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Isomonodromic differential equations and differential categories

Sergey Gorchinskiy and Alexey Ovchinnikov

ABSTRACT

We study isomonodromicity of systems of parameterized linear differential equations and related conju-gacy properties of linear differential algebraic groups bymeans of differential categories. We prove thatisomonodromicity is equivalent to isomonodromicity with respect to each parameter separately under afiltered-linearly closed assumption on the field of functions of parameters. This result cannot be furtherstrengthened by weakening the requirement on the parameters as we show by giving a counterexam-ple. Also, we show that isomonodromicity is equivalent to conjugacy to constants of the associatedparameterized differential Galois group, extending a result of P. Cassidy and M. Singer, which we alsoprove categorically. We illustrate our main results by a series of examples, using, in particular, a relationbetween Gauss–Manin connection and parameterized differential Galois groups.

1. Introduction

A system of parameterized linear differential equations isa system of linear differential equations whose coeffi-cients are functions of principal variablesx1, . . . , xn and parameterst1, . . . , td and derivations only with respect tox1, . . . , xn appear in the system. We say that such a system is isomonodromic if it can be extended to a consistentsystem of linear differential equations with derivations with respect to all ofx1, . . . , xn, t1, . . . , td. That is, onerequires that the extended system satisfies all integrability conditions with respect to the principal and paramet-ric variables. In this paper, we study such isomonodromic systems via the parameterized Picard–Vessiot (PPV)theory [5] and differential Tannakian categories [15, 37, 38, 23, 22, 2].

To verify isomonodromicity of a system of parameterized linear differential equations, say, with one principalvariablex andd parameterst1, . . . , td explicitly means to findd extra matrices that satisfy

(d+12

)integrability

conditions [5, Definition 3.8]. We improve this by showing that it is enoughto find matrices that satisfy onlydintegrability conditions for pairs of derivations(∂x, ∂ti) under a filtered-linearly closed assumption (Definition3.7)on the field of functions of parameters (Theorem6.3and Remark6.4). Namely, the existence of the latter matricesimplies the existence of (possibly, different) matrices that satisfy all

(d+12

)integrability conditions. This result is

non-trivial not only because of the method of proof (which uses differential categories [15] and CDG-algebras [39])but also because it is counterintuitive. The initial explicit steps for this result restricted to2 × 2 systems with theparameterized differential Galois group Zariski dense inSL2 can be found in [13, Proposition 4.4] (see also [47,Theorem 1.3, Chapter 2]). Note that the condition on the fieldto be filtered-linearly closed is, indeed, necessary asis shown in Example6.7. This example is based on iterated integrals.

A similar but more specialized question was treated in [19, 20] for the case of rational functions in the principalvariable. Using analytic methods, it is shown that, ford extra matrices of a certain special type,d integrabilityconditions imply all

(d+12

)integrability conditions for the same matrices. Additionally, it is proved in [19, 20] that,

if the differential equation is isomonodromic (when restricted only to rational functions in the principal variable),then one can choose extra matrices of the special type discussed above (for more details, see Section6.2).

Given a system of parameterized linear differential equations, the PPV theory associates a parameterized dif-ferential Galois group, which can be represented by groups of invertible matrices whose entries are in the field ofconstants, that is, the field of functions of the parameterst1, . . . , td. Moreover, these groups are linear differentialalgebraic groups (LDAGs), that is, groups of matrices satisfying a system of polynomial differential equations with

2010 Mathematics Subject Classification12H05 (primary), 12H20, 13N10, 20G05, 20H20, 34M15, 34M56,37K20 (secondary)Keywords:differential Tannakian categories, isomonodromic differential equations, differential Galois theory

S. Gorchinskiy was supported by the grants RFBR 11-01-00145-a, NSh-4713.2010.1, MK-4881.2011.1, and AG Laboratory GU-HSE,RF government grant, ag. 11 11.G34.31.0023..

A. Ovchinnikov was supported by the grants: NSF CCF-0952591and PSC-CUNY No. 60001-40 41.

SERGEY GORCHINSKIY AND ALEXEY OVCHINNIKOV

respect of the parametric derivations [3, 4, 25, 36, 38]. Using descent for connections (Lemma3.11), we prove inTheorem6.6that, under the filtered-linearly closed assumption on the field of constants, a system of parameterizedlinear differential equations is isomonodromic (Definition 6.1) if and only if its Galois group is conjugate (possibly,over an extension field of the field of constants) to a group of matrices whose entries are constant functions in theparameters. This extends the corresponding result in [5], which required the field of constants to be differentiallyclosed. Recall that a differential field is differentially closed if it contains solutions of all consistent systems ofpolynomial differential equations with coefficients in thefield. Note that, even in the case of a differentially closedfield of constants, our proof, based on differential Tannakian categories, is different from the one given in [5].

We construct examples showing that, in general, one really needs to take an extension of the field of constants toobtain the above conjugacy (Examples6.8and6.9). The construction of the examples uses an explicit description ofGalois groups for PPV extensions defined by integrals (Propositions5.2and5.4), which seems to have interest in itsown right. Namely, we interpret such differential Galois groups in terms of Gauss–Manin connections (Section5).More concretely, the examples involve the incomplete Gamma-function and the Legendre family of elliptic curves.Note that the relation between the PPV theory and Gauss–Manin connection was also elaborated in [46].

Recall that the Galois groups in the PPV theory are LDAGs. As noted above, isomonodromicity correspondsto conjugation to constants for LDAGs. In this way, our Theorem4.4corresponds to Theorem6.3and says that ifa LDAG is conjugate to groups of matrices whose entries are constants with respect to each derivation separately,then there is a common conjugation matrix, under the filtered-linearly and linearly closed assumption on the differ-ential field. This matrix may have entries in a Picard–Vessiot extension of the base field. We construct an exampleshowing that, in general, one needs to take a Picard–Vessiotextension (Example4.9).

As an application, we obtain a generalization of [33, Theorem 3.14], which characterizes semisimple categoriesof representations of LDAGs in the case when the ground field is differentially closed and has only one derivation.In Theorem4.6, we improve this result by showing a more general statement without these inconvenient restrictionsto differentially closed fields and the case of just one derivation.

Our method is based on the new notion of differential objectsin differential categories (Definition3.1). Weprove that there is a differential structure on an objectX in a differential category over a differential field(k,Dk)if and only if there is a differential structure onX with respect to any derivation∂ ∈ Dk, provided that(k,Dk)is filtered-linearly closed (Proposition3.10). This result is applied to both isomonodromic differential equationsand LDAGs. We show in Example4.7 that this result is not true over an arbitrary differential field already forthe category of representations of a LDAG overQ(t1, t1). The example uses the Heisenberg group. Note that theapplication to isomonodromic differential equations requires that we work with arbitrary differential categories,not just with differential Tannakian categories or categories of representations of LDAGs (see also the discussionat the end of Section2).

In [26], Landesman initiated the parameterized differential Galois theory in a more general setting based onKolchin’s axiomatic development of the differential algebraic group theory [25]. Galois theories in which Galoisgroups are LDAGs also appear in [16, 17, 18, 11, 10, 12, 51, 50] with the initial algorithm given in [13] and analyticaspects studied in [34, 35]. The representation theory for LDAGs was also developed in[36, 32], and the relationswith factoring partial differential equations was discussed in [6]. Analytic aspects of isomonodromic differentialequations were studied by many authors, let us mention [19, 20, 30, 31]. See also a survey [41] of Bolibrukh’sresults on isomonodromicity and the references given there.

The paper is organized as follows. We start by recalling the basic definitions and properties of differential alge-braic groups, differential Tannakian categories, and the PPV theory in Section2. Most of our notation is introducedin this section. The following section contains our main technical tools, Proposition3.10 and Proposition3.12.Section4 deals with conjugating linear differential algebraic groups to constants over not necessarily differentiallyclosed fields. The results from Section5, where we also establish a relation with Gauss–Manin connection, areused in order to construct non-trivial examples to Theorem6.6. In Section6, we show our main results on isomon-

2

ISOMONODROMIC DIFFERENTIAL EQUATIONS AND DIFFERENTIAL CATEGORIES

odromic systems of parameterized linear differential equations as well as illustrate them with examples justifyingthe necessity of the hypotheses in our main result. Also, we give an analytic interpretation of our results includingthe reasons that support of the conclusion of the above example.

The authors thank P. Cassidy, L. Di Vizio, A. Its, B. Malgrange, A. Minchenko, O. Mokhov, T. Scanlon,M. Singer, and D. Trushin for very helpful conversations andcomments. We are also very grateful to the refereefor the suggestions.

2. Notation and preliminaries

Most of the notation and notions that we use are taken from [15]. All rings are assumed to be commutative andhaving a unit.

In the paper,(k,Dk) stays for a differential field of zero characteristic, that is, k is a field andDk is a finite-dimensionalk-vector space with a Lie bracket and ak-linear map of Lie rings

Dk → Der(k, k)

that satisfies a compatibility condition (see [15, Definition 3.1]). For example, if∂1, . . . , ∂n denote commutingderivations fromk to itself (possibly, some of them are zero), then(k,Dk) with

Dk := k · ∂1 ⊕ . . . ⊕ k · ∂n

is a differential field. LetDk in the superscript denote takingDk-constants, that is, the elements annihilated by all∂ ∈ Dk. Putk0 := kDk . PutΩk := D∨

k . We have the de Rham complexΩ•k

0 −−−−→ Ω0k

d−−−−→ Ω1

kd

−−−−→ Ω2k

d−−−−→ . . . ,

whereΩik := ∧i

kΩk, i > 1, and we putΩ0k := k (see [15, Remark 3.4]). Note thatd is k0-linear andd d = 0.

Denote the category of sets bySets. Denote the category ofk-vector spaces byVect(k). Denote the categoryof k-algebras byAlg(k). Denote the category ofDk-modules overk byDMod(k,Dk) (see [15, Definition 3.19]).Denote the category ofDk-algebras overk by DAlg(k,Dk) (see [15, Definition 3.12]). Denote the ring ofDk-polynomials in differential indeterminatesy1, . . . , yn by

ky1, . . . , yn.

We say that a (possibly, infinite-dimensional)Dk-moduleM is trivial if the multiplication map

k ⊗k0 MDk →M

is an isomorphism (by [40, Lemma 1.7], this map is always injective).

We say that a differential field(k,Dk) is linearly Dk-closedif (k,Dk) has no non-trivial Picard–Vessiot ex-tensions, that is, all finite-dimensionalDk-modules overk are trivial (see also [29, 43, 42], [28, Section 3], [25,Section 0.5] for the existence and use of such fields, and [1] for analogues for difference fields). One can alsoiteratively apply [44, Embedding Theorem] to realize such fields (if they are countable) as germs of meromorphicfunctions indimk(Dk) variables.

A functorX : DAlg(k,Dk) → Sets is corepresented by aDk-algebraA if there is a functorial isomorphism

X(R) ∼= HomDk(A,R)

for anyDk-algebraR. A linearDk-group is a group-valued functor onDAlg(k,Dk) that is corepresented by aDk-finitely generatedDk-Hopf algebra. Given a (pro-)linearDk-group, denote the category of finite-dimensionalrepresentations ofG as a (pro-)algebraic group overk by Rep(G).

Given a functorX : Alg(k) → Sets, one traditionally denotes also its composition with the forgetful func-tor DAlg(k,Dk) → Alg(k) by X. If X is corepresentable onAlg(k), thenX is also corepresentable on

3

SERGEY GORCHINSKIY AND ALEXEY OVCHINNIKOV

DAlg(k,Dk). In other words, the forgetful functorDAlg(k,Dk) → Alg(k) has a left adjoint (for example,see [14, Section 1.2]), which is usually called a prolongation. In particular, we have a corepresentable functor

An : R 7→ R⊕n,

whereR is aDk-algebra. Also, given a finite-dimensionalk-vector spaceV , we have a linearDk-group

GL(V ) : R 7→ AutR(R⊗k V ).

Given a functorY : Alg(k0) → Sets, letY c denote its composition with the functor ofDk-invariants

DAlg(k,Dk) → Alg(k0), R 7→ RDk .

We say that functors of typeY c areconstant. If Y is corepresented by ak0-algebraB, thenY c is corepresented by

k ⊗k0 B (1)

with the naturalDk-structure, whereDk acts by zero onB. Denote the latterDk-algebra byBc and also callit constant. IfH is a linear algebraic group, thenHc is a constant linearDk-group. In particular, we have acorepresentable functor

(An)c : R 7→(RDk

)⊕n,

whereR is aDk-algebra. Also, given a finite-dimensionalk0-vector spaceV0, we have the linearDk-group

GL(V0)c : R 7→ AutRDk

(RDk ⊗k0 V0

).

It follows that there is a morphism of linearDk-groupsGL(V0)c → GL(V ), whereV := k ⊗k0 V0.

Note that aDk-algebraA is constant if and only ifA is trivial as aDk-module. ADk-finitely generatedDk-algebraA is constant if and only if there is an isomorphism ofDk-algebras

A ∼= ky1, . . . , yn/I,

whereI ⊂ ky1, . . . , yn is aDk-ideal such that, for all∂ ∈ Dk andi, 1 6 i 6 n, the differential polynomial∂yiis in I.

For a more explicit description of constant algebras, consider a functor

X : DAlg(k,Dk) → Sets

corepresented by a reducedDk-finitely generatedDk-algebra. Then, by the differential Nullstellensatz (see [24,Theorem IV.2.1]),X is constant if and only if there is a Kolchin closed embeddingX ⊂ An over (k,Dk) suchthat, for aDk-closed fieldU overk (equivalently, for anyU as above), we have

X(U) ⊂ Un0 , U0 := UDk ,

that is, all points inX ⊂ An have constant coordinates.

Given aDk-objectX overk (e.g., aDk-module, aDk-algebra, a linearDk-group) and aDk-field l overk, letXl denote theDl-object overl obtained by the extension of scalars from(k,Dk) to (l,Dl), whereDl := l ⊗k Dk.

One finds the definition of a parameterized differential fieldin [15, Section 3.3]. Recall that, for a parameterizeddifferential field(K,DK) over(k,Dk), one has aK-linear map

DK → K ⊗k Dk,

called a structure map. This defines a differential field(K,DK/k

), whereDK/k is the kernel of the structure map.

Also, one hasKDK/k = k. For example, if

k = C(t), K = C(t, x), Dk = k · ∂t, and DK = K · ∂x ⊕K · ∂t,

then(K,DK) is a parameterized differential field over(k,Dk) with DK/k = K · ∂x.

4

ISOMONODROMIC DIFFERENTIAL EQUATIONS AND DIFFERENTIAL CATEGORIES

Given a finite-dimensionalDK/k-moduleN overK, one has the notion of a parameterized Picard–Vessiot(PPV) extensionL for N , whereL is aDK-field overK. This was first defined in [5] (see also [15, Definition3.27] for the present approach to parameterized differential fields). Ifk isDk-closed, then a PPV extension existsfor anyN as above (see [5, Theorem 3.5(1)]). Given a PPV extensionL, one shows that the group-valued functor

GalDK (L/K) : DAlg(k,Dk) → Sets, R 7→ AutDK (R⊗k A/R⊗k K)

is a linearDk-group (see [15, Lemma 8.2]), which is called the parameterized differential Galois group ofL overK, whereA is the PPV ring associated toL (see [15, Definition 3.28]).

The main notion defined in [15] is that of a differential category. ADk-categoryC overk is an abeliank-lineartensor category together with exactk-linear endofunctorsAt1C andAt2C , called Atiyah functors, that satisfy a list ofaxioms (see [15, Sections 4.2,4.3]). In particular, for any objectX in C, there is a functorial exact sequence

0 −−−−→ Ωk ⊗k X −−−−→ At1C(X)πX−−−−→ X −−−−→ 0,

where, as above,Ωk = D∨k , and a functorial embedding

At2C(X) ⊂ At1C(At1C(X)

).

We have the equality

Sym2k Ωk ⊗k X = At2C(X) ∩ (Ωk ⊗k Ωk ⊗k X) ⊂ At1C

(At1C(X)

), (2)

and both compositions

At2C(X) −−−−→ At1C(At1C(X)

) At1C(πX)

−−−−−→ At1C(X), (3)

At2C(X) −−−−→ At1C(At1C(X)

) πAt1(X)−−−−−→ At1C(X) (4)

are surjective (see [15, Lemma 4.14,Proposition 4.18]). For example,Vect(k) has a canonicalDk-structure givenby the usual Atiyah extension (see [15, Example 4.7]). This induces a canonicalDk-structure onRep(G) for a(pro-) linearDk-group (see [15, Example 4.8]). Another important example is the category

DMod(K,DK/k)

ofDK/k-modules overK, where(K,DK) is a parameterized differential field over(k,Dk). The categoryDMod(K,DK/k)is k-linear and has a canonicalDk-structure (see [15, Theorem 5.1]). In [15, Section 4.4], the authors investigatedifferential Tannakian categories. Any neutralDk-Tannakian category with aDk-fiber functor is equivalent toRep(G) with the forgetful functor, whereG is a (pro-)linearDk-group. Note that the categoryDMod(K,DK/k)is not necessarily aDk-Tannakian category (even without the requirement of beingneutral), because the categoryVect(K,DK) does not necessarily have a structure of aDk-category.

3. Dk-objects inDk-categories

In this section, we defineDk-objects in abstractDk-categories. This notion and its main property given in Proposi-tion 3.10are used in Sections4 and6 for applications to linearDk-groups and isomonodromic parameterized lineardifferential equations, respectively. As we will further see in Example4.7, the filtered-linearly closed assumptionof the proposition cannot be removed.

Let C be aDk-category overk, X be an object inC. The following definition is a variation of [15, Definition3.34].

DEFINITION 3.1. ADk-connectiononX is a section

sX : X → At1C(X)

5

SERGEY GORCHINSKIY AND ALEXEY OVCHINNIKOV

of πX . A Dk-connectionsX is aDk-structureif the image of the composition

XsX−−−−→ At1C(X)

At1C(sX)

−−−−−→ At1C(At1C(X)

)

is contained inAt2C(X). A Dk-objectin C is an object together with aDk-structure on it.

EXAMPLE 3.2. To give aDk-connection on ak-vector spaceV as an object inC = Vect(k) is the same as to givea usual connection onV , that is, a map

∇V : V → Ωk ⊗k V

that satisfies the Liebniz rule. ADk-object inVect(k) is the same as aDk-module (see [15, Proposition 3.41]).

Let us give an equivalent condition for the existence of aDk-connection. For any∂ ∈ Dk, theDk-categoryChas a canonical structure of a∂-category by [15, Proposition 4.12(i)]. Explicitly, a calculation shows that, for eachobjectX in C, we have

At1C,∂(X) = At1C(X)/U, where U := Ker(∂ ⊗ idX : Ωk ⊗k X → X)

andAt1C,∂ is the Atiyah functor that corresponds to the∂-category structure onC. Denote the quotient morphismby

α∂ : At1C(X) → At1C,∂(X).

SinceU is contained inΩk ⊗k X, the morphismπX : At1C(X) → X factors throughα∂ . That is, we obtain amorphism

πX,∂ : At1C,∂(X) → X

such thatπX,∂ α∂ = πX . By definition, a∂-connection onX is a section ofπX,∂ .

PROPOSITION3.3. There is aDk-connection onX if and only if there is a basis∂1, . . . , ∂d in Dk overk such thatfor anyi, there is a∂i-connection onX.

Proof. The existence of aDk-connection onX implies the existence of a∂-connection onX for any∂ ∈ Dk bythe explicit construction ofAt1C,∂ given above.

Now let us show the reverse implication. The morphismsα∂i , 1 6 i 6 d, defined above give a morphism

α : At1C(X) → Z

such thatπ α = πX , where

Z := At1C,∂1(X) ×X . . .×X At1C,∂d(X)π

−→ X.

is the fibred product inC. Thus, we have the following commutative diagram:

0 −−−−→ Ker(πX) −−−−→ At1C(X)πX−−−−→ X −−−−→ 0

y α

y idX

y

0 −−−−→ Ker(π) −−−−→ Zπ

−−−−→ X −−−−→ 0

Since∂1, . . . , ∂d is a basis ofDk overk, the map

d⊕

i=1

∂i : Ωk → k⊕d

is an isomorphism. It follows that the restriction ofα toKer(πX) = Ωk ⊗k X is an isomorphism

Ker(πX)⊕i(∂i⊗idX)−−−−−−−→ Ker(π) =

d⊕i=1

Ker(πX,∂i) =d⊕

i=1X.

6

ISOMONODROMIC DIFFERENTIAL EQUATIONS AND DIFFERENTIAL CATEGORIES

Thus,α itself is an isomorphism. Hence, given sectionssi of the morphismsπi for all i, 1 6 i 6 d, we obtain asectionsX of πX .

In what follows, we address the question whether the existence of aDk-connection onX implies the existenceof aDk-structure onX. It will be convenient to use the following notion first introduced in [39]. Recall that, for agraded associative algebra

A• =⊕

i

Ai,

the commutator is defined by the formula

[a, b] := a · b− (−1)deg(a) deg(b)b · a

for homogenous elementsa, b ∈ A•.

DEFINITION 3.4. ACDG-structureon a graded associative algebraA• over a fieldF is a pair(d, h), where

d : Ai → Ai+1

is a collection ofF -linear maps that satisfy the graded Leibniz rule

d(a · b) = d(a) · b+ (−1)deg(a)a · d(b)

for all homogenous elementsa, b ∈ A•, andh ∈ A2 is such that

(d d)(·) = [h, · ], d(h) = 0.

Given a CDG-structure(d, h) onA• and an elementa ∈ A1, we obtain a new CDG-structure with

d′ = d+ [a, · ], h′ = h+ d(a) + a2. (5)

By definition, the CDG-structures(d, h) and(d′, h′) areequivalent.

EXAMPLE 3.5.

(i) The pair(d, 0) defines a CDG-structure on the graded associative algebraΩ•k overk0, whered denotes the

differential in the de Rham complex.

(ii) Let V be ak-vector space,∇V be aDk-connection onV . We obtain maps

∇V : Ωik ⊗k V → Ωi+1

k ⊗k V, ∇V (ω ⊗ v) := dω ⊗ v + (−1)iω ∧ ∇V (v)

and a CDG-structure(d, h) on the graded associative algebraΩ•k ⊗k Endk(V ) overk0 with

d(a) := (idΩk∧ a) ∇V − (−1)i∇V a, a ∈ Ωi

k ⊗k Endk(V ) = Homk(V,Ωik ⊗k V ),

h := ∇V ∇V ∈ Ω2k ⊗k Endk(V ) = Homk(V,Ω

2k ⊗k V ).

The conditiond(h) = 0 is classically called the second Bianchi identity. Note that h vanishes if and only ifthe connection∇V is aDk-structure onV . The natural embedding

Ω•k ⊂ Ω•

k ⊗k Endk(V )

given byidV ∈ Endk(V ) commutes withd. Thus, the notationd in the CDG-structure onΩ•k ⊗k Endk(V )

does not lead to a contradiction.

There is a notion of a morphism between differential fields(k,Dk) → (K,DK), which generalizesDk-fieldsoverk (see [15, Definition 3.6]). In particular, we have an injectivek-linear mapΩk → ΩK . Given such a mor-phism, one defines differential functors fromDk-categories overk to DK-categories overK (see [15, Defini-tion 4.9]). For example, ifC is a Tannakian category, then there is a faithful differential functorC → Vect(K) foraDk-fieldK overk. The following result generalizes Example3.5(ii ).

7

SERGEY GORCHINSKIY AND ALEXEY OVCHINNIKOV

LEMMA 3.6. Let sX be aDk-connection onX. Suppose that there is a morphism of differential fields(k,Dk) →(K,DK) and a faithful differential functorF : C → Vect(K). Then the following is true:

(i) sX defines a CDG-structure(d, h) on the graded associative algebraΩ•k ⊗k EndC(X) overk0;

(ii) h vanishes if and only ifsX is aDk-structure;

(iii) given a CDG-structure(d′, h′) on Ω•k ⊗k EndC(X), there is aDk-connections′X on X such that(d′, h′)

corresponds tos′X if and only if (d′, h′) is equivalent to(d, h).

Proof. First let us show (i). The sectionsX defines a map

∇ : EndC(X) → Ωk ⊗k EndC(X), ∇(a) := sX a−At1C(a) sX .

In other words,∇(a) measures non-commutativity of the diagram

XsX−−−−→ At1C(X)

a

y At1(a)

y

XsX−−−−→ At1C(X).

One checks that∇ is aDk-connection on thek-algebraEndC(X). By the (graded) Leibniz rule, this extendsuniquely to a collection ofk0-linear maps

d : Ωik ⊗k EndC(X) → Ωi+1

k ⊗k EndC(X).

Next, let us defineh ∈ Ω2k ⊗k EndC(X). Put

Y := Ker(At1C(πX)− πAt1(X) : At

1C

(At1C(X)

)→ At1C(X)

).

We claim that the image of the composition

At1C(sX) sX : X → At1C(At1C(X)

)

is contained inY . To prove this, recall thatπX sX = idX . SinceAt1C is a functor, we have

At1C(πX) At1C(sX) = idAt1(X),

whence

At1C(πX) At1C(sX) sX = sX . (6)

Since the morphismAt1C(X)πX−→ X is functorial inX, the following diagram commutes:

At1C(X)At1(sX)−−−−−→ At1C

(At1C(X)

)

πX

y πAt1(X)

y

XsX−−−−→ At1C(X).

Hence, we have

πAt1(X) At1C(sX) sX = sX πX sX = sX . (7)

Combining (6) and (7), we obtain the following equality of morphisms fromX to At1C(At1C(X)

):

At1C(πX) At1C(sX) sX = πAt1(X) At1C(sX) sX .

Thus, the image ofAt1C(sX) sX is contained inY .

SinceF : C → Vect(K) is faithful, we have thatAt2C(X) ⊂ Y (see [15, Remark 4.21(iii)]). By the construc-tion of Y , we have the following exact sequence

0 −−−−→ Ωk ⊗k Ωk ⊗k X −−−−→ YAt1(πX)−−−−−→ At1C(X) −−−−→ 0.

8

ISOMONODROMIC DIFFERENTIAL EQUATIONS AND DIFFERENTIAL CATEGORIES

By (2) and (3) (see Section2), we obtain an isomorphism

Ω2k ⊗k X

∼−→ Y/At2C(X).

Put

h ∈ Ω2k ⊗k EndC(X) = HomC(X,Ω

2k ⊗k X)

to be the composition

XAt1(sX)sX−−−−−−−→ Y −−−−→ Y/At2C(X)

∼−−−−→ Ω2

k ⊗k X.

It remains to show the identities

d d = [h, · ], d(h) = 0. (8)

One can show that ifC is the category of vector spaces over a differential field, then d andh constructedas above coincide with those defined in Example3.5 (ii ). Further, the constructions ofd andh commute withdifferential functors. More explicitly, consider the differential functorF : C → Vect(K). The morphism ofdifferential fields(k,Dk) → (K,DK) defines a homomorphism of graded algebrasΩ•

k → Ω•K , which commutes

with the de Rham differentiald (see [15, Definition 3.6]). The functorF induces a homomorphism of gradedalgebras

α : Ω•k ⊗k EndC(X) → Ω•

K ⊗K EndK (F (X)) .

The connectionsX onX defines aDK -connection on theK-vector spaceF (X) such thatα commutes withd andpreservesh. SinceF is faithful,α is injective. Thus, we obtain (8) by Example3.5(ii ) applied toK-vector spaces.This finishes the proof of (i).

Further, (ii ) follows from the construction ofh. To prove (iii ), note that any otherDk-connection onX isgiven by

s′X = sX + a, (9)

where

a ∈ Ω1k ⊗k EndC(X)

is an arbitrary element. We need to show that the corresponding CDG-structure(d′, h′) on Ω•k ⊗k EndC(X)

satisfies (5). As above, by the injectivity of the algebra homomorphismα, it is enough to consider the caseC = Vect(K), in which the required follows from Example3.5(ii ).

It follows from Lemma3.6 that if dimk(Dk) = 1 andC satisfies the condition from Lemma3.6, then anyDk-connectionsX on an objectX in C is aDk-structure onX.

One can give a different definition of aDk-category so that Lemma3.6holds for anyDk-category in this newsense. Namely, one should require the compatibility condition from [15, Remark 4.21(i)] and also the pentagoncondition forΨ in notation from there. The latter condition involves consideration of the third jet-ringP 3

k .

DEFINITION 3.7. We say that a differential field(k,Dk) is filtered-linearly closedif there is a sequence ofk-vectorsubspaces closed under the Lie bracket

0 = D0 ⊂ D1 ⊂ . . . ⊂ Dd−1 ⊂ Dd = Dk

such that for anyi, 0 6 i 6 d− 1, we have

dimk (Di+1/Di) = 1

andk is linearlyDi-closed.

Note that, in Definition3.7, we do not require thatk be linearlyDk-closed, that is, a filtered-linearly closedfield is not necessarily linearly closed.

9

SERGEY GORCHINSKIY AND ALEXEY OVCHINNIKOV

EXAMPLE 3.8.

(i) If dimk(Dk) = 1, then(k,Dk) is filtered-linearly closed.

(ii) If k is Dk-closed, then(k,Dk) is filtered-linearly closed. Indeed, sincek is Dk-closed, the natural mapDk → Der(k, k) is injective. By [25, p. 12, Proposition 6], there is a commuting basis∂1, . . . , ∂d in Dk overk and we put

Di := spank〈∂1, . . . , ∂i〉.

Again, sincek isDk-closed, we see thatk is linearlyDi-closed.

LEMMA 3.9. Let A be a finite-dimensional associative algebra overk. Suppose that(k,Dk) is filtered-linearlyclosed. Then any CDG-structure onΩ•

k ⊗k A is equivalent to a CDG-structure withh = 0.

Proof. We use induction ond := dimk(D). The cased = 1 is automatic. Let us make the inductive step fromd− 1 to d. Consider the differential fields(k,Di) from Definition3.7and putΩ•

i to be the corresponding de Rhamcomplexes. Also, put

Ω := Ker (Ωd → Ωd−1) .

SinceDd−1 is a Lie subring inDd, we have a morphism of differential fields(k,Dd) → (k,Dd−1) (see [15,Definition 3.6]). Thus, we obtain a morphism of graded associative algebras

Ω•d → Ω•

d−1,

which commutes with the de Rham differentiald and whose kernel is the ideal generated byΩ. Thus, the idealgenerated byΩ in Ω•

d is ad-ideal. Further, we have the morphism of graded associativealgebras

Ω•d ⊗k A→ Ω•

d−1 ⊗k A,

whose kernelI• is the graded ideal generated byΩ in Ω•d ⊗k A. Sinced from the CDG-structure onΩ•

d ⊗k Asatisfies the graded Leibnitz rule and the natural homomorphism Ω•

d → Ω•d ⊗k A commutes withd, we deduce

thatI• is also ad-ideal. Consequently,d induces a mapd′ on the graded associative algebraΩ•d−1⊗kA. It follows

that this defines a CDG-structure(d′, h′) onΩ•d−1 ⊗k A with h′ being the image ofh under the natural map

Ω2d ⊗k A→ Ω2

d−1 ⊗k A.

By the inductive hypothesis, we may assume thath′ = 0, whenceh ∈ I2, whereI2 is the second degree part ofI•.

PutV := Ω⊗k A. Sincedimk(Ω) = 1, we have that

Ii = Ωi−1d−1 ⊗k V, i > 1, and I• · I• = 0.

Sinceh ∈ I2, we see that the compositiond d = [h, · ] vanishes onI•. We obtain a(Dd−1)-module structure onthe finite-dimensionalk-vector spaceV with ∇V being the mapd : I1 → I2. Moreover, the element

h ∈ Ω1d−1 ⊗k V

satisfies∇V (h) = 0 by the second Bianchi identity (see Example3.5(ii )).

Sincek is linearly(Dd−1)-closed, we see that there isa ∈ V such that∇V (a) = −h, or, equivalently, there isa ∈ I1 with d(a) = −h. Sincea · a = 0, the CDG-structure(d + [a, ·], h + d(a) + a · a) satisfies the requiredcondition.

Combining Lemma3.6and Lemma3.9, we obtain the following result, which is used for applications to linearDk-groups and isomonodromic parameterized linear differential equations in Sections4 and6, respectively.

PROPOSITION3.10. Suppose that(k,Dk) is filtered-linearly closed and there is a morphism of differential fields(k,Dk) → (K,DK) together with a faithful differential functorC → Vect(K). Then there is aDk-connection onan objectX in C if and only if there isDk-structure onX.

10

ISOMONODROMIC DIFFERENTIAL EQUATIONS AND DIFFERENTIAL CATEGORIES

Below in Example4.7, we show that Proposition3.10is not true over an arbitrary field(k,Dk). The categoryC in this example isRep(G) for a linearDk-groupG.

Suppose thatC is finite, that is, all Hom-spaces inC are finite-dimensional overk and all objects have finitelength (see [49]). For example, ifC satisfies the condition from Lemma3.6, then it is finite. Letl be aDk-fieldoverk. Recall from [49] that there is an abelianl-linear tensor categoryl⊗k C, calledextension of scalars category,together with an exactk-linear tensor functor

l ⊗k − : C → l ⊗k C .

For short, putD := l ⊗k C andY := l ⊗k X. By [15, Proposition 4.12(i)], there is a canonicalDl-structure onDwith

At1D(Y ) ∼= l ⊗k At1C(X).

Besides,D is a (not full) subcategory in the categoryInd(C) of ind-objects inC and there is a canonical morphismX → Y in Ind(C). For example, ifC = Vectfg(k) is the category of finite-dimensionalk-vector spaces, thenD = Vectfg(l) andl ⊗k − is the usual extension of scalars functor.

LEMMA 3.11. In the above notation and assumptions, given aDk-field l overk, there is aDk-connection onXin C if and only if there is aDl-connection onY in D.

Proof. Applying the functorl⊗k−, we see that aDk-connection onX leads to aDl-connection onY . Conversely,assume that there is aDl-connectionsY on Y . Choose ak-linear mapλ : l → k such that the composition

k → lλ

−→ k is the identity. Then the composition in the categoryInd(C)

X −−−−→ YsY−−−−→ At1D(Y )

∼−−−−→ l ⊗k At

1C(X)

λ⊗idAt1(X)−−−−−−−→ At1C(X)

defines aDk-connection onX in C.

In general theDk-connection onX constructed in the proof of Lemma3.11 can be not aDk-structure. IfC = Vectfg(k), then the connection matrices forX are obtained by applyingλ to the connection matrices forY .Combining Proposition3.10and Lemma3.11, we obtain the following result.

PROPOSITION 3.12. Let l be aDk-field overk. Suppose that(k,Dk) is filtered-linearly closed and there is amorphism of differential fields(k,Dk) → (K,DK) together with a faithful differential functorC → Vect(K).Then there is aDk-structure onX in C if and only if there is aDl-structure onl ⊗k X in l ⊗k C.

4. Linear differential algebraic groups and conjugation

In this section, we show how Proposition3.10 can be applied to linear differential algebraic groups. Themainresults here are in Theorem4.4and Theorem4.6. The behavior of conjugation under extensions of scalars isillus-trated in Section4.2. In particular, Example4.9shows that the assumption on the ground field made in Theorem4.4cannot be relaxed. Also, Example4.7 demonstrates that Proposition3.10is not true over an arbitrary differentialfield, and will be further used in Example6.7 to justify the need in the filtered-linearly closed assumption in themain result of the paper, Theorem6.3.

4.1 Main results

LetG be a linearDk-group overk andV be a faithful finite-dimensional representation ofG.

LetA be aDk-Hopf algebra overk that corresponds toG. A Dk-connection onV as an object inC = Rep(G)(see Definition3.1) is aDk-connection onV as ak-vector space such that the coaction map

V → V ⊗k A

11

SERGEY GORCHINSKIY AND ALEXEY OVCHINNIKOV

is a morphism ofk-vector spaces withDk-connections. Equivalently, for anyDk-algebraR, the action of the groupG(R) onR⊗k V commutes with theDk-connection.

A Dk-connection onV in Rep(G) is aDk-structure if and only ifV is aDk-module. In this case, we also saythatV is aDk-representationof G.

DEFINITION 4.1. We say thatG is conjugate to a constant subgroup inGL(V ) if there is ak0-vector spaceV0 andan isomorphismk ⊗k0 V0

∼= V of k-vector spaces such that there is an embedding inGL(V ):

G ⊂ GL(V0)c

(see Section2 for the definition ofGL(V0)c).

Note that ifG is conjugate to a constant subgroup inGL(V ), thenG is constant: there is an algebraic subgroupG0 ⊂ GL(V0) such that the isomorphism

k ⊗k0 V0∼= V

induces the equalityG = (G0)c in GL(V ). We say thatG is conjugate to a reductive constant subgroup inGL(V )

if G0 is reductive.

For an explicit description of Definition4.1, choose a basis inV overk. ThenGL(V ) ∼= GLn(k) for somen.By the differential Nullstellensatz (see [24, Theorem IV.2.1]),G is conjugate to a constant subgroup inGL(V ) ifand only of there is an elementg ∈ GLn(k) such that, for aDk-closed fieldU overk (equivalently, for anyU asabove), we have

g−1G(U)g ⊂ GLn(U0), U0 := UDk .

EXAMPLE 4.2. Letk = k0(t),Dk = k · ∂t,G ⊂ Ga be given by the linear equation

∂2t u = 0, u ∈ Ga,

and letV be a faithful representation ofG given by the faithful upper-triangular two-dimensional representationof Ga. Then

G ∼=(G2

a

)c

is constant, because∂2t (1) = ∂2t (t) = 0. On the other hand,G is not conjugate to a constant subgroup inGL(V ),because there are no faithful two-dimensional representations of the linear algebraic groupG2

a overk0. This showsthat a constant linearDk-group is not necessarily conjugate to a constant subgroup in GL(V ) for a faithful repre-sentationV of G.

The following result is also proved in [36, Corollary 1] but only for the case of a differentially closed field withone derivation.

PROPOSITION4.3. TheDk-groupG is conjugate to a constant subgroup inGL(V ) if and only if there is aDk-structure onV in Rep(G) such thatV is a trivialDk-module.

Proof. If G is conjugate to a constant subgroup, then the isomorphismk ⊗k0 V0∼= V defines aDk-structure on

V in Rep(G), whereV0 is as in Definition4.1. Conversely, suppose that we are given aDk-structure onV thatsatisfies the hypothesis of the proposition. Then putV0 := V Dk .

Combining Proposition3.10, Proposition3.3, and Proposition4.3, we obtain the following result.

THEOREM 4.4. Suppose that(k,Dk) is filtered-linearly closed (see Definition3.7) and linearlyDk-closed (see thecomment following Definition3.7). ThenG is conjugate to aDk-constant subgroup inGL(V ) if and only if thereis a (possibly, non-commuting) basis∂1, . . . , ∂d in Dk overk such that, for alli, 1 6 i 6 d, G is conjugate to a∂i-constant subgroup inGL(V ).

12

ISOMONODROMIC DIFFERENTIAL EQUATIONS AND DIFFERENTIAL CATEGORIES

Let l be aDk-field overk, Dl := l ⊗k Dk. We havel ⊗k Rep(G) ∼= Rep(Gl). Combining Proposition3.12and Proposition4.3, we obtain the following result.

PROPOSITION 4.5. Suppose that(k,Dk) is filtered-linearly closed andl is linearlyDl-closed. Then the linearDl-groupGl is conjugate to aDl-constant subgroup inGL(Vl) if and only if there is aDk-structure onV inRep(G).

The following result is also proved in [33, Theorem 3.14] but just for the case of a differentially closed fieldwith one derivation.

THEOREM 4.6. Suppose that(l,Dl) is filtered-linearly closed and linearlyDl-closed. Then the linearDl-groupGl

is conjugate to a reductive constant subgroup inGL(Vl) if and only if the categoryRep(G) is semisimple.

Proof. We will use the following fact: given a field extensionE ⊂ F , a Hopf algebraA overE corresponds toa reductive linear algebraic group overE if and only if this holds for the extension of scalarsAF overF (see [9,Remarque 2.1.3(ii)]).

Assume thatGl is conjugate to a reductive constant subgroup inGL(Vl). By the fact above, this implies thatGis a reductive algebraic group overk (with theDk-structure forgotten). Sincechar k = 0, we obtain thatRep(G)is semisimple (see [48, Chapter 2]).

Now assume thatRep(G) is semisimple. Then there is aDk-connection onV as all exact sequences inRep(G) are split. This induces aDl-connection onVl in Rep(Gl). By Proposition3.10, there is aDl-structureonVl in Rep(Gl). By Proposition4.3,Gl is conjugate to a constant subgroup inGL(Vl). Hence,k[G] is a finitelygenerated algebra overk. SinceRep(G) is semisimple andchar k = 0, we obtain thatG is reductive as an al-gebraic group overk. Again, by [9, Remarque 2.1.3(ii)], this implies thatGl is conjugate to a reductive constantsubgroup inGL(Vl).

4.2 Examples

First, we provide a non-trivial example to Proposition3.10.

EXAMPLE 4.7. Let

k := Q(t1, t2) and Dk := k · ∂t1 ⊕ k · ∂t2 .

Let V be a3-dimensionalk-vector space with a basise := (e1, e2, e3). Consider theDk-connection∇V on Vgiven by

∇V (e) := −dt1 ⊗ e · B1 − dt2 ⊗ e · B2,

where

B1 :=

0 1

t10

0 0 00 0 0

, B2 :=

0 0 00 0 1

t20 0 0

.

That is, we have

∂ti(e) = −e ·Bi.

Note that

∂t2B1 − ∂t1B2 − [B2, B1] =1

t1t2· ε, where ε :=

0 0 10 0 00 0 0

. (10)

In particular, theDk-connection∇V onV is not aDk-structure onV . Further, consider the unipotent subgroupU

13

SERGEY GORCHINSKIY AND ALEXEY OVCHINNIKOV

in GL(V ) that consists of matrices of the following form (with respect to the basise):

g(u1, u2, v) :=

1 u1 v0 1 u20 0 1

.

LetG be the linearDk-subgroup inU given by the equations

∂tiuj = 0, i = 1, 2, j = 1, 2,

∂t1v =1

t1· u2, ∂t2v = −

1

t2u1.

Note that these equations are equivalent to the equations

∂tig(u1, u2, v) + [g(u1, u2, v), Bi] = 0, i = 1, 2.

This means that the action ofG onV commutes with the action ofDk (see also the discussion following Lemma4.8),that is,∇V is aDk-connection onV as an object inRep(G).

Let us show that there is noDk-structure onV in Rep(G). Assume the converse. By (9) (see Section3), thismeans that there existC1, C2 ∈ EndG(V ) such that

∂t2A1 − ∂t1A2 − [A2, A1] = 0, Ai := Bi + Ci, i = 1, 2. (11)

A calculation shows that we have an isomorphism (via choosing the basise)

EndG(V ) ∼= k · Id⊕ k · ε. (12)

Since

[Bi, ε] = 0, i = 1, 2,

we see that (10) and (12) imply that (11) holds if and only if there existf1, f2 ∈ k such that

1

t1t2+ ∂t2f1 − ∂t1f2 = 0. (13)

This implies that the coefficient oft−11 with values inQ(t2) of the function

1

t1t2+ ∂t2f1

vanishes. Therefore, we have

1

t2+ ∂t2(a−1) = 0, where f1 =

∑

i

aiti1, ai ∈ Q(t2).

This gives a contradiction. Thus, we see that Proposition3.10is not true over an arbitrary field(k,Dk).

Next, we describe two types ofDk-subgroups inGLn(k) that are not constant overk but are conjugate toconstant subgroups inGLn(l) over l, wherel is a Picard–Vessiot extension ofk. LetM be a finite-dimensionalDk-module overk.

LEMMA 4.8. The group-valued functor

GLDk(M) : DAlg(k,Dk) → Sets, R 7→ AutDkR (R ⊗k M)

is represented by a linearDk-group.

Proof. The corresponding finitely generatedDk-Hopf algebra is the Hopf algebra of the algebraic groupGL(M)overk with theDk-structure obtained by the localization over the determinant of theDk-structure on the symmetricalgebra of theDk-moduleEndk(M) ∼=M∨ ⊗k M .

14

ISOMONODROMIC DIFFERENTIAL EQUATIONS AND DIFFERENTIAL CATEGORIES

For an explicit description ofGLDk(M), choose a basise = (e1, . . . , en) in M over k. ThenGL(M) ∼=GLn(k). For each∂ ∈ Dk, denote the corresponding connection(n× n)-matrix byA∂ , that is, we have

∂(e) = −e · A∂ .

By definition,GLDk(M) consists of invertible(n×n)-matrices such that the corresponding gauge transformationpreserves the connection matricesA∂ for all ∂ ∈ Dk. Thus,GLDk(M) is given by the differential equations

∂g + [g,A∂ ] = 0, g ∈ GLn(k), ∂ ∈ Dk.

Note thatGLDk(M) is a closed linearDk-subgroup in the linearDk-groupGL(M) andM is faithful aDk-representation ofGLDk(M). A morphism of linearDk-groups

G→ GLDk(M)

corresponds to aDk-representationV of G such thatV ∼=M asDk-modules.

It follows directly from the proof of Lemma4.8 that the linearDk-groupGLDk(M) is constant if and only iftheDk-module

M∨ ⊗k M

is trivial, because a submodule of a trivialDk-module is trivial and the determinant is aDk-constant inSymnk(M

∨⊗k

M). Besides, ifM is trivial, thenGLDk(M) is conjugate to a constant subgroup inGL(M) (the converse is nottrue already fordimk(M) = 1).

EXAMPLE 4.9. Consider the differential field

k = Q(t1, t2), Dk := k · ∂t1 ⊕ k · ∂t2

and theDk-moduleM := 1⊕ L, where

L = k · e, ∂t1(e) = 0, ∂t2(e) = e.

Since theDk-module

M∨ ⊗k M ∼= 1⊕ L⊕ L∨ ⊕ 1

is not trivial, the linearDk-group

G := GLDk(M) ⊂ GL2(k)

is not constant and henceforthG is not conjugate to a constant subgroup inGL2(k). Put

∂1 := t1∂t1 , ∂2 := t1∂t1 + ∂t2 .

Then[∂1, ∂2] = 0 andL is a trivial ∂2-module as

∂2(t−11 · e

)= 0.

Therefore,M is a trivial ∂i-module fori = 1, 2. By Proposition4.3, G is conjugate to a∂i-constant subgroup inGL2(k) separately with respect to eachi. This shows that Theorem4.4is not true for an arbitrary(k,Dk).

The following type of non-constant groups will be used in Section 6 in order to construct non-trivial examplesto Theorem6.6.

LEMMA 4.10. The group valued functor

MDk : DAlg(k,Dk) → Sets, R 7→ (R⊗k M)Dk

is represented by a linearDk-group.

Proof. The corresponding finitely generatedDk-Hopf algebra is the Hopf algebra of(M,+), that is, the symmetricalgebra ofM∨, with the inducedDk-structure (see also [40, Lemma 2.16]).

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SERGEY GORCHINSKIY AND ALEXEY OVCHINNIKOV

For an explicit description ofMDk , choose a basis inM overk. Then

(M,+) ∼= Gna

for somen. For each∂ ∈ Dk, denote the corresponding connection(n× n)-matrix byA∂ . ThenMDk is given bythe differential equations

∂y = A∂ · y, y ∈ Gna , ∂ ∈ Dk.

Note thatMDk is a closed linearDk-subgroup in the linearDk-groupM . One can show that a linearDk-groupG is isomorphic to(Gn

a)c over someDk-field l overk if and only ifG ∼=MDk for somen-dimensionalDk-module

M overk (see also [3, Proposition 11]).

Assume thatDk = k · ∂ for a derivation∂ : k → k andk0 6= k. Let m ∈ M be a cyclic vector (see [40,Definition 2.8]). Let0 6= D be a linear∂-operator with coefficients ink of the smallest order such thatDm = 0,which exists becauseM is finite-dimensional overk. Then

(M∨

)Dk

is isomorphic to theDk-subgroup inGa given by the equation

Du = 0, u ∈ Ga,

because they represent the same functor (see [40, Lemma 2.16]).

Remark4.11.

(i) There is a faithfulDk-representationVM of MDk defined as follows: as aDk-module,VM isM ⊕1, and theaction ofMDk is given by

m : (n, c) 7→ (n+ c ·m, c),

wherem ∈MDk , n ∈M , andc ∈ k. We have an exact sequence ofDk-representations ofMDk

0 −−−−→ M −−−−→ VM −−−−→ 1 −−−−→ 0,

whereMDk acts trivially on theDk-modulesM and1.

(ii) One can show that there is a bijection between morphismsof linearDk-groupsG→MDk and isomorphismclasses of exact sequences ofDk-representations ofG

0 −−−−→ M −−−−→ V −−−−→ 1 −−−−→ 0,

whereG acts trivially on theDk-modulesM and1. An argument shows that this implies that, for a linearDk-groupG, the categoryRep(G) isDk-equivalent toRep

(MDk

)if and only ifG ∼=MDk .

5. Gauss–Manin connection and parameterized differentialGalois groups

The main results of this section, Proposition5.2and Proposition5.4, are used in Section6.3 in order to constructnon-trivial examples to Theorem6.6. The constructions and results of this section seem to have also their owninterest in the parameterized differential Galois theory.

5.1 Gauss–Manin connectionWe define algebraically a Gauss–Manin connection, which is used to describe a parameterized differential Galoisgroup of integrals in Section5.2. For this, we use the Gauss–Manin connection onH1 only, so that the reader mayput i = 1 in what follows if desired.

For any differential field(K,DK), let H i(K,DK) denote the cohomology groups of the de Rham complexΩ•K (see Section2). That is, we have

H i(K,DK) := Ker(ΩiK

d−→ Ωi+1

K

)/Im

(Ωi−1K

d−→ Ωi

K

), i > 1,

16

ISOMONODROMIC DIFFERENTIAL EQUATIONS AND DIFFERENTIAL CATEGORIES

andH0(K,DK) = KDK . Recall that, for∂ ∈ DK , the Lie derivative is defined as follows (see [15, Section 3.10]):

L∂ = d i∂ + i∂ d : ΩiK → Ωi

K ,

where

i∂ : ΩiK → Ωi−1

K , ω 7→a 7→ ω(∂ ∧ a), a ∈ ∧i−1

K DK

, i > 1

andi∂ = 0 for i = 0. In particular,

L∂(a) = ∂(a) for any a ∈ K.

It follows from the definition that the Lie derivative commutes withd, acts as zero onH i(K,DK), satisfies theLeibniz rule

L∂(ω ∧ η) = L∂(ω) ∧ η + ω ∧ L∂(η)

for all ω ∈ ΩiK , η ∈ Ωj

K , i, j > 0, and we have

L[∂,δ] = [L∂ , Lδ] for all ∂, δ ∈ DK .

Let now (K,DK) be a parameterized differential field over(k,Dk). We have the relative de Rham complexΩ•K/k, where

ΩK/k := D∨K/k.

For short, put

H i(K/k) := H i(K,DK/k), i > 0.

ThenH i(K/k) arek-vector spaces, because the differential onΩ•K/k is k-linear. Moreover, there is a canonical

Dk-structure onH i(K/k), called aGauss–Manin connectionand constructed as follows.

For∂ ∈ Dk, let ∂ ∈ DK be any lift of1 ⊗ ∂ with respect to the structure mapDK → K ⊗k Dk. One checksthat the action ofL∂ onΩK preservesΩk. Since the kernel

C• := Ker(Ω•K → Ω•

K/k

)

is generated byΩk as an ideal inΩ•K with respect to the wedge product andL∂ satisfies the Leibniz rule as

mentioned above, we see that the action ofL∂ onΩ•K preserves the subcomplexC•. Therefore,L∂ is well-defined

on the quotientΩ•K/k. SinceDK/k acts as zero onH i(K/k), we obtain a well-defined action ofDk onH i(K/k).

Finally, one checks that the corresponding map

Dk → EndZ(H i(K/k)

)

is k-linear, whenceH i(K/k) is aDk-module.

Explicitly, for anyω ∈ ΩiK/k with dω = 0, we have

∂[ω] =[L∂(ω)

], (14)

whereω ∈ ΩiK is any lift of ω with respect to the mapΩi

K → ΩiK/k, and the brackets denote taking the class in

H i(K/k). The preceding discussion shows that∂[ω] is well-defined.

EXAMPLE 5.1.

(i) We haveH0(K/k) = KDK/k = k with the usualDk-structure.

(ii) Suppose thatDK/k = K · ∂x. Then

ΩK/k = Ω1K/k = K · ωx with ωx(∂x) = 1, da = ∂x(a) · ωx

for anya ∈ K, andΩiK/k = 0 for i > 2. Hence, there is an isomorphism

K/(∂xK)∼

−→ H1(K/k), [a] 7→ [a · ωx],

17

SERGEY GORCHINSKIY AND ALEXEY OVCHINNIKOV

wherea ∈ K. Under the above isomorphism, the Gauss–Manin connection onH1(K/k) corresponds to theDk-structure onK/(∂xK) given by

∂[a] = [∂(a)], ∂ ∈ Dk,

where, as above,∂ ∈ DK is any lift of 1⊗ ∂ with respect to the structure mapDK → K ⊗k Dk.

(iii) Suppose in addition to (ii ) thatK = k(x) and∂x(x) = 1. Then, for any class inK/(∂xK), there is a uniquerepresentative of the form

n∑

i=1

bix− ci

, bi, ci ∈ k, bi 6= 0.

Since

∂

[n∑

i=1

bix− ci

]=

[n∑

i=1

∂bix− ci

],

we obtain isomorphisms ofDk-modules⊕c∈k

k ∼= K/(∂xK) ∼= H1(K/k).

5.2 PPV extensions defined by integrals

As above, let(K,DK) be a parameterized differential field over(k,Dk). Given an elementω ∈ ΩK/k with dω = 0,the equationdy = ω corresponds to a consistent system of (non-homogenous) linear differential equations in theunknowny

δ(y) = ω(δ), δ ∈ DK/k.

Note that Lemma4.10remains valid if one assumes thatM is aDk-finitely generated module overk instead ofbeing finite-dimensional overk. We use this generality in the following statement. Its special case appears in [46,Lemma 2.3].

PROPOSITION5.2. LetL be a PPV extension ofK for the system of linear differential equations that correspondsto the equationdy = ω, whereω ∈ ΩK with dω = 0 (see above). LetM be theDk-submodule inH1(K/k)generated by[ω] (see Section5.1). Then there is an isomorphism of linearDk-groups (see Lemma4.10and theremark preceding the proposition)

GalDK (L/K) ∼=(M∨

)Dk .

Proof. The proof is in the spirit of the Kummer and Artin–Schreier theories, for example, see [27, Chapter VI,§8].LetR be aDk-algebra. The natural map

α : R⊗k H1(K/k) → R⊗k H

1(L/k)

is a morphism ofDk-modules. SinceL contains a solution of the equationdy = ω, we haveα([ω]) = 0. Therefore,for anyη ∈ R⊗k ΩK/k with dη = 0 and[η] ∈ R⊗k M , we have

α([η]) = 0.

Thus, the equationdy = η has a solution inR ⊗k L. Let∫η ∈ R ⊗k L denote any of these solutions. For each

g ∈ GalDK (L/K)(R), consider the map

φg : R⊗k M → R, [η] 7→ g(∫η)−

∫η.

One checks thatφg([η]) is well-defined, that is, does not depend on the choices ofη and∫η for a given[η], and

belongs to

R = (R⊗k L)DK/k .

18

ISOMONODROMIC DIFFERENTIAL EQUATIONS AND DIFFERENTIAL CATEGORIES

Further,

φg ∈(M∨

)Dk ,

that is,φg is aDk-map: for any∂ ∈ Dk and its lift ∂ ∈ DK , we have

∂(φg([η])) = ∂(g(∫η))− ∂

(∫η)= g

(∂(∫η))− ∂

(∫η)= g

(∫L∂η

)−

∫L∂η = φg(∂[η]),

because the restriction of∂ from R ⊗k L to R is ∂, g commutes with∂, d commutes withL∂ , and by (14) (seeSection5.1). Also, for allg, h ∈ GalDK (L/K)(R), we have

hg(∫η)− h(

∫η) = g(

∫η)−

∫η,

as the right-hand side belongs toR and is Galois invariant. Therefore,

φhg = φh + φg.

Summing up, we obtain a morphism of linearDk-groups

φ : GalDK (L/K) →(M∨

)Dk .

SinceL isDK-generated overK by∫ω, we see thatφ is injective. Suppose thatφ is not surjective. Then there is

a non-zero element[η] ∈M such that, for anyDk-algebraR overk and anyg ∈ GalDK (L/K)(R), we have

φg([η]) = 0.

Equivalently, for anyg ∈ GalDK (L/K)(R), we have

g(∫η)=

∫η,

whence∫η ∈ K and[η] = 0 in H1(K/k), which is a contradiction. Thus,φ is an isomorphism.

The fact that the parameterized differential Galois group in Proposition5.2does not depend of the PPV exten-sion corresponds directly to Remark4.11(ii ).

EXAMPLE 5.3.

(i) Assume thatDk = k · ∂t. Let 0 6= D be a linear∂t-operator with coefficients ink of the smallest order suchthat

D[ω] = 0 in H1(K/k).

If there is no non-zeroD with D[ω] = 0, then we putD := 0. Proposition5.2and the discussion followingLemma4.10imply thatGal(L/K) is isomorphic to theDk-subgroup inGa given by the equation

Du = 0, u ∈ Ga.

(ii) We use the notation of Example5.1 (iii ). By Proposition5.2, the parameterized differential Galois group ofthe equationdy = ω with

ω =

n∑

i=1

bix− ci

· ωx, bi, ci ∈ k, bi 6= 0,

is isomorphic to(Gna)

c. This is also explained in [5, Example 7.1].

Surprisingly, the description of the parameterized differential Galois group given in Proposition5.2 allows toprove the existence of a PPV extension. For simplicity, suppose thatDk = k · ∂t and letD be as in Example5.3(i).Let ∂t ∈ DK be a lift of1⊗ ∂t with respect to the structure mapDK → K ⊗k Dk and let a linearDK-operatorDbe the corresponding lift ofD. SinceD[ω] = 0, there isa ∈ K such that

LD(ω) = da (15)

19

SERGEY GORCHINSKIY AND ALEXEY OVCHINNIKOV

by (14) and the preceding discussion (see Section5.1). An equation similar to (15) was considered in [46] and [7].Consider theDK-algebra

R := Ky/(

dy − ω, Dy − a)DK

, (16)

where(Σ)DKdenotes theDK-differential ideal generated byΣ, anddy − ω means the collection

δ(y)− ω(δ), δ ∈ DK/k.

Note thatR is isomorphic as aK-algebra to the ring of polynomials overK (possibly, of countably many variables).In particular,R is a domain.

PROPOSITION5.4. In the above notation, the fieldL := Frac(R) is a PPV extension ofK for the equationdy = ω.

Proof. Let l be aDk-field overk and suppose that the proposition is true for the parameterized field

Kl := Frac(l ⊗k K)

over l (see [15, Section 8.2] for the extension ofDK/k-constants in parameterized differential fields). That is,suppose that

Ll := Frac(l ⊗k R)

is a PPV extension ofKl for the equationdy = ω. Therefore,LDK/k

l = l. On the other hand, by [15, Corollary8.9], we have that

LDK/k

l = l ⊗k LDK/k ,

whenceLDK/k = k and we obtain the needed result forL. Thus, we may assume that(k,Dk) is differentiallyclosed.

Now suppose that the proposition is true fora and leta′ ∈ K be another element such that

LD(ω) = da′.

Thena′ = a+ b with b ∈ k. Since(k,Dk) is differentially closed, there isc ∈ k such thatDc = b. This defines anisomorphism

R→ Ky/(

dy − ω, Dy − a′)

DK

, y 7→ y − c.

Thus, it is enough to show that there is at least onea ∈ K with LD(ω) = da such that the proposition is true fora.

Again, since(k,Dk) is differentially closed, there is a PPV extensionE of K for the equationdy = ω by [5,Theorem 3.5(1)]. Letz ∈ E be a solution of the latter equation. Consider the subringS in E that isDK-generatedby z. We have that

LD(ω) = d

(Dz

).

SinceEDK/k = k, we see thatDz ∈ K. Put

a := Dz.

Then we obtain a surjectiveDK -morphismf : R→ S sendingy to z.

By Proposition5.2 and Example5.3(i), GalDK (E/K) is isomorphic to theDk-subgroup inGa given by theequation

Du = 0, u ∈ Ga.

It follows from the proof of Proposition5.2that the action ofGalDK (E/K) onE is given by the formula

u : z 7→ z + u.

20

ISOMONODROMIC DIFFERENTIAL EQUATIONS AND DIFFERENTIAL CATEGORIES

LetG be the extension of scalars fromk toK of GalDK (E/K) as a (pro-)algebraic group overk. It follows fromthe PPV theory thatSpec(S) is a torsor underG overK (see [5, Section 9.4]). By the explicit description ofR,Spec(R) is also a torsor underG andf corresponds to a closed embeddingSpec(S) → Spec(R) ofG-torsors. Weconclude thatf is an isomorphism, which proves the proposition for the above choice ofa.

6. Isomonodromic differential equations

In this section, we show how Proposition3.10can be applied to isomonodromic parameterized linear differentialequations. The main results here are in Theorem6.3and Theorem6.6. Section6.2provides an analytic interpreta-tion of our results. The main illustrating examples are in Section 6.3(see also Example6.5).

6.1 Main resultsLet (K,DK) be a parameterized differential field over(k,Dk) andN be a finite-dimensionalDK/k-moduleoverK.

DEFINITION 6.1. We say thatN is isomonodromicif there is aDK -structure onN such that its restriction fromDK toDK/k is equal to the initialDK/k-structure onN .

This is called complete integrability in [5, Definition 3.8], but we preferred to use the terminology slightly morecommon in differential equations for this notion (see also Section6.2).

PROPOSITION6.2. A finite-dimensionalDK/k-moduleN is isomonodromic if and only if there is aDk-structureonN in DMod

(K,DK/k

).

Proof. We use facts about the Atiyah functorAt1 in DMod(K,DK/k

)that can be found in [15, Section 5.1]. We

have the equality of sets (see [15, equation (17)])

At1(N) =

n⊗ 1 +

∑

i

ni ⊗ ωi ∈ N ⊕ (N ⊗K ΩK)∣∣ ∀δ ∈ DK/k, δ(n) =

∑

i

ωi(δ)ni

(we are not specifyingK-linear andDK structures onAt1(N) here). Further,

At1(N) ⊂ At1K(N),

whereAt1K denotes the Atiyah functor inVect(K). Assume that there is aDk-structuresN : N → At1(N) onNin DMod

(K,DK/k

). Since the forgetful functor

DMod(K,DK/k

)→ Vect(K)

is differential (see [15, Theorem 5.1]), the composition

NsN−−−−→ At1(N) −−−−→ At1K(N)

defines aDK -structure onN that extends the givenDK/k-structure. Conversely, assume thatN is isomonodromic.Since theDK-structure extends the givenDK/k-structure, we see that the mapN → At1K(N) factors throughAt1(N) by the construction ofAt1, which gives the needed splittingsN .

For each∂ ∈ Dk, we have a parameterized differential field(K,DK,∂) over (k,Dk) with DK,∂ being thepreimage ofK ⊗ ∂ with respect to the structure mapDK → K ⊗k Dk. By definition, a finite-dimensionalDK/k-moduleN is ∂-isomonodromic if and only if it is isomonodromic over(K,DK,∂). By Proposition6.2,this is equivalent to the existence of a∂-structure onN in DMod(K,DK/k). Note that we have a morphismof differential fields(k,Dk) → (K,DK) (while there is a no a fixedDk-field structure onK) and the forgetfulfunctorDMod(K,DK/k) → Vect(K) is a faithful differential functor (see [15, Theorem 5.1]). Thus, combiningProposition3.10, Proposition3.3, and Proposition6.2, we obtain the following result.

21

SERGEY GORCHINSKIY AND ALEXEY OVCHINNIKOV

THEOREM 6.3. Suppose that(k,Dk) is filtered-linearly closed. ThenN is isomonodromic if and only if there is a(possibly, non-commuting) basis∂1, . . . , ∂d in Dk overk such thatN is ∂i-isomonodromic for alli.

It seems hard to prove Theorem6.3only in terms ofDK/k-modules instead ofDk-categories. The latter allowsone to work purely with the fieldk instead ofK and use its differential properties. This is the advantage of ourapproach.

Remark6.4. Let us explain Theorem6.3more explicitly in the case mentioned in the introduction:dimK(DK/k) =1 anddimk(Dk) = d. All matrices below have entries from a fieldK of functions in the variablesx, t1, . . . , td.Assume that the subfieldk ⊂ K of function int1, . . . , td has the following property: all linear differential equationsthat involve∂t1 , . . . , ∂ti , 1 6 i 6 d− 1, have a fundamental solution matrix with entries fromk. LetA be a matrixwith entries fromK. Suppose that there are matricesB1, . . . , Bd that satisfy

∂tiA− ∂xBi = [Bi, A]. (17)

Then Theorem6.3asserts the existence of matricesA1, . . . , Ad such that

∂tiA− ∂xAi = [Ai, A] (18)

and, for alli, j, 1 6 i, j 6 d, we have

∂tiAj − ∂tjAi = [Ai, Aj ]. (19)

EXAMPLE 6.5. We will see that, in general, theAi’s in the above have to be different from the originalBi’s. Letd > 2 and(n × n)-matricesA,A1, . . . Ad with entries inK as above satisfy all the integrability conditions (18)and (19). Define

B1 := A1, . . . , Bd−1 := Ad−1, Bd := Ad + diag(t1),

wherediag(t1) denotes the diagonal(n × n)-matrix with t1 on the diagonal. Then the new set of matricesA,B1, . . . , Bd will still satisfy (17) but will not satisfy the integrability condition for the pair of derivations∂t1and∂td .

Let U be aDk-closure ofk,KU := Frac(U ⊗kK). By [15, Proposition 8.11], we have

U ⊗k DMod(K,DK/k) ∼= DMod(KU ,DKU/U )

Below, we extend [5, Proposition 3.9(1)] to the case when(k,Dk) is not necessarily differentially closed, whichwe also prove categorically.

THEOREM 6.6. In the above notation letL be a PPV extension forNKU(which exists by [5, Theorem 3.5(1)]),

V := NDK/k

L , and letGalDK (L/KU ) ⊂ GL(V ) be the parameterized differential Galois group ofL overKU . Sup-pose that(k,Dk) is filtered-linearly closed. ThenN is isomonodromic if and only ifGalDK (L/KU ) is conjugateto a constant subgroup inGL(V ).

Proof. Let C be the subcategory inDMod(KU ,DKU/U ) that isDk-tensor generated byNKU(see [15, Defini-

tion 4.19]). Recall thatL defines aDk-fiber functor

ω : C → Vect(U)

such thatω(NKU) = V andGalDK (L/KU ) is the associatedDk-group (see [15, Theorem 5.5]). More precisely,

there is an equivalence ofDk-categories

C ∼= Rep(GalDK (L/KU )

)

sendingNKUtoV . Thus, combining Proposition6.2, Proposition3.12, and Proposition4.3, we obtain the required

result.

22

ISOMONODROMIC DIFFERENTIAL EQUATIONS AND DIFFERENTIAL CATEGORIES

6.2 Analytic interpretation

We will now explain in more detail the relation between the analytic notion of isomonodromicity and Definition6.1.Letf : X → S be a holomorphic submersion between connected complex analytic manifolds with connected fiberssuch thatf is topologically locally trivial. LetE be a holomorphic vector bundle onX and∇X/S be a relative flatholomorphic connection onE overS (that is, the connection∇X/S is defined only along vector fields onX thatare tangent to the fibers off ). For a subsetΣ ⊂ S, put

XΣ := f−1(Σ).

In particular,Xs denotes the fiber off at a points ∈ S.

LetU be a sufficiently small open neighborhood of a points ∈ S such that there is a smooth isomorphism

φ : U ×Xs∼

−→ XU

whose restriction tos×Xs coincides with the embeddingXs → XU . This gives a collection of smooth isomor-phisms

φst : Xs∼

−→ Xt,

wheret ∈ U , and a sectionσ : U → XU . Also, choose a trivialization

ψ : Cn × U∼−→ σ∗E.

Then the connection∇X/S defines a family of relative monodromy representationsρt, t ∈ U , as the composition

π1 (Xs, σ(s))∼

−−−−→ π1 (Xt, σ(t)) −−−−→ GLn ((σ∗E)t)

∼−−−−→ GLn ((σ

∗E)s)∼

−−−−→ GLn(C).

The isomorphism classes of the representationsρt do not depend on the choices ofφ andψ.

We say that(E,∇X/S) is analytically isomonodromicif the isomorphism classes of the relative monodromyrepresentationsρs are locally constant overS (for example, see [41, Section 1]). It is shown in [41, Proof of 1.2(1),first step] that(E,∇X/S) is analytically isomonodromic if and only if, for any points ∈ S, there is an openneighborhoods ∈ U ⊂ S such that∇X/S extends to a flat holomorphic connection onE overXU (see also [45,Theorem A.5.2.3] for the case of one-dimensional fibers). This is a version of Definition6.1in the analytic context.

Let us give an analytic interpretation of Theorem6.3. By definition,(E,∇X/S) is analytically isomonodromicalong a holomorphic vector fieldv onS if and only if the relative monodromy representations are locally constantalong (local) holomorphic curves onS that are tangent tov. Thus,(E,∇X/S) is analytically isomonodromic if andonly if it is analytically isomonodromic alongd transversal vector fields onS, whered := dim(S). Combining thiswith the property of analytic isomonodromicity discussed above, we obtain an analytic proof of following weakerversion of Theorem6.3:

Let k (respectively,Dk) be the field of meromorphic functions (respectively, the space of meromorphic vectorfields) onS. Analogously, defineK andDK for X in place ofS. Assume that a finite-dimensionalDK/k-moduleN satisfies the partial isomonodromicity condition from Theorem 6.3. Then there is a points ∈ S such thatN is isomonodromic over the parameterized differential fieldKs over ks, whereks is the field of meromorphicfunctions on open neighborhoods ofs in S andKs is the field of meromorphic functions on open subsets inXwhose intersection withXs is dense inXs.

In general, one cannot replaceKs by the field of meromorphic functions along allXs. However, the resultsfrom [19, 20] allow to similarly treat the latter case when the fibers off are complex projective lines with finite setsof points removed. Finally, the need of replacingk by ks reflects the requirement for(k,Dk) to be filtered-linearlyclosed in Theorem6.3.

23

SERGEY GORCHINSKIY AND ALEXEY OVCHINNIKOV

6.3 Examples

First, we provide a non-trivial example to Theorem6.3 showing that its statement is not true for an arbitrary field(k,Dk). Namely, in the notation Example4.7, we construct a parameterized fieldK over the field(k,Dk) and aPPV extensionK ⊂ L such thatGalDK (L/K) ∼= G and the solution space corresponds to the representationV .We are very grateful to M. Singer, who suggested a general method for constructing PPV extensions with a givenparameterized differential Galois group to us.

EXAMPLE 6.7. The following example is based on iterated integrals. Letk := Q(t1, t2) andDk := k ·∂t1 ⊕k ·∂t2 .Let

F := k(∂ix∂

j1t1 ∂

j2t2 Im

), i, j1, j2 > 0, m = 1, 2,

be the field of∂x, ∂t1 , ∂t2-rational functions in the differential indeterminatesI1 andI2 overk. Put

DF := F · ∂x ⊕ F · ∂t1 ⊕ F · ∂t2 ,

and do the analogous for the other fields that appear in what follows. Then(F,DF ) is a parameterized differentialfield over(k,Dk). LetL be a PPV extension ofF for the equation

∂x(y) = ∂xI1 · I2 (20)

and letI ∈ L be a solution of this equation (for example, see Proposition5.4for the existence ofL). A calculationshows that there are no elementsa ∈ F and linearDk-operatorsD with coefficients ink such that

∂x(a) = D(∂xI1 · I2).

By Proposition5.2, we see that

GalDF (L/F ) ∼= Ga

and, therefore, the elements∂iI ∈ L, i > 0, are algebraically independent overF . Let

K ⊂ L

be the∂x, ∂t1 , ∂t2-subfield generated by

∂xIm, ∂t1Im, ∂t2Im, m = 1, 2,

J1 := ∂t1I − ∂t1I1 · I2 −1

t1I2, J2 := ∂t2I − ∂t2I1 · I2 +

1

t2I1.

SinceI satisfies (20) andJ1, J2 ∈ K, for all (i, j1, j2) 6= (0, 0, 0), we have

∂ix∂j1t1 ∂

j2t2 (I) ∈ K(I2).

Therefore,

L = K(I1, I2, I).

One can show thatI1, I2, I are algebraically independent overK using a characteristic set argument with respectto any orderly ranking of the derivatives withI > I1 > I2 [24, Sections I.8–10]. Put

fi := ∂xIi ∈ K, i = 1, 2

and consider the equation

∂x(y) = A∂x · y, y := t(y1, y2, y3), A∂x :=

0 f1 00 0 f20 0 0

, (21)

24

ISOMONODROMIC DIFFERENTIAL EQUATIONS AND DIFFERENTIAL CATEGORIES

Then

Φ :=

1 I1 I0 1 I20 0 1

is the fundamental matrix for the equation (21), that is,I is the iterated integral∫x

(f1 ·

∫x f2

). HenceL is a PPV

extension ofK for the equation (21).

In what follows,U andG are as in Example4.7. We see thatGalDK (L/K) is a linearDk-subgroup inU ,whereU acts onΦ by multiplication on the right. Explicitly, we have

g(u1, u2, v)(Ii) = Ii + ui, g(u1, u2, v)(I) = I + I1u2 + v.

A calculation shows thatK ⊂ LG. By a dimension argument, we conclude thatGalDK (L/K) = G. By Exam-ple 4.7, the equation (21) is not isomonodromic. On the other hand, this equation is∂ti -isomonodromic,i = 1, 2with the corresponding matrices given by

Bi := Φ · Bi · Φ−1 + ∂tiΦ · Φ−1, i = 1, 2,

where

B1 :=

0 1

t10

0 0 00 0 0

, B2 :=

0 0 00 0 1

t20 0 0

.

More explicitly,

B1 :=

0 1

t1+ ∂t1I1 J1

0 0 ∂t1I20 0 0

, B2 =

0 ∂t2I1 J20 0 1

t2+ ∂t2I2

0 0 0

Thus, we see that Theorem6.3is not true for an arbitrary field(k,Dk).

The purpose of the rest of the section is to show that, in Theorem 6.6, one really needs to take the extensionof scalars fromk to U in order to obtain conjugacy to a constant group. Namely, we construct examples of anisomonodromicDK/k-moduleN such that there are PPV extensions ofK for N , but, for any PPV extensionL,the parameterized differential Galois groupGalDK (L/K) is not a constant group and, thus,GalDK (L/K) is notconjugate to a constant subgroup inGLn(k).

The idea of the examples is as follows. We construct a parameterized differential field(K,DK) over (k,Dk)andω ∈ ΩK/k with dω = 0 such that theDk-submodule

M ⊂ H1(K/k)

generated by[ω] is finite-dimensional overk and is not trivial as aDk-module. By Propositions5.2and5.4, thereare PPV extensions ofK for the equationdy = ω and any of them has the parameterized differential Galois group

isomorphic to(M∨)Dk . SinceRep((M∨)Dk

)is Dk-equivalent to aDk-subcategory inDMod

(K,DK/k

),

the (faithful)Dk-representationVM∨ of (M∨)Dk (see Remark4.11(i)) corresponds to an isomonodromicDK/k-moduleN (see Proposition6.2).

Let us give an explicit description. Suppose that

Dk = k · ∂t, DK = K · ∂x ⊕K · ∂t, and [∂x, ∂t] = 0.

Letωx ∈ ΩK/k be such thatωx(∂x) = 1. Thenω = b · ωx with b ∈ K. Suppose that there exists a non-zero monic

25

SERGEY GORCHINSKIY AND ALEXEY OVCHINNIKOV

linear∂t-operatorD as in Example5.3(i). Explicitly,

D = ∂nt −n−1∑

i=0

ci∂it , ci ∈ k,

is of the smallest order such that there isa ∈ K with D(b) = ∂x(a) (see Example5.1(ii )). One can show that thedifferential moduleN defined above corresponds to the following system of linear differential equations:

∂x(y) = A∂x · y, y := t(y0, . . . , yn), A∂x :=

0 0 . . . 0b 0 . . . 0

∂t(b) 0 . . . 0... . . .

∂n−1t (b) 0 . . . 0

.

By Proposition5.4, there is a PPV extensionL of K for N andL = K(z, ∂t(z), . . .), where∂x(z) = b. ByProposition5.2and its proof combined with Example5.3(i), the morphism of linearDk-groups

GalDK (L/K) → Ga, g 7→ g(z) − z

induces an isomorphism

GalDK (L/K)∼

−→ u ∈ Ga |Du = 0 ⊂ Ga.

The∂x-moduleN is isomonodromic with

A∂t :=

0 0 0 . . . 00 0 1 . . . 0...

.... . .

0 0 0 . . . 1a c0 c1 . . . cn−1

.

Now we give concrete examples withk = Q(t) andK being a generated by functions int andx. We constructb ∈ K such that there exists a linear∂t-operatorD as above and the equationDu = 0 in u is non-trivial overQ(t).

EXAMPLE 6.8. This examples comes from the algebraic independence ofthe derivatives of the incomplete Gamma-function (see [21]). Put

E := Q(t, x, log x, xt−1e−x

), DE := E · ∂x ⊕E · ∂t.

By Proposition5.4, there is a PPV extensionL of E for the equation

∂x(y) = xt−1e−x. (22)

As noted in [5, Example 7.2], by [21], there is an isomorphism

GalDE (L/E) ∼= Ga.

Let γ ∈ K be a solution of (22) and put

K := E(∂t(γ)− γ, ∂2t (γ)− ∂t(γ), . . .

)⊂ L.

SinceGalDE(L/E) ∼= Ga, the parameterized Galois theory implies thatγ /∈ K. The element

b := xt−1e−x ∈ K

satisfies

D(b) = ∂x(a), D := ∂t − 1, a := ∂t(γ)− γ ∈ K.

The operatorD is of the smallest order, becauseb /∈ ∂x(K) asγ /∈ K. Note thatK is of infinite transcendencedegree overQ(t, x), becauseGalDE(K/E) ∼= Ga.

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ISOMONODROMIC DIFFERENTIAL EQUATIONS AND DIFFERENTIAL CATEGORIES

EXAMPLE 6.9. This example comes from the Gauss–Manin connection forthe Legendre family of elliptic curves.Namely, putK := Q(t, x, z), wherez2 = x(x− 1)(x − t). Then the element

b :=1

z∈ K

satisfies

D(b) = ∂x(a), D := −2t(t− 1)∂2t − (4t− 2)∂t −1

2, a :=

z

(x− t)2∈ K.

The operatorD is of the smallest order (for example, this follows from a monodromy argument, see [8, Sec-tion 2.10]).

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Sergey Gorchinskiy gorchins@mi.ras.ruSteklov Mathematical Institute, Gubkina str. 8, Moscow, 119991, Russia

Alexey Ovchinnikov aovchinnikov@qc.cuny.eduCUNY Queens College, Department of Mathematics, 65-30 Kissena Blvd, Flushing, NY 11367, USACUNY Graduate Center, Department of Mathematics, 365 FifthAvenue, New York, NY 10016, USA

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