with polar divisor (at most)Don a holomorphic vector bundle E→ X is by definition a C-linear morphism E → E ⊗Ω1X(D) satisfyingthe Leibnitz rule. Here the coefficients of the matrix connections are meromorphic one-forms with polardivisor at most D. The monodromy representation of such a meromorphic connection is by definition themonodromy representation of the holomorphic connection (E,∇)|X\Σ, where Σ ⊂ X is the support of D.Remark (Relation to the lecture series on holomorphic foliations). An integrable connection ∇ on avector bundle π : E→ X defines a foliation of dimension dimCX on the total space E of the vector bundle,where integral curves are simply horizontal sections. This particular type of foliation is transversal to thefibers of the projection π : E→ X. Those are of course preferred transversals with respect to holonomy. Ifγ is a path in X from x0 to x1, the induced holonomy map between the fibers Ex0 and Ex1 is constructedby analytic continuation of horizontal sections. The holonomy group with respect to Ex0 corresponds tothe image of the monodromy representation ρ∇ with base-point x0.

16

2.6 Riemann-Hilbert correspondence

Let X be a complex manifold and let p ∈ X.We denote by

Con(X)

the category of holomorphic vector bundles E over X, endowed with integrable holomorphic con-nections ∇. Morphisms are simply morphisms of vector bundles compatible with the connections.

Theorem 2.18 (Riemann-Hilbert correspondence, version 1): The functor

Con(X)→ LocSys(X) ,

which to an integrable connection (E,∇) associates its sheaf of horizontal sections ker∇, is anequivalence of categories.

Sketch of proof. For details, we again refer to [12, Chap 2, §3]. Let us just indicate how to constructa vector bundle with connection from a local system L. Set E := OX ⊗C L and choose ∇ such thatits connection matrices are zero for any frame of E given by a frame of L. For all other frames,define the connection matrices by gauge transformation.

Theorem 2.19 (Riemann-Hilbert correspondence, version 2): If X is connected, the functor

Con(X)→ Rep(X,p) ,

which to a flat connection (E,∇) associates its monodromy representation ρ∇ (with base-point p),is an equivalence of categories.

Proof. This follows by transitivity from Theorem 2.8, combined with Theorem 2.18.

Remark.

• We have a natural bijectionFlat holomorphic connections ∇on trivial vector bundles E→ X,

modulo isomorphisms (E,∇) ' (E′ ,∇′)

'

Integrable systems dY = −ΩYover X, modulo global

holomorphic gauge transformations

• If X is a smooth algebraic variety over C, one may, instead of holomorphic (E,∇) over X, consider

flat algebraic connections on algebraic vector bundles over X. The monodromy representation isthen defined via analytification. For the Riemann-Hilbert correspondence to hold in that setting,in the case of non-compactX, one needs to add a regularity condition “at infinity” to the consideredflat algebraic connections on vector bundles over X (see [5]).

17

3 Meromorphic systems, local theory

Main references for this lecture:

? Ilyashenko/Yakovenko, Lectures on analytic differential equations, chapter 3 [7].? Singer/van der Put, Galois theory of linear differential equations, chapter 2 [14].? Wasow, Asymptotic expansions for ordinary differential equations, chapter 3 [16].

3.1 Some terminology for meromorphic systems

From now on, we will also consider meromorphic systems

Y ′ = AY

over complex domains U ⊂ C. Here A ∈ Mr×r(M(U )) is a matrix function over U , whose coeffi-cients are meromorphic functions.

• A singularity of such a meromorphic system is a point in U corresponding to a pole of (atleast one of the coefficients of) A.

• For technical reasons, one usually fixes a discrete subset Σ ⊂ U , and speaks of a meromor-phic system Y ′ = AY over U with polar locus Σ if the “true polar locus” (i.e., the set of polesof A) is contained in Σ.

• The monodromy of a meromorphic system Y ′ = AY over U with polar locus Σ is by defini-tion the monodromy of the holomorphic system Y ′ = AY over U \Σ.

• For meromorphic systems, one may consider two types of gauge transformations:

Definition 3.1: Let Y ′ = AY be a meromorphic system over U with polar locus Σ.

• A holomorphic gauge transformation overU ′ is the change of variable Y = ΦZ, yieldinga meromorphic system Z ′ = BZ over U ′ with polar locus Σ∩U ′, where

Φ ∈GLr(O(U ′)) ,

i.e. Φ is a holomorphic matrix function over U ′ with non-vanishing determinant.

• A meromorphic gauge transformation over U ′ is the change of variable Y = ΦZ, yield-ing a meromorphic system Z ′ = BZ over U ′ with polar locus Σ∩U ′, where

Φ ∈GLr(M(U ′)) ,

i.e., Φ is a meromorphic matrix function over U ′ whose determinant is not identically zeroon any connected component of U ′, such that moreover,

Φ |U ′\Σ ∈ Mr×r(O(U ′ \Σ)) and Φ−1|U ′\Σ ∈ Mr×r(O(U ′ \Σ)) .

Example 6. For U = C and polar locus {0}, a global gauge transformation of the form

Ψ =(x 00 1

)18

is meromorphic, but not holomorphic as a gauge transformation.

• As usual, two meromorphic systems over U with polar locus Σ are said to be holomorphi-cally (resp. meromorphically) gauge equivalent, if they are related via a global holomor-phic (resp. meromorphic) gauge transformation. Note that the monodromy [ρ] is well de-fined for a meromorphic gauge equivalence class of meromorphic system over U with polarlocus Σ.

We will now focus on the local theory of meromorphic systems near a singularity1. We maytherefore consider meromorphic systems defined over a complex disc, and (at most) one singular-ity, at the center of the disc. Up to choosing a convenient coordinate on C, we may assume thisdisc to be centered at 0. In other words, we shall consider meromorphic systems of the form

Y ′ =A

xp+1Y ,

where A is a holomorphic matrix function of size r×r defined on a disc D ⊂C of radius R centeredat 0, and p ∈N∗ is an integer.

Definition 3.2: With respect to the above notation, if A(0) , 0, then the integer p is called thePoincaré rank of the singularity {x = 0}. A singularity of Poincaré rank 0 (i.e., a simple pole) iscalled a Fuchsian singularity.

• The Poincaré rank of a singularity is preserved by (global) holomorphic gauge transforma-tions. For meromorphic gauge transformations, this is not true in general. For example forany k ∈N, the Fuchsian system

Y ′ =1x

(0 10 0

)Y

over D with polar locus {0} is meromorphically gauge equivalent, via Y =(xp 00 1

)Z, to a

system where the singularity at {x = 0} is of Poincaré rank p.

• Singularities may even disappear by global meromorphic gauge transformations. Such sin-gularities are called apparent singularities. An example is given by

Y ′ =1x2

(x2 x4

1 −2x

)Y

over D with polar locus {0}, which is meromorphically gauge equivalent, via Y =(1 00 1x2

)Z,

to

Z ′ =(1 11 1

)Z .

The monodromy of a meromorphic system over U with only apparent singularities is neces-sarily trivial, and any germ of solution of such a system can be meromorphically continuedalong any path in U .

1Our main focus here lies on regular singularities. For the more involved local theory of irregular singularities, werefer for example to [7, Chap 3, §20], [12, Chap 2, §5-6], [16, Chap. 4].

19

3.2 Fuchsian singularities and their classification

Let us consider a Fuchsian system of the form

Y ′ =AxY (5)

over D with polar locus {0}, for some A ∈Mr×r(O(D)). We shall denote by

A(x) =∞∑k=0

Akxk

the power series expansion of A.

Definition 3.3: With respect to the above notation, A0 = A(0) will be called the residue at 0 ofthe Fuchsian system (5) over D. The singularity {x = 0} of the Fuchsian system (5) is said to beresonant if the residue A0 admits two eigenvalues λ,µ ∈ C such that

λ−µ ∈Z \ {0} .

Otherwise, the singularity is said to be non-resonant.

Our aim is to classify Fuchsian systems of the form (5) up to holomorphic and meromorphicgauge equivalence. The above notion of resonancy plays an important role in this context, due tothe following lemma of elementary matrix-theory. For M ∈Mr×rC we define the linear map

ad(M) :{

Mr×rC → Mr×rCX 7→ MX −XM

}.

Lemma 3.4: Let M ∈Mr×rC.If the eigenvalues of M, counted with multiplicity, are

µ1, . . . ,µr ,

then the eigenvalues of ad(M), counted with multiplicity, are

µi −µj , (i, j) ∈ {1, . . . , r}2 .

Proof. • First step: the case of semisimple matrices.If M is the diagonal matrix M = diag(µ1, . . . ,µr ), then with respect to the canonical basis(Eij )i,j∈{1,...,r} of Mr×rC, we have ad(M)(Eij ) = (µi−µj )Eij . Hence the results holds for diagonalmatrices M. By conjugation, one deduces the result for diagonalisable matrices M.

• Second step: the general case.Let M =Ms +Mn be the Dunford decomposition of M into diagonalisable (semisimple) partMs and nilpotent part Mn with MsMn =MnMs. Then ad(M) = ad(Ms) + ad(Mn). One readilychecks that ad(Ms) is semisimple, ad(Mn) is nilpotent and that

ad(Ms) ◦ ad(Mn) = ad(Mn) ◦ ad(Ms) .

20

In other words, ad(M) = ad(Ms) + ad(Mn) is the Dunford decomposition of ad(M). Hencethe eigenvalues of ad(M), counted with multiplicity, are those of ad(Ms). These are of therequired form by the first step of the proof.

Theorem 3.5 (Classification in the non-resonant case): Let

Y ′ =AxY

be a Fuchsian system over D as in (5). If it is non-resonant, then this system is holomorphicallygauge equivalent to the Euler system

Ŷ ′ =A0xŶ

over D.

Proof. • Note firstly that for Φ ∈ GLr(O(D)), the two Fuchsian systems in the statement arerelated via Y = ΦŶ if and only if

Φ−1AxΦ −Φ−1Φ ′ = A0

x,

or, equivalently,xΦ ′ = AΦ −ΦA0 . (6)

• Consider a formal matrix power series

Φ =∑k≥0

Φkxk

with coefficients Φk ∈Mr×rC andΦ0 := Ir .

This Φ formally satisfies (6) if and only if

(ad(A0)− (k + 1)id)(Φk+1) = −k∑`=0

Ak+1−`Φ` ∀k ∈N . (7)

Since the Fuchsian system

Y ′ =AxY

is non-resonant by assumption, Lemma 3.4 ensures that for any k ∈N, the linear map

(ad(A0)− (k + 1)id) : Mr×rC→Mr×rC

is invertible. Hence given the power series expansion of A together with Φ0, . . . ,Φk , thereexists a unique matrix Φk+1 satisfying (7). We conclude by induction that there exists aunique formal matrix power series Φ =

∑k≥0Φkx

k with Φ0 = Ir that formally satisfies (6).

21

• Since xΦ ′ = AΦ −ΦA0 is a system of first order linear differential equations (with respectto the r2 coefficients of the matrix Φ), with only a simple pole, by Theorem 1.6, the formalmatrix solution Φ determined before converges on D. We conclude that there is a uniqueΦ ∈Mr×r(O(D)) with Φ(0) = Ir satisfying (6).

• It remains to prove that det(Φ) is non-vanishing on D. Since det(Φ)(0) , 0, there is a discD′ with positive radius centered at 0, which is contained in D and on which det(Φ) doesnot vanish. On D′, we have a well-defined holomorphic matrix function Ψ := Φ−1. Since(Φ−1)′ = −Φ−1Φ ′Φ−1, the matrix Ψ satisfies

xΨ ′ = A0Ψ −ΨA.

Since this is again a meromorphic system over D of linear differential equations (linear withrespect to the coefficients of Ψ ) with only a simple pole at 0, by Theorem 1.6, the functionΨ on D′ extends holomorphically to D. Since

ΦΨ ≡ Ir

on D′ and Ψ is holomorphic, this equation remains satisfied on D. In particular, the deter-minant of Φ is non-vanishing over D.

Definition 3.6: A Fuchsian system Y ′ = BxY of rank r over D, with polar locus {0}, is said to beof Levelt normal form if

B =∑k≥0

Bkxk

such that for any k ∈N, the matrix Bk ∈Mr×rC satisfies

ad(bs)(Bk) = kBk ,

where B0 = bs + bn is the Dunford decomposition of B0 into semisimple part bs and nilpotentpart bn with bsbn = bnbs.

Remark. A matrix-function B as above is necessarily polynomial.

Example 7. Let n1,n2 ∈N and β,c1, c2, c3 ∈C. The Fuchsian system Y ′ = BxY over D, where

B =

β +n1 +n2 c2xn2 c3xn1+n2

0 β +n1 c1xn10 0 β

,is of Levelt normal form.

22

Lemma 3.7: Let bs ∈Mr×rC be semisimple.Then there exists a unique matrix L ∈Mr×rC such that

• L is semisimple, commutes with bs, has only integer eigenvalues and

• for any eigenvalue µ of bs −L, one has

Proof. The proof of Theorem 3.8 is a refinement of the proof for the non-resonant case in Theo-rem 3.5. As before, it is sufficient to find a formal matrix power series

Φ =∑k≥0

Φkxk

with Φ0 = Ir such that formally, for a convenient choice of B =∑k≥0Bkx

k , one has

xΦ ′ = AΦ −ΦB. (8)

At level k = 0, we setΦ0 := Ir and B0 = A0 .

Once all terms of Φ and B up to level k are determined, at level k + 1, equation (8) amounts to anequation of the form

(ad(A0)− (k + 1)id)(Φk+1) = Rk −Bk+1 , (9)

where Rk is a matrix determined uniquely from A0, . . . ,Ak+1,Φ0, . . . ,Φk and B1, . . . ,Bk . We distin-guish two cases.

Case 1): k + 1 is not an eigenvalue of ad(A0).In this case, we set Bk+1 = 0 and determine the unique possible Φk+1 satisfying (9).

Case 2): k + 1 is an eigenvalue of ad(A0).In this case, we consider the decomposition

Mr×rC =⊕β∈C

Wβ , where Wβ = ker(ad(bs)− β · id) .

Here, bs is the semisimple part of B0 = A0. Since A0 commutes with bs, we have that ad(A0)commutes with ad(bs). Therefore, each eigenspace Wβ of ad(bs) is stable under ad(A0). Inorder to solve equation (8), we may therefore reason on each eigenspace Wβ seperately.Let β ∈C.

Case 2.1): β , k + 1.In this case, analogously to case 1), we may set Bk+1|Wβ = 0 and determine the uniquepossible Φk+1|Wβ satisfying (9) in restriction to Wβ .Case 2.2): β = k + 1.In this case, equation (9) restricted to Wβ becomes

(ad(A0)− ad(bs))(Φk+1|Wβ ) = Rk |Wβ −Bk+1|Wβ . (10)

Note that ad(A0) − ad(bs) = ad(bn) is nilpotent. However, we may choose Bk+1|Wβ suchthat Rk |Wβ − Bk+1|Wβ lies in the image of ad(bn)|Wβ . In particular, we may then choosesome Φk+1|Wβ satisfying (10).

Note that a thusly constructed Bk+1 satisfies ad(bs)(Bk+1) = (k + 1)Bk+1.

Remark. The construction of non-zero coefficients Bk of a convenient matrix B =∑k≥0Bkx

k as aboveis completed after finitely many steps, and the number of these steps is bounded above by the maximalinteger difference of eigenvalues of A0.

24

Corollary 3.9: Let

Y ′ =AxY

be a Fuchsian system over D with polar locus {0} as in (5). Then this system is is meromorphicallygauge equivalent to an Euler system over D with polar locus {0}.

Proof. By Theorem 3.8, there exists Φ ∈ GLr(O(D)) such that by setting Y = ΦŶ , we obtain aFuchsian system Ŷ ′ = Bx Ŷ of Levelt normal form. By Lemma 3.7, there exists a semisimple matrixL ∈Mr×rC with integer eigenvalues, such that such that B = xLB(1)x−L. Note that xL = exp(L ln(x))is of the form

xL = P −1 ·diag(x`1 , . . . ,x`r

)· P

for some P ∈ GLrC and (`1, . . . , `r ) ∈ Zr . In particular, xL is a meromorphic matrix function on Dwhich is holomorphic on D∗ = D \ {0} and such that the determinant of xL is (holomorphic and)non-vanishing in restriction to D∗. Setting Ψ := ΦxL, via Y = Ψ Z, from Y ′ = Ax Y we obtain

Z ′ =B(1)−L

xZ ,

which is an Euler system.

Remark. • Euler systems up to holomorphic gauge equivalence are classified by the conjugacy classof their residue.

• One can show (see the remark after Proposition 3.12 in the next section), that Euler systems upto meromorphic gauge equivalence are classified by their monodromy [ρ].

• In particular, in the resonant case, a Fuchsian system Y ′ = Ax Y might not be meromorphicallygauge equivalent to the Euler system Y ′ = A0x Y given by the residue.

3.3 Regular singularities of meromorphic systems

By a sector of opening α > 0 and radius R > 0 we mean

Sα,θ,R :={x ∈C

∣∣∣∣∣ 0 < |x| < R, arg(x) ∈ ]θ − α2 ,θ + α2 [ }for some θ ∈R.

Definition 3.10: Let F ∈Mm×n(O(Sα,θ,R)). We say that F has moderate growth as |x| → 0 iffor all α′ ,R′ ∈ R with 0 < α′ < α and 0 < R′ < R there exists a constant c > 0 and an integerN ∈N such that

||F(x)|| ≤ c|x|−N ∀ x ∈ Sα′ ,θ,R′ .

We will now consider a meromorphic system of the form

Y ′ =A

xp+1Y , (11)

over D = D(0,R) ⊂C and holomorphic on D∗ = D \ {0}, of Poincaré rank p at {x = 0}.

25

Note that as long as α < 2π, for any θ ∈R, the sector Sα,θ,R is simply connected and containedin D∗. In particular, the meromorphic system (11) admits a fundamental solution over Sα,θ,R.

Definition 3.11: The singularity {x = 0} of Y ′ = Axp+1

Y as in (11) is said to be regular if for anysector Sα,θ,R of opening α < 2π, some (and hence any) fundamental solution defined on Sα,θ,Rhas moderate growth as |x| → 0.

Example 8. • The singularity {x = 0} of the differential equation y′ = − 1x2 y is irregular. Indeed, thesolution e

1x has exponential growth as |x| → 0.

• The singularity {x = 0} of Poincaré rank 1 of the meromorphic system

Y ′ =(

0 11x2 −

1x

)Y

is regular. Indeed, the fundamental solution(x 1x1 − 1x2

)has moderate growth as |x| → 0.

• Any Euler system Y ′ = A0x Y with constant A0 ∈Mr×r(C) is regular singular.Indeed, there is a (multivalued) fundamental solution of the form Φ = exp(A0 ln(x)). Let

A0 = as + an

be the Dunford decomposition of A0. Then

exp(A0 ln(x)) = exp(as ln(x))exp(an ln(x)) .

Since an is nilpotent, the coefficients of exp(an ln(x)) are polynomials in ln(x). If N +δ > 0 for anyeigenvalue δ of as, then xNΦ tends to 0 as |x| → 0.

• Fuchsian singularites are regular singular.Indeed, we have seen that each Fuchsian system over D with polar locus {0} is meromorphicallygauge equivalent to some Euler system. As one can easily check, regularity of the singularity{x = 0} is invariant under meromorphic gauge equivalence.

Proposition 3.12: A meromorphic system

Y ′ =A

xp+1Y

over D with polar locus {0} is regular singular if and only if it is meromorphically gauge equiva-lent to a Fuchsian system over D.

Proof. The “if”-part of the statement has been settled in Example 8.In order to prove the “only if”-part, let us consider Y ′ = A

xp+1Y as in (11) and assume that it is

regular singular.

26

Let ΦA be a fundamental solution of this system on some sector S ⊂ D∗ of opening α < 2πand denote by ρ the monodromy representation of Y ′ = A

xp+1Y over D∗ with respect toΦA and

some base-point x0 ∈ S. Denote M := ρ(γ), where γ is the standard generator of π1(D∗,x0).There exists a constant matrix B ∈Mr×r(C) such that exp(2iπB) = M. The monodromy rep-resentation of the Euler system Z ′ = BxZ over D with respect to the fundamental solutionΦB = xB on S and the base-point x0 is then identical to the above ρ.

Because it is univalued, the holomorphic matrix function

Φ = ΦA ·Φ−1B ∈ GLr(O(S))

extends holomorphically to an invertible matrix function over D∗.

By assumption, ΦA has moderate growth as |x| → 0 on any sector of opening less than 2π.One can easily check that Φ−1B has moderate growth as |x| → 0 on any sector of opening lessthan 2π. It follows that Φ has moderate growth as |x| → 0 on D∗. Hence Φ extends to ameromorphic matrix function on D.

As one can easily check, the meromorphic systems Y ′ = Axp+1

Y and Z ′ = BxZ over D arerelated, via Y = ΦZ, by the meromorphic gauge transformation Φ ∈ GLr(M(D)).

Remark. By the proof of the above proposition, any regular singular meromorphic system over D withpolar locus {0} is meromorphically gauge equivalent to any Euler system over D with polar locus {0} thathas the same monodromy. In particular, two regular singular meromorphic systems over D with polarlocus {0} are meromorphically gauge equivalent if and only if they have the same monodromy.

Exercise 4. Show that in the case of rank r = 1, a singularity is regular if and only if it is Fuchsian.

3.4 Regular singularities of differential equations

Using only the tools presented so far, it is rather hard in general to decide whether a singularityof a system is regular. As we will now see, this is much easier for singular linear differentialequations of order r.

We consider a meromorphic linear differential equation

y(r) = a0y + · · ·+ ar−1y(r−1) (12)

of order r over D, with polar locus {0}.

Definition 3.13 (Regular singularities for differential equations): The singularity {x = 0}of (12) is said to be regular if for any sector Sα,θ,R of opening α < 2π, any solution defined onthis sector has moderate growth as |x| → 0.

As we have seen, to (12) one can associate a linear system Y ′ = AY of rank r by setting Y =t(y,y′ , . . . , y(r−1)

). The matrix A then is of the particular form

A =

0 1 . . . 0

. . .. . .

.... . . 1

a0 a1 . . . ar−1

.

27

Conversely, when A is of the above so-called compagnon form, then to Y ′ = AY we can associatea linear differential equation of order r as above by considering y := y1.

Proposition 3.14: The singularity {x = 0} of (12) is regular if and only if it is a regular singularityof the associated compagnon system.

Proof. The “if”-part of this proposition is obvious. For the “only if”-part, it suffices to show thatif some holomorphic function defined on a sector in D has moderate growth as |x| → 0, then allits derivatives have moderate growth as |x| → 0. This will be proven by the following lemma.

Lemma 3.15: Let f ∈ O(S), where S = Sα,θ,R. If f has moderate growth as |x| → 0, then f ′ hasmoderate growth as |x| → 0.

Proof. Consider a chain of smaller sectors S ′′ ⊂ S ′ ⊂ S, where S ′ := Sα′ ,θ,R′ and S ′′ := Sα′′ ,θ,R′′ with0 < α′′ < α′ < α and 0 < R′′ < R′ < R. By assumption, for some integer N and some constant c > 0,we have

|xN f (x)| ≤ c ∀x ∈ S ′ .

By elementary geometry, there is a constant C > 0 such that

d(x,∂S ′) ≥ C|x| ∀x ∈ S ′′ .

By Cauchy estimates, it follows that∣∣∣∣(xN f (x))′∣∣∣∣ ≤ cd(x,∂S) ≤ cC|x| ∀ x ∈ S ′′ .One deduces that ∣∣∣f ′∣∣∣ ≤ cC +Nc

xN+1∀ x ∈ S ′′ .

Since for any sector S ′′ smaller that S in the above sense, one can find a sector S ′ with S ′′ ⊂ S ′ ⊂ Sin the above sense, the statement follows.

Theorem 3.16 (Fuchs’ criterion): The singularity {x = 0} of (12) is regular if and only if

ai =bixr−i

∀ i ∈ {0, . . . , r − 1} .

with b0, . . . , br−1 ∈ O(D) holomorphic.

Proof. Let us first prove the “if”-part of the statement. We consider a meromorphic differentialequation of order r over D of the form

y(r) =b0xr−0

y +b1xr−1

y + · · ·+ br−1x1

y(r−1) (13)

with holomorphic bi ’s. Here, one uses another type of compagnon system. Namely, one sets

Y = t(x0y,x1y′ . . . ,xr−1y(r−1)) .

28

From (13) we then obtain Y ′ = BxY with

B =

0 0 . . . . . . 0

0 1. . .

....... . .

. . .. . .

......

. . . (r − 2) 00 . . . . . . 0 (r − 1)

+

0 1 0 . . . 0...

. . .. . .

. . ....

.... . .

. . . 00 . . . . . . 0 1b0 b1 . . . br−2 br−1

.

This system is Fuchsian and in particular, regular singular. It follows that (13) is regular singular.In order to prove the “only if”-part of the statement, one uses another type of differential

operator, namely

δ := x∂∂x.

As one can easily check, any meromorphic differential equation of the form (12) may be writtenuniquely as

δry = c0δ0y + · · ·+ cr−1δr−1y (14)

where the ci ’s are meromorphic functions over D, holomorphic over D∗. Moreover, setting bi =xr−iai , one gets that these b0, . . . , br−1 are all holomorphic over D if and only if the associatedc0, . . . , cr−1 are all holomorphic over D.

One proceeds by induction on r ∈N∗ to prove that if a meromorphic linear differential equa-tion of the form (14), with polar locus (at most) {0} is regular singular, then all ci ’s are holomorphicover D. Note that the case r = 1 corresponds to Exercise 4. For details on the quite technical in-duction process, we refer to [13, Chap 9 §5], see also [15, Chap. 2 C, §12].

Example 9. From Fuchs’ criterion, we can see directly that the Bessel differential equation

x2y′′ = (α2 − x2)y − xy′

(with α ∈C) is regular singular at 0.

3.5 Cyclic vectors

We shall now introduce the method of cyclic vectors, which is, roughly speaking, a method ofusing meromorphic gauge transformations to bring a meromorphic system to compagnon form.Recall that for meromorphic systems of compagnon form, Fuchs’ criterion allows us to decidewhether a singularity is regular or irregular.

Definition 3.17: Let Y ′ = AY be a meromorphic system of rank r over a connected domain U .Let Λ0 be a non-zero element of M1×r(O(U )). For i ∈ {0, . . . , r − 1} define recursively

Λi+1 =Λi ·A+Λ′i .

Then set

P :=

Λ0...

Λr−1

.If det(P ) is not identically zero, then Λ0 is called a cyclic vector for Y ′ = AY .

29

Assume we have a cyclic vector Λ0 for Y ′ = AY . Then

P ∈GLr(M(U )) ,

and setting Y = P −1Z yields Z ′ = BZ with B = (PA+ P ′)P −1 , which is of compagnon form

B =

0 1 . . . 0

. . .. . .

.... . . 1

b0 b1 . . . br−1

.Note however that the combined polar locus of P and P −1 might not be limited to the polar

locus of A, i.e. P −1 is not necessarily a meromorphic gauge transformation in the strict sense thatwe defined. However, the singularties of the new system Z ′ = BZ that are in addition to those ofY ′ = AY are all apparent.

Example 10. For A =(a 00 b

), the constant vector Λ0 = (1,1) is a cyclic vector if a , b. Indeed, we

obtain

P =(1 1a b

), B =

(0 1

a′b−ab′b−a − ab b+ a+

b′−a′b−a

).

Actually, in this example, a constant vector Λ0 = (λ1,λ2) is a cyclic vector if and only if λ1λ2(b−a) , 0.

Theorem 3.18: Let A ∈Mr×rC(x) be a matrix with rational coefficients. Then there exists a cyclicvector Λ0 with coefficients in Cr [x] for the meromorphic system Y ′ = AY on C.

Proof. The existence of a cyclic vector has been proven in [4]. As a refinement of Cope’s result forour context, the above theorem has been established in [11, Chap I, §6].

In practice, for Y ′ = AY with rational A, one may use Barkatou’s algorithm [2, p. 194] in orderto find a cyclic vector:

Step 1: Choose a vector Λ0 with random polynomial coefficients.

Step 2: Test whether Λ0 is a cyclic vector.

Step 3: If yes, then the output of the algorithm is Λ0. Else, go back to Step 1.

Note that the subset of (Cr [x])r defining cyclic vectors is Zariski open. By the existence theoremfor polynomial cyclic vectors, this subset is moreover dense. So an arbitrary element of (Cr [x])r ismost often already a cyclic vector and Barkatou’s algorithm usually terminates in just a few steps.

4 Fuchsian systems over P1, global theory

Main references for this lecture:

? Anosov/Bolibruch, The Riemann-Hilbert problem [1].

? Sabbah, Déformations isomonodromiques et variétés de Frobenius, chapter 4 [12].

? Sauloy, Differential Galois Theory through Riemann-Hilbert Correspondence, chap. 12 [13].

30

4.1 Fuchsian systems over P1

LetA ∈Mr×r(C(x))

be a matrix with rational coefficients. Then the meromorphic system

Y ′ = AY

over C can also be seen as a meromorphic system over P1 = Cx ∪ {∞}. Indeed, if we denote byx̃ = 1x the natural coordinate on P

1 \ {0}, then setting Z(x̃) = Y (x) yields

Z ′ = − 1x̃2A(1x̃

)Z .

Conversely, if a meromorphic system over C extends to a meromorphic system over P1 via theabove relation, then the matrix of the system has to be rational.

Let us fix an ordered setΣ = {x1, . . . ,xn,∞} ⊂ P1

of n+ 1 distinct points in P1, one of them being∞.

Lemma 4.1: Let Y ′ = AY be a meromorphic system over P1 as above. Assume this system isFuchsian (i.e. it has only Fuchsian singularities, including at∞), and has polar locus (at most) Σ.Then the rational matrix A is of the form

A =n∑i=1

Aix − xi

(15)

with constant Ai ∈Mr×rC. Moreover, the residue at∞ of Y ′ = AY then is given by

A∞ = −n∑i=1

Ai .

4.2 Local and global monodromy

We may fix convenient generators γi of the fundamental group of P1 \ Σ with respect to somebase-point x0, such that

π1(P1 \Σ,x0) =

〈γ1, . . . ,γn,γ∞

∣∣∣ γ∞.γn . . .γ1 = 1〉 .Moreover, we may assume that these γi are all of the form

γi = αi .δi .α−1i ,

as follows.

• For each i ∈ {1, . . . ,n,∞}, choose a disc D′i around xi and a slightly bigger disc Di around xi ,such that the Di ’s are pairwise disjoint.

• For each index i, denote by δi the closed path running counterclockwise over the boundary∂D′i , with some endpoint x0i ∈ ∂D

′i .

31

• For each index i, αi is some path in P1 \Σ from x0 to x0i .

Let us denote

FS :=

A = (A1, . . . ,An,A∞) ∈ (Mr×rC)n+1∣∣∣∣∣∣∣ A∞ +

n∑i=1

Ai = 0

/GLrC

,

where the quotient is seen with respect to the action of GLrC on (Mr×rC)n+1 by simultaneous

conjugation, i.e. M ∈ GLrC acts as (A1, . . . ,An,A∞) 7→ (M−1A1M,. . . ,M−1AnM,M−1A∞M). Notethat since global holomorphic functions on P1 are constant, we have

GLr(OP1(P1)) = GLrC .

On the other hand, constant gauge transformations simply act by matrix conjugation on the matrixof a meromorphic system. In other words, (15) yields a bijection

FS '

Fuchsian systems of rank r over P1

with polar locus (at most) Σ,modulo global holomorphic gauge transformations

.Let us denote

AR :={M = (M1, . . . ,Mn,M∞) ∈ (GLrC)n+1

∣∣∣M1 · · ·Mn ·M∞ = Ir}/GLrC ,where again GLrC acts by simultaneous conjugation. We have a natural bijection

AR '

Anti-representations

ρ : π1(P1 \Σ,x0)→GLrCmodulo conjugation

,obtained by associating, to an anti-representation ρ, the n+ 1-tuple

(M1, . . . ,Mn,M∞) = (ρ(γ1), . . . ,ρ(γn),ρ(γ∞)) .

With respect to the above identifications, we can define the monodromy map

Mon :{FS → AR[A] 7→ [ρA] .

(16)

The monodromy problem asks to determine explicitly the image Mon([A]) for a given A =(A1, . . . ,An,A∞). If [M] = Mon([A]), the for each i ∈ {1, . . . ,n,∞}, the conjugacy class of the matrixMi corresponds to the monodromy of the system

Y ′ =

n∑i=1

Aix − xi

Yrestricted to Di . Hence we know that

• If Ai is non-resonant, then [Mi] = [exp(2iπAi)].

• Otherwise, one can still determine [Mi] by computing a possible Levelt normal form of thesystem over Di with polar locus {xi}.

32

However, one can not in general determine [M] = [(M1, . . . ,Mn,M∞)] from ([M1], . . . , [Mn], [M∞]).Cases where this is possible are the trivial ones n = 1 and r = 1, which we are going to excludehereafter, and, with some work, the case (r,n) = (2,2), closely related to the hypergeometric differ-ential equation

y′′ =γ − (α + β + 1)x

x(x − 1)y′ −

αβ

x(x − 1)y

with constant α,β,γ ∈C.

Definition 4.2: A (n+ 1)-tuple M ∈ GLrC is called reducible if there exists a sub-vector spaceV ⊂Cr of intermediate dimension 0 < dim(V ) < r such that Mi ·V ⊂ V for each i ∈ {1, . . . ,n+ 1}.Otherwise, M is called irreducible.

Note that reducibility is a well defined notion for the conjugacy class [M] under the action ofGLrC by simultaneous conjugation.

The multiplicative Deligne-Simpson problem asks which (n+ 1)-tuples

(C1, . . . ,Cn,C∞) ,

where each Ci ⊂GLrC is a conjugacy class of matrix, arise via Ci = [Mi] from irreducible elements[M] ∈ AR. We refer to [9] for a survey.

4.3 The Riemann-Hilbert problem

Recall the monodromy mapMon : FS→ AR,

which to [A] ∈ FS, seen as the gauge equivalence class of the Fuchsian system Y ′ =(∑n

i=1Aix−xi

)Y

over P1, associates the conjugacy class [M] ∈ AR, where Mi = ρ(γi) for the monodromy represen-tation ρ of the Fuchsian system with respect to some fundamental solution and base point.

Remark. Note that Mon is not injective. Indeed, one may conjugate different elements in FS by mero-morphic gauge transformations over P1 with polar locus Σ, without changing the monodromy.

The classical Riemann-Hilbert problem asks : is Mon is surjective ?

This problem has a long history:

• It appeared as problem number 21 in Hilbert’s list of problems presented at the ICM in Parisin 1900.

• In 1907, Plemelj published an article providing a positive answer to the Riemann-Hilbertproblem.

• However, about seventy years later (published in 1983), Treibich-Kohn discovered a gap inPlemelj’s proof.

• By a result of Dekkers in 1979, the answer to Hilbert’s twenty-first problem is still affirma-tive in the case of rank r = 2.

• As shown by Bolibruch in the 1990’s, Mon is not surjective for r ≥ 3 (if n ≥ 2).

33

Since Bolibruch’s counterexamples, the natural question is to find necessary and sufficientconditions for a [M] ∈ AR to be in the image of Mon. The most important sufficient conditionsthat have been established are the following:

• Mi is semisimple for at least one i ∈ {1, . . . ,n,∞}. This is actually Plemelj’s result, correctedby Treibich-Kohn.

• [M] is irreducible. This has been proven independently by Bolibruch and Kostov.

We are not going to present the proofs of these results. They can however be seen as a refinementof the proof of the following theorem.

Theorem 4.3: Letρ : π1(P

1 \Σ,x0)→GLrC

be an antirepresentation. Then there exists a meromorphic system

Y ′ = AY

over P1 with polar locus Σ = {x1, . . . ,xn,∞}, such that

• the monodromy of this system is [ρ],

• the singularities x1, . . . ,xn of this system are all Fuchsian

• the singularity∞ of this system is regular singular.

Proof. • By Riemann-Hilbert correspondence, there exists a holomorphic vector bundle E0over P1 \ Σ endowed with a holomorphic connection ∇0, such that the monodromy ρ0 of(E0,∇0) with base-point x0 is isomorphic to ρ.

• For each i ∈ {1, . . . ,n,∞}, there exists a holomorphic vector bundle Ei over the disc Di aroundxi , endowed with a meromorphic connection ∇i with polar divisor 1 · {xi} such that the mon-odromy of the holomorphic connection (Ei ,∇i)|D∗i , where D

∗i := Di \ {xi}, is, with respect to

the base-point x0i , isomorphic to

ρi :{π1(D∗i ,x0i) → GLrC

δi 7→ ρ(γi) .

Indeed, choose for example an Euler system Y ′ = Bix−xi Y over Di with polar locus {xi} andmonodromy [ρi], let Ei be a trivial vector bundle over Di and let ∇i : E i → E i ⊗Ω1Di (1 · {xi})be given, with respect to the some global frame ei of Ei , by the connection matrix

Ωi := −Bix − xi

dx .

• We may “glue” the connections (E0,∇0), (E1,∇1), . . . , (En,∇n), (E∞,∇∞) to a vector bundle Eover P1 endowed with a meromorphic connection

∇ : E → E ⊗Ω1P1(Σ) ,

34

where we abusively also denote by Σ the divisor 1 · {x1} + · · · + 1 · {xn} + 1 · {∞}. Indeed, foreach i ∈ {1, . . . ,n,∞}, the monodromies with respect to x0i of the holomorphic connections(Ei ,∇i)|D∗i and (E

0,∇0)|D∗i are isomorphic. Hence by Riemann-Hilbert correspondence, thereexist isomorphisms

ψi : Ei |D∗i → E

0|D∗icompatible with the respective holomorphic connections. We may set

E :=(E0qE1q ·· ·qEnqE∞

)∼

to be the disjoint union of all the previously chosen vector bundles, quotiented by the equiv-alence relation declaring, for any x ∈D∗i , vi ∈ E

ix and v0 ∈ E0x ,

vi ∼ v0 if v0 = ψi(vi) .Since for any i, the connections ∇0 and ∇i are compatible with respect to ψi , this provides awell-defined meromorphic connection ∇ on E satisfying moreover:

– ∇ has only simple poles, and polar locus Σ,– the monodromy of the holomorphic connection (E,∇)|P1\Σ is isomorphic to ρ = ρ0.

• Although the vector bundles Ei are all trivial by the Grauert-Röhrl theorem, the vector bun-dle E over P1 obtained by conveniently gluing them might not be. However, by the Birkhoff-Grothendieck theorem, the holomorphic vector bundle E over P1 admits two holomorphicframes e over C and ê over P1 \ {0} such that over C∗, these frames are related via

e = ê ·Φ , where Φ =

x̃k1 0

. . .

0 x̃kr

for some (k1, . . . , kr ) ∈ Zr . Let us denote by Ω and Ω̂ respectively the connection matrices of∇ with respect to e and ê. Recall that they are related over C∗ by

Ω = Φ−1Ω̂Φ +Φ−1dΦ . (17)

• We may write Ω = −Adx for some A ∈Mr×r(MP1(C)). We claim that the systemY ′ = AY

can be seen as a system over P1 which satisfies all the requirements of the statement. Indeed,

– By construction, the monodromy of the holomorphic system Y ′ = AY over P1 \Σ is [ρ].– Since Ω̂, Φ and Φ−1 are meromorphic over P1 \ {0}, it follows from (17) that Ω extends

to a meromorphic matrix-valued one-form over P1. Hence A has rational coefficients.– By construction, the poles x1, . . . ,xn of A are (at most) simple poles.– The meromorphic system

Ỹ ′ = ÃỸ

over P1 \ {0} with polar locus Σ obtained from Y ′ = AY by setting Ỹ (x̃) = Y(

1x

)has a

regular singularity at {x̃ = 0} = {x =∞}. Indeed, by (17), it is meromorphically gauge-equivalent, via Ỹ = ΦŶ , to the Fuchsian system

Ŷ ′ = Â(x̃)Ŷ

over P1 \ {0}, where Ω̂ = −Âdx̃.

35

References

[1] D. Anosov, A. Bolibruch, The Riemann-Hilbert problem, A Publication from the Steklov Insti-tute of Mathematics 22 , 2013.

[2] M. Barkatou, An algorithm for computing a companion block diagonal form for a system of lineardifferential equations, Applicable algebra in engineering, communication and computing 4.3 ,pp. 185–195, 1993.

[3] C. Behrenstein and R. Gay, Complex variables, an introduction, Graduate Texts in Mathematics125, Springer, 1991.

[4] F. T. Cope, Formal Solutions of Irregular Linear Differential Equations. Part II , American Journalof Mathematics 58 (1), pp. 130–140, 1936.

[5] P. Deligne, Équations différentielles à points singuliers réguliers, Springer 163, 2006.

[6] O. Forster Lectures on Riemann Surfaces, Graduate Texts in Mathematics 81, Springer, 1999.

[7] Y. Ilyashenko and S. Yakovenko, Lectures on Analytic Differential Equations, American Mathe-matical Society 86, 2008.

[8] V. Kleptsyn and B. Rabinovic, Analytic classification of Fuchsian singular points, MathematicalNotes 76 (3-4), pp. 248–357, 2004.

[9] V. Kostov, Kostov, The Deligne-Simpson problem–a survey, Journal of Algebra, 281(1), pp. 83–108, 2004.

[10] C. Mitschi and D. Sauzin, Divergent Series, Summability and Resurgence I, Lecture Notes inMathematics 2153, Springer, 2016.

[11] J.-P. Ramis, Théorèmes d’indices Gevrey pour les équations difféerentielles ordinaires, Memoirs ofthe AMS 296, American Mathematical Society, 1984.

[12] C. Sabbah, Déformations isomonodromiques et variétés de Frobenius, Savoirs Actuels, EDP Sci-ences, CNRS Editions, 2002.

[13] J. Sauloy, Differential Galois Theory through Riemann-Hilbert Correspondence, Graduate Studiesin Mathematics 177, American Mathematical Society, 2017.

[14] M. Singer and M. van der Put, Galois theory of linear differential equations, Grundlehren derMathematischen Wissenschaften 328, Springer, 2012

[15] C. Wagschal, Fonctions holomorphes, Équations différentielles, Méthodes, Hermann Éditeursdes Sciences et de l’Art, 2003.

[16] W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Dover Publications,2018.

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Holomorphic systems of first order LDE on complex domainsDefinitionsBasic existence and uniqueness theoremWronskianFundamental solutionsConvergence of formal solutionsAnalytic continuation

Monodromy and Riemann-Hilbert correspondenceMonodromy of holomorphic systemsHolomorphic gauge transformationsLocal systemsHolomorphic vector bundles and connectionsIntegrable connectionsRiemann-Hilbert correspondence

Meromorphic systems, local theorySome terminology for meromorphic systemsFuchsian singularities and their classificationRegular singularities of meromorphic systemsRegular singularities of differential equationsCyclic vectors

Fuchsian systems over ¶1, global theoryFuchsian systems over ¶1Local and global monodromyThe Riemann-Hilbert problem