Degeneration from difference to differential Okamoto spaces for the sixth Painlev\'e equation

In the current paper we study the $q$-analogue introduced by Jimbo and Sakai of the well known Painlev\'e VI differential equation. We explain how it can be deduced from a $q$-analogue of Schlesinger equations and show that for a convenient change of variables and auxiliary parameters, it admits a $q$-analogue of Hamiltonian formulation. This allows us to show that Sakai's $q$-analogue of Okamoto space of initial conditions for $qP_\mathrm{VI}$ admits the differential Okamoto space \emph{via} some natural limit process.


Introduction
In [JS96], Jimbo and Sakai have introduced a q-analogue of the Painlevé VI equation, namely the following system of q-difference equations: (qP JS,VI ) : where κ 1 , κ 2 , ϑ 1 , ϑ 2 , a 1 , a 2 , a 3 , a 4 ∈ C * are parameters subject to the relation Here q is a complex parameter that is neither zero nor one, and σ q,t is the operator which to a function f (t) associates f (q • t).The q-derivative ∂ q,t := σ q,t − 1 (q − 1)t formally converges, when q → 1, to the classical derivative ∂ t (differentiation with respect to t).It has been shown in [JS96] that the classical Painlevé VI equation may be obtained by some limit process, when q goes to 1, from its q-analogue.More precisely, by a series of changes of variables and parameters, qP JS,VI formally yields a certain system of differential equations with eight complex parameters, subject to one relation.As one can easily check, one can then 2010 Mathematics Subject Classification.14D05, 14F35, 34M56, 39A13.This work took place at IRMA.It was supported by ANR-13-JS01-0002-01 and ANR-19-CE40-0008.
further normalize these parameters to a quadruple θ = (θ 0 , θ 1 , θ t , θ ∞ ) of complex parameters such that this system of differential equations is the non-autonomous Hamiltonian system (P VI ) : where H θ VI (y, Z, t) is given by This non-autonomous Hamiltonian system (P VI ) is actually the one discovered in [Oka86] that, when reformulated as a single second order differential equation in y, yields the sixth Painlevé differential equation with auxiliary parameters θ.Given a generic initial condition, i.e. y 0 ∈ C \ {0, 1, t 0 }, and Z 0 ∈ C, Cauchy's theorem implies the existence and uniqueness of a germ at t 0 = 0, 1 of associated holomorphic solution of (P VI ).As shown in [Oka86], it is moreover possible to give a meaning to solutions including non-generic initial conditions.More precisely, Okamoto's space of initial conditions at a fixed time t 0 ∈ C \ {0, 1} is the second Hirzebruch surface F 2 blown up in eight points, whose position is encoded by θ and t 0 , minus a divisor formed by five irreducible components of self-intersection number (−2) related to each other according to the following intersection diagram: Here each node represents an irreducible component, and nodes share a common edge if and only if they intersect each other.For each point in Okamoto's space of initial conditions at t 0 , there exists a unique associated germ (at t 0 ) of meromorphic solution of (P VI ).A q-analogue of Okamoto's space for (qP JS,VI ) for some fixed generic time t 0 was found in [Sak01].It is given by F 0 = P 1 ×P 1 , blown up in eight points, whose positions are encoded by a 1 , a 2 , a 3 , a 4 , κ 1 , κ 2 , ϑ 1 , ϑ 2 and t 0 , minus a divisor formed of four irreducible components of self-intersection number (−2), arranged according to the following intersection diagram: For each point in Sakai's q-analogue of Okamoto's space of initial conditions at t 0 , there exists a unique discrete solution of (qP JS,VI ), which, roughly speaking, encodes the values at q Z t 0 that a meromorphic solution with prescribed value at t 0 , if it exists, should interpolate.The questions adressed in the present paper are the following.
Q1) How can Okamoto's space of initial values at t 0 for (P VI ) be obtained via a natural limit process from its discrete analogue?Q2) How can meromorphic solutions of (P VI ) be obtained via a natural limit process from their discrete analogue?
Let us first answer question (Q1) informally.What we will obtain in Section 4 is that one of the four irreducible components of the boundary of the q-Okamoto space at t 0 does degenerate at the limit q → 1: for q = 1, it is no longer irreducible, but is itself the union of three irreducible components of self-intersection (−2), one of which coincides with the limit of a non-degenerating one, and the other two of which intersect only the latter.So in terms of intersection diagrams, the informal answer to question (Q1) is the following.
The key to this result (see Section 4.3 for a precise formulation) is to coveniently identify normalizations and changes of variables in (qP JS,VI ) before considering the limit process, such that the limit when q → 1 is (P VI ).To this end, we retrace, with some alterations, the method by which in [JS96], the q-difference equation (qP JS,VI ) has been obtained from a q-analogue of isomonodromic deformations.Here the isomonodromy condition to be considered concerns certain families, parametrized by a time variable t, of q-Fuchsian systems of rank 2, which for fixed q are of the form where x is the standard coordinate on C ⊂ P 1 .As shown in [JS96], under certain generic conditions, for given spectral parameters κ 1 , κ 2 , ϑ 1 , ϑ 2 , a 1 , a 2 , a 3 , a 4 ∈ C * as above, such a family can be encoded by a triple of functions (w(t), y(t), z(t)).We show in Section 1.3 that a convenient change of variables and parameters (including a normalization) that is both compatible with the definition of the considered q-Fuchsian systems in [JS96] and that yields tracefree differential Fuchsian systems at the limit when q → 1 is the following setting: (1) λ = κ 2 w q − 1 , Z = (y−ta 1 )(y−ta 2 ) q(y−1)(y−t)z − 1 (q − 1)y , κ 2 , κ 2 = 1 + (q − 1) θ∞ 2 .In Section 2.2, we both simplify and generalize the definition of q-isomonodromy in [JS96] into the requirement that there exists a certain matrix B(x, t), depending rationally on x, such that the system of q-difference equations σ q,x Y (x, t) = A(x, t)Y (x, t) σ q,t Y (x, t) = B(x, t)Y (x, t) satisfies the q-analogue of integrability, namely σ q,t A • B = σ q,x B • A .We further introduce a sufficient condition for q-isomonodromy, which we call q-Schlesinger isomonodromy.We show that under some generic conditions such as the non-resonancy condition ϑ 1 ϑ 2 ∈ q Z * and κ 1 κ 2 ∈ q Z * , this q-Schlesinger isomonodromy of σ q,x Y = AY is equivalent to the following q-Schlesinger equations (ta 1 −1)(ta 2 −1) • (qI 2 + B 0 ) A 1 I 2 + (qta 1 −1)(qta 2 −1) qt−1 where .
Moreover, we show in Section 2.3 that when the spectral values are functions of q given by (2), then these q-Schlesinger equations yield the usual (differential) Schlesinger equation at the limit q → 1.When A is expressed with respect to (w(t), y(t), z(t)) and the spectral parameters, then these q-Schlesinger equations are (generically) equivalent to a system of q-difference equations given by (qP JS,VI ) and an additional equation for σ q,t w.However, our weaker definition of qisomonodromy is actually (generically) equivalent to (qP JS,VI ).We conclude that the change of variables and parameters (1) combined with (2) is a natural setting for the study of confluence of the q-Painlevé VI equation.And indeed, for generic values of q, the change of variable (1) defines a biregular transformation of Sakai's q-Okamoto space (see Section 4.2), such that the obtained modified q-Okamoto space yields, for spectral values (2), the differential Okamoto space when q → 1 (see Section 4.3).Under convenient assumptions on q and the spectral data, some meromorphic solutions of (qP JS,VI ) defined in a convenient sectorial neighborhood of t = 0 have been constructed in [Man10,Ohy09].On the other hand, contrary to the differential setting, the following question remains open: for a given generic value of t 0 and generic initial condition (y(t 0 ), z(t 0 )), does there exist an associated meromorphic solution of (qP JS,VI ), defined on a connected subset of C * stable under multiplication by q ±1 ?Of course the answer is likely to depend on particular choices of q and the spectral parameters.For example, when q is a n-th root of unity, then a necessary condition for the existence of meromorphic solution is that the spectral parameters are chosen in a way such that the n-th iterate of (qP JS,VI ) is the identity.The question whether meromorphic solutions of (qP JS,VI ) over convenient set, parametrized by q, admit a limit when q → 1 also seems difficult and remain open.
Much more abordable is the question of the existence of discrete solutions, essentially solved in [Sak01, Prop.1], which, roughly speaking, encode the values at q Z t 0 that meromorphic solutions with prescribed value at t 0 , if they exist, should interpolate.More precisely, a discrete solution with initial value (y 0 , z 0 ) ∈ C * ×C * at t 0 ∈ C * is a sequence (y ℓ , z ℓ , t ℓ ) ℓ∈Z of points in P 1 ×P 1 ×C * such that for ℓ = 0, we have t ℓ = q ℓ t 0 and (y ℓ , z ℓ ) = (f ℓ (y 0 , z 0 , t 0 , q), g ℓ (y 0 , z 0 , t 0 , q)), where f ℓ , g ℓ are the rational functions invariables y, z, t, q such that the ℓ-th iterate of (qP JS,VI ) is of the form σ ℓ q,t y = f ℓ (y, z, t, q) σ ℓ q,t z = g ℓ (y, z, t, q) .We refer to Section 3.2 for more details.It is shown in [Sak01, Prop.1] that discrete solutions with initial value in C * × C * at t 0 are well defined, in particular they exist and are unique, if t 0 and the spectral values are generic.Moreover, under this assumption, one can consider a space of initial values bigger than C * × C * , namely the q-Okamoto space.We specify in Section 4.2 which are the special values for t 0 and the spectral parameters that need to be excluded here.Note that a discrete solution, as a sequence, does make sense even if q is a root of unity.
Of course there is an analogous notion of discrete solution for the modified q-Painlevé VI equation obtained by applying the change of variables and parameters (1), (2) to (qP JS,VI ).We prove in Section 3.3 that the therby obtained system of q-difference equations is a q-analogue of Hamiltonian system.More precisely, it is given by , where H θ VI is the Hamiltonian from the (differential) (P VI ) and for i ∈ {1, 2}, R θ i is some rational function such that R θ i | q=1 is well defined and does not have poles outside the polar locus of H θ VI .From this we deduce, also in Section 3.3, the answers to question (Q2).For t 0 ∈ C * and (y 0 , Z 0 ) ∈ (C \ {0, 1, t 0 }) × C, the sequence (y ℓ (q), Z ℓ (q), q ℓ t) ℓ∈N of triples defining the corresponding discrete solution, but seen as rational functions of q, is well defined, and encodes in some precise manner the Taylor series coefficients of the unique solution of (P VI ) with initial condition (y 0 , Z 0 ) at t 0 .
Each of the four sections following this introduction is decomposed into three parts.Each time, in the first part we briefly recall some notions and known results in the differential case.In the second part, their q-analogues are discussed, and in the third part, confluence is adressed.Concerning the q-analogues, we usually recall some results from [JS96] and [Sak01], complemented by some precisions that we deemed helpful, and to which we add new results.We finish by an appendix, see Section 5, explaining how our notion of q-isomonodromy is related to the one in [JS96].
In this paper, we adopt the following (standard) notation.GL 2 (R) the ring of invertible 2 × 2 matrices with coefficients in a ring R. SL 2 (R) the ring of invertible 2 × 2 matrices of determinant 1 with coefficients in R. M 2 (R) the algebra of 2 × 2 matrices with coefficients in a ring R. sl 2 (M ) the vector space of 2 × 2 matrices of trace 0 with coefficients in M , where M is a C-module.I 2 the identity matrix in M 2 (R) A (i,j)  the (i, j)-entry of a matrix A.

O(U )
the ring of holomorphic functions on some complex domain the field of meromorphic functions on U .k[x 1 , . . ., x n ] the ring of polynomials in n variables, named x 1 , . . ., x n , with coefficients in a field k. k(x 1 , . . ., x n ) the fraction field of k[x 1 , . . ., x n ].
We would like to emphasize that some results in the sequel require stronger assumptions on the complex variable q than q = 0, 1.These assumptions will of course be duly specified when needed.We choose not to accumulate these requirements along the way towards the q-Painlevé VI equation (which in and by itself is well-defined for q = 0, 1), in order to get the full picture of possible q-Okamoto spaces.

Fuchsian systems
We consider a linear partial differential equation of the form Here x is the standard coordinate on C, seen as a subset of P 1 = C ∪ {∞}, and t is the standard coordinate on an open connected subset U ⊂ C \ {0, 1}.
Definition 1.1.We shall say that (4) is a family of sl 2 -Fuchsian systems with spectral data θ if the following hold: • for all i ∈ {0, 1, t} and all t ∈ U , we have • the residue A ∞ := −A 0 − A 1 − A t at infinity is constant and normalized as follows: With this normalization, the (1, 2) entry of x(x− 1)(x− t)A, seen as an element of O(U )[x], is a polynomial of degree at most one.Let us assume that it has degree one and define a non-zero holomorphic function λ(t) ∈ O(U ) and a meromorphic function y(t) ∈ M(U ) by ( 5) Assuming moreover that y(t) ≡ 0, 1, t, we may define a meromorphic function Z(t) ∈ M(U ) by ( 6) The next Lemma shows that the matrix A is determined by the triple (λ, y, Z).
Lemma 1.2.If a family of sl 2 -Fuchsian systems (4) with spectral data θ, with θ ∞ = 0, gives rise to as above, then the coefficients of the matrix A necessarily are the following functions of λ, y, Z, x, t and θ: Here we denote Proof.This lemma can be deduced from the formulae in [JM81, p. 443-444] by considering the tensor product of Remark 1.3.If we have an arbitrary meromorphic triple (λ, y, Z) as in (7), then via the formulae in Lemma 1.2 we can associate a family of Fuchsian systems.Note however that the coefficients of the matrix functions A 0 , A 1 and A t then are meromorphic functions of t.If one wants to obtain holomorphic coefficients, one might have to restrict to the complement of a discrete subset in U .Indeed, for example the product λy needs to be holomorphic.
Remark 1.4.As explained in [Lor16, Sec.4], the condition θ ∞ = 0 can actually be overcome if one works in a (conjugated) setting where Remark 1.5.In order to motivate our choice of notation, let us indicate that with respect to confluence, will be led to consider Θ i satisfying some relation with the θ i from the differential context, and Θ i satisfying Θ i = 1/Θ i (see Section 1.3).
We say that (Θ, Θ) satisfies the non-resonancy condition if If ζ is a standard coordinate in a complex domain that is stable under multiplication by q and 1 q , then we define the following operators on functions of ζ: We consider families of linear q-difference systems of the form (10) or, equivalently, by setting Here x is again the standard coordinate on C, and t is the standard coordinate on an open connected subset D of C * .
Definition 1.6.We shall say that (10) is a family of q-Fuchsian systems with spectral data (Θ , Θ) if the following hold: • for all t ∈ D, we have Spec(A 0 (t)) = Θ 0 , Θ 0 , Remark 1.7.The entries of the matrix (x − 1)(x − t)A(x, t) ∈ M 2 (O(D)[x]) have degree two.Then, the determinant of the latter, is a degree four polynomial.The The assumption that that two zeros of p are proportional to t, and the two others are independent of t will be needed for instance in the proof of Proposition 2.9.
Remark 1.8.We chose here to slighlty modify the notation from [JS96] because we are mainly interested at the limit when q → 1. Towards this goal, it is worth mentioning that our variables satisfy A (1,2) (x, t) = λ(t)(x−y(t)) (x−1)(x−t) , and A (1,1) (y(t), t) = Z(t).More details are given in Section 1.3.Analogously to the differential case of families of sl 2 -Fuchsian systems, we have the following lemma, which is a slight adaptation of the formulas in [JS96,p. 4].
Proof.The general form of A in (15), together with the equation for z 1 in (16), its precisely what is needed in order for A ∞ to be of the required normalized form, and for λ, y, Z to satisfy the equalities (13) and (14).Via evaluation at x = 0, 1, t, y, the equation with arbitrary α is equivalent to the remaining equations in (16).More precisely, the successive evaluations at y, 0, 1, t give the lines 2, 5, 4, and 3 of (16).In particular, using (8), we have motivates the following definition.
Definition 1.10.Let f ∈ C(g, x, q) such that {q = 1} is not an irreducible component of the polar divisor of f , i.e. f (g, x, 1) is a well defined rational function.Then we say that the q-difference equation ∂ q,x g = f (g, x, q) discretises the differential equation ∂ x g = f (g, x, 1).This definition generalizes in the obvious way to the case of systems of rational q-difference equations in several variables.It can also be generalized, in a more subtle way, to the case when the base field is not C, but for example the field of meromorphic functions on some domain.The term confluence is used when the inverse phenomenon occurs: when objects associated to a discretized differential equation (most importantly, solutions), yield the corresponding object of the differential equation by some limit process as q → 1. Confluence is widely studied, see for instance [Sau00,Zha02,DVZ09,Dre15,Dre17].Before confluence can even be adressed, one of course needs to identify the appropriate discretization.The aim of the current section is to do so for families of sl 2 -Fuchsian systems ∂ x Y = A(x, t)Y as in Definition 1.1.Note that the naive approach of setting A(x, t, q) = I 2 + (q − 1)xA(x, t) does in general not yield a q-Fuchsian system as in Definition 1.6.Instead, we will consider A(x, t, q) given by a triple of meromorphic functions as in Lemma 1.9, but with an additional parameter q, and study when A = A−I 2 (q−1)x admits a limit as q → 1.Here, in a first step, we ignore the difficulty of the coefficients of A being meromorphic functions with respect to t, by simply considering λ, y, Z as additional variables.
(1) The divisor {q = 1} in C 6 x,λ,y,Z,t,q is not an irreducible component of the polar divisor of A and the therefore well-defined rational matrix function A| q=1 equals A(x, λ, y, Z, t).In other words, lim q→1 A(x, λ, y, Z, t, q) = A(x, λ, y, Z, t) .
(2) Up to permuting the roles of Θ i and Θ i for i ∈ {0, 1, t}, the following holds as q → 1: From the particular form of A it follows that A can be decomposed as where A 0 , A 1 , A t do not depend on x.It follows that A can be decomposed as where A 0 , A 1 , A t do not depend on x.Similarly, we may denote by A 0 , A 1 , A t the residues of A with respect to x = 0, 1, t.We denote Then (1) holds if and only if, for each i ∈ {0, 1, t, ∞}, we have lim q→1 A i (x, λ, y, Z, t, q) = A i (x, λ, y, Z, t).
Assume (1) holds.We have Since A ∞ is of normal form, we deduce the estimates for Θ ∞ and Θ ∞ .Since Spec(A 0 ) = {Θ 0 , Θ 0 }, we have Since Spec(A 0 ) = {θ 0 /2, −θ 0 /2}, we deduce the estimates for Θ 0 and Θ 0 as in the statement (up to interchanging their roles).Recall that det . We therefore have det Since det(A i ) = −θ 2 i /4 for i ∈ {1, t}, we deduce the estimates for Θ i and Θ i as in the statement (up to interchanging their roles).Hence (1) ⇒ (2).Conversely, the above calculations show that if (2) holds and the limit lim q→1 A is a well defined element of M 2 (C(x, λ, y, Z, t)), then this limit is of the required form.Moreover, it is straightforward to check (with some more effort), that if (2) holds, then the limit is well defined.Here one needs to use the Taylor series expansion of the Θ i 's and Θ i 's up to order O(q − 1) 3 and use the relation on the thereby appearing second order terms imposed by the equality Θ Note that we have arranged the general definition of (λ, y, Z) associated to a matrix A(x, t) as in Definition 1.6, such that for the matrix A = A−I 2 (q−1)x , equations ( 13) and ( 14) may be written as This definition is analogous to the general definition in equations ( 5) and ( 6) of (λ, y, Z) associated to a matrix A(x, t) as in Definition 1.1.Indeed, recall that these equations were given by Therefore, we expect ∂ q,x Y = A(x, t, q)Y as in Definition 1.6, but with an additional parameter q, to be an appropriate discretization of family of sl 2 -Fuchsian systems ∂ x Y = A(x, t)Y as in Definition 1.1, where moreover (λ, y, Z) are well defined, if the spectral data (Θ, Θ)(q) satisfy (18) and if, in some convenient sense, we have Let us now explain what we shall mean by this convenient sense.Let Q ⊂ C \ {0, 1} be a connected, not necessarily open, subset, with 1 in its closure.Consider a connected open subset D ⊂ C * and let f (t, q) be a function such that for each fixed q ∈ Q sufficiently close to 1, we have a well-defined meromorphic function t → f (t, q) in M(D).We say that f ∈ M(D) is the limit of f as q → 1 if for generic values (i.e.outside a proper closed analytic subset) of t ∈ D, we have lim Analogously, if we have a reduced rational function φ (x, (f , where each coefficient f ℓ (t, q) is as above and lim q→1 f ℓ (t, q) = f ℓ (t) ∈ M(D), then we say that Let Θ(q) and Θ(q) be two quadrupels of elements of C(q) such that Θ ∞ = Θ ∞ and such that equations (8) and (18) hold.Let (λ(t, q), y(t, q), Z(t, q)) be a triple of meromorphic functions in neighborhood of D × Q ⊂ C 2 such that λy(y − 1)(y − t)(1 + (q − 1)yZ) does not vanish identically on D × Q and let (λ(t), y(t), Z(t)) be a triple of meromorphic functions on D such that λy(y − 1)(y − t) does not vanish identically.Then for A = A−I 2 (q−1)x and A as in Lemmas 1.9 and 1.2 respectively, we have On the other hand, let us consider the rational function By definition of A, A and by Proposition 1.11, the affine part of the polar divisor of L is contained in and does not contain {q = 1}.Hence if (19) holds, then [(x, t, q) → (q − 1)L(x, λ(t, q), y(t, q), Z(t, q), t, q)] = 0 .
Let us now prove the "only if" part of the statement.If (20) holds, then the limit of the (1, 2) coefficient of A(x, t, q) yields the (1, 2) coefficient of A as q → 1.From the explicit formulae, we deduce and thus lim q→1 (y(t, q), λ(t, q)) = (y(t), λ(t)).By assumption, we have Since lim q→1 y(t, q) = y(t), we deduce On the other hand, again from lim q→1 y(t, q) = y(t), we get The addition of the limits ( 23) and ( 24) yields lim q→1 Z(t, q) = Z(t).
According to the above proposition, under some generic hypotheses, for a convenient choice of spectral value functions (Θ, Θ)(q), equation (18) provides a convenient setting for the discretization of families of sl 2 -Fuchsian systems as in Definition 1.1.Here the convenient conditions the spectral value functions must satisfy are the following (up to permutation of the roles of Θ i and Θ i for i ∈ {0, 1, t}): A simple way to achieve these conditions is to choose the following setting: This convention Θ i Θ i = 1 can be seen as a q-analogue of the tracefreeness of the differential Fuchsian systems we consider.Note that if Θ i (q) is analytic in a neighborhood of 1 and Θ i (q) = 1 + (q − 1) independently of the particular value of the second order term in the Taylor series expansion of Θ i .

Schlesinger equations
x − t over P 1 .An important invariant of such a system is its monodromy, defined as follows.
In a neighborhood V of a point x 0 ∈ C \ {0, 1, t}, this system admits a fundamental solution Y, i.e. a holomorphic function Y : does not depend on the choice of the fundamental solution Y near x 0 and is referred as the monodromy of the Fuchsian system.Note that the monodromy does not depend on the choice of the base point x 0 in the following sense.If x 1 ∈ C\{0, 1, t}, we may choose a path γ 1 from x 0 to x 1 in C\{0, 1, t}, yielding an isomorphism τ γ 1 : The representation ρ 1 := ρ • τ γ 1 then is the monodromy representation with respect to the fundamental solution Y γ 1 , and the conjugacy class [ρ 1 ] does not depend on the choice of the path γ 1 .
Definition 2.2.We say that (25) is isomonodromic if one of the two following equivalent properties hold.
(2) The system (25) can locally be completed into a Lax pair, i.e. any point t 0 ∈ U admits a neighborhood ∆ where there exists B ∈ sl 2 (O(∆)(x)) such that the following holds: • the polar locus of B is contained in • and the system of differential equations over P 1 × ∆ satisfies the Lax equation That these properties are indeed equivalent can deduced from [Bol97, Thm.2].Moreover, considering the special case of non-resonant spectral data in [Bol97, Thm.3], we have the following lemma.
Lemma 2.3.Assume that the spectral data θ satisfy the non-resonancy condition (3).Under that condition, if the system (25) can be completed into a Lax pair over P 1 × ∆ via a certain matrix function B, then this matrix is of the form where C ∈ sl 2 (O(∆)) is a tracefree diagonal matrix function.
A key ingredient in the proof of the above classical results are the well-known Schlesinger equations.They yield particular types of ismonodromic deformations.
Definition 2.4.We say that (25) is Schlesinger isomonodromic if one of the two following equivalent properties hold.
(1) The system (25) can be completed into a Lax pair via the matrix (2) The residues A i (t) of (25) satisfy the Schlesinger equations In order to see that these properties are equivalent, it suffices to compare the residues at x = 0, 1, t of the Lax equation in the case B = − At(t) x−t .Of course Schlesinger isomonodromic families are isomonodromic.As is immediate to check, the converse holds locally up to conjugation: Lemma 2.5.Assume that (25) is isomonodromic and non-resonant.Let us consider C(t) = diag(c(t), −c(t)) ∈ sl 2 (O(∆)) be the tracefree diagonal matrix appearing in Lemma 2.3.Let t 0 ∈ U and let ∆ be a sufficiently small neighborhood of t 0 such that there exists a nonvanishing holomorphic function µ ∈ O(∆) such that µ ′ (t) = c(t)µ(t).Then, the gauge transformation Y = M Y with M (t) = diag µ(t), 1 µ(t) yields a family of sl 2 -Fuchsian systems which is Schlesinger isomonodromic.
By definition, the family (25) is Schlesinger isomonodromic if and only if the entries of A satisfy a certain list of differential equations.Since these entries may be written in terms of the triple (λ, y, Z), see Lemma 1.2, we may translate the differential equations in terms of the entries of A into differential equations in terms of the triple (λ, y, Z).This will lead to the sixth Painlevé equation, according to the following classical results, due to R. Fuchs [Fuc07].
Proposition 2.6.Let ∂ x Y = A(x, t)Y be a family of Fuchsian sl 2 -systems with spectral data θ as in Definition 1.1, giving rise to a triple (λ, y, Z) as in (7).
, and λ is a solution of Proof.Since this is less often detailed in the literature, let us explain how to verify this result by direct computation.Since A ′ ∞ (t) = 0, we have Then, the Schlesinger equations ( 26) are equivalent to the vanishing of the following two matrices Substituting the explicit values of the entries the A i 's given by Lemma 1.2, one easily checks that Sch With some (computer assisted) effort, one checks that Sch (1,1) 0
Conversely, if (y, Z) is a solution of (27), then locally in U * one may choose a local nonvanishing solution λ of (28).One obtains a family of Fuchsian sl 2 -systems ∂ x Y = A Y given by ( λ, y, Z) as in Lemma 1.2.This family of systems is Schlesinger isomonodromic by Proposition 2.6 and conjugated to ∂ x Y = AY by a diagonal gauge transformation of the form Y = M Y with M = diag(µ, 1/µ) satisfying λµ 2 = λ.Hence the initial system is isomonodromic.

2.2.
A discrete analogue.We will now define a convenient q-analogue of isomonodromy, which will lead to a q-analogue of Schlesinger equations for families of Fuchsian linear q-difference systems.Analogously to the differential case, we will define q-isomonodromy by the existence of the q-analogue of the Lax pair.This definition differs from the approach to q-isomonodromy used for example in [JS96].The relation between the two will be explained in Section 5. Let q ∈ C \ {0, 1} .From now on, we make the additional asumption that D is a connected open subset of C * , which is stable under multiplication by q and 1 q .Let (29) be a family of q-Fuchsian systems with spectral data (Θ , Θ) as in Definition 1.6.
Definition 2.8.We say that the system (29) is q-isomonodromic if it can be completed into a q-Lax pair, i.e. there exists B ∈ GL 2 (M(D)(x)) such that the system satisfies the q-Lax equation We will now establish the first step towards the q-analogue of Lemma 2.3: under the assumption that q is not a root of unity, if (29) is non-resonant and can be completed into a q-Lax pair via a matrix function B, then B has a very particular shape.
Proof.Let us write .
With this notation, equation (30) reads From the expression of the determinant of A(x, t) we have , with 0 ≡ c(t) ∈ M(D) and a i , b j being elements of the algebraic closure M(D) of M(D) such that a i (t) ≡ b j (t) for all i, j.Applying the determinant in the both sides of (32), we find that which simplifies as follows: . At x = ∞, the left hand side behaves like x n 1 −n 2 −2 , while the right hand side behaves like q n 1 −n 2 −2 x n 1 −n 2 −2 .Since q is not a root of unity, we must have n 1 = n 2 + 2. We may thus rewrite the equality as It follows easily that f ∈ M(D)(x) is constant in x, forcing n 2 = 0 and, up to renumbering the a i 's, that a 1 = qtΘ t and a 2 = qtΘ t .In particular, We may rewrite (32) as follows. ( Let us now show that B(x, t) is of the form x • B ∞ (t) + B 0 (t), i.e. let us show that the function x → B(x, t) has only one possible pole, at x = ∞, and that this is at most a simple pole.
Taking the determinant in both sides yield det(R must actually be invertible and 1 = q 2k−2 .Since q is not a root of unity, we have k = 1.Hence B(x, t) has a simple pole at From the particular form of A and (33) we know that both A(x, qt) −1 and A(x, t) are finite and invertible at x = 1 q α, i.e.A(α/q, t), A(α/q, qt) −1 ∈ GL 2 (M(D)).Indeed, the only possible poles of A(x, t) are x = tΘ t and x = tΘ t and the only possible poles of A(x, qt) −1 are x = Θ 1 and x = Θ 1 .Hence by (35), B(x, t) has a pole at x = 1 q α as well.By induction, B(x, t) has a pole at x = 1 q n α for each n ∈ N. Yet by assumption, B(x, t) is a rational function of x and can therefore only have finitely many poles.
• Let α(t) ∈ {Θ 1 , Θ 1 , tΘ t , tΘ t } • q Z >0 .Assume for a contradiction that x = α is a pole of B(x, t).Then x = qα is a pole of B(x/q, t).We may rewrite equations ( 35) and (33) as The same reasoning as before shows that B(x, t) then has a pole at x = qα as well.Again this leads by induction to an infinite number of poles, and therefore, a contradiction.• Finally, we treat the case α = 0. We have . Equation (34) yields det(R 0 (t)) = q 2 t 2 c(t)Θ t Θ t = 0, so that R 0 is actually invertible.Since Spec(A 0 (t)) = {Θ 0 , Θ 0 }, we have det(A 0 (t)) = det(A 0 (qt)) and we may take the determinant in both sides of R 0 (t) = q k A −1 0 (qt)R 0 (t)A 0 (t) and find q 2k = 1.Since q is not a root of unity, we deduce k = 0.
We have now proven that B(x, t) has a simple pole at x = ∞ and is finite and non-zero everywhere else.More precisely, we have proven that holds for the diagonal matrix C(t) := R ∞ (t) ∈ GL 2 (M(D)) and the matrix B 0 := C(t) −1 R 0 (t) ∈ GL 2 (M(D)).Moreover, from (34) we get det(xI 2 + B 0 (t)) = (x − qtΘ t )(x − qtΘ t ), yielding the sought expression for the eigenvalues of B 0 .
We will see in the proof of the following proposition that the matrix B 0 , as well as the matrix C up to a scalar multiple, are uniquely defined by (30).However, once these matrices are obtained, (30) still imposes strong conditions on the matrix A, which will yield the q-Painlevé VI equation.
Proposition 2.10.Assume that q ∈ e 2iπQ .Let (Θ, Θ) ∈ (C * ) 4 × (C * ) 4 such that Θ ∞ = Θ ∞ and such that the relation (8) as well as the non-resonancy condition (9) hold.Let (λ, y, Z) ∈ (M(D)) 3 .Denote and assume that y, (y − 1), (y ) are all well defined meromorphic functions on D and are each not identically zero.Let A(x, t) = A 0 (t) + x x−1 A 1 (t) + x t(x−t) A t (t) be defined by (λ, y, Z) as in Lemma 1.9 and assume that the coefficients of A 0 , A 1 , A t are holomorphic on a domain D * ⊂ D stable under multiplication by q and 1/q.Let us denote The following are equivalent.
Proof.Let us first show that (1) ⇔ (2).Since q is not a root of unity, by Proposition 2.9 and the non-resonancy assumption, (1) is equivalent to the existence of B 0 , C ∈ GL 2 (M(D)) with C diagonal, such that for B given by (31), we have (σ q,t A(x, t)) B(x, t) = (σ q,x B(x, t)) A(x, t).
We may rewrite , where Hence (1) is equivalent to the vanishing R i = 0 for all i ∈ {0, 1, 2, 3, 4, 5} for some B 0 , C ∈ GL 2 (M(D)) with C diagonal.Note that by Proposition 2.9, the matrix B 0 must have eigenvalues We may therefore omit R 3 in the following.For i ∈ {0, 1}, the vanishing of R i is equivalent to the equation for σ q,t A i as in the statement, with general B 0 , C. Substituting these equations into R 2 = 0 yields the equation for σ q,t A t as in the statement, again with general B 0 , C. However, R 4 = 0 is equivalent to B 0 being as in the statement.Note that this matrix B 0 is well-defined and invertible under the assumptions.As one can check by direct computation, this B 0 solves R 5 = 0. Hence (1) is equivalent to the existence of a diagonal matrix C ∈ GL 2 (M(D)) such that the equations in (2) hold for B 0 as in the statement.If C is such a convenient matrix, then for any f ∈ M(D) non-vanishing, f C is also convenient.Hence we may require that C is of the form diag(c, 1).Since (q − 1)λ = (1 − t)A (1,2) 1 − tA (1,2) 0 , we must have With the equations in (2) for the σ q,t A i 's, this is equivalent to c being as in the statement.We conclude that (1) ⇔ (2).
Therefore, σ q,t Z is as in the statement.
Let us now show that (3) ⇒ (2).Note that for each i ∈ {0, 1, t}, the matrices A i may be expressed as functions of λ, X and y.If (3) holds, then the matrices C −1 σ q,t A i C, with C as in (2), can also be expressed as functions of λ, X and y.It its straightforward to check (with computer assistance) that the equations in (2) then are satisfied.
In analogy with the differential case, we give a name to the particular case when a family can be completed into a Lax pair via a matrix B as in (31) with C(t) = I 2 : Definition 2.11.We say that the family σ q,x Y = A(x, t)Y of Fuchsian systems is q-Schlesinger isomonodromic if it can be completed into a q-Lax pair via a matrix B ∈ GL 2 (M(D)(x)) of the form Let us now say a few words about whether, analogously to the differential setting, a family of q-Fuchsian systems which is q-isomonodromic can be made q-Schlesinger isomonodromic via a gauge transformation.Let A ∈ GL 2 (O(D)(x)) be as in (29) and assume that the family of q-Fuchsian systems ∂ q,x Y = A(x, t)Y can be completed into a Lax pair via a matrix B ∈ GL 2 (M(D)(x)) of the form (31), with C ∈ GL 2 (M(D)) diagonal.Assume there exists M ∈ GL 2 (M(D)) which is diagonal and solves the q-difference equation Since M does not depend on x, performing the gauge transformation Y = M Y yields the family σ q,x Y = A(x, t) Y given by Since M is diagonal, up to shrinking D to the domain of holomorphy of the coefficients of A ∈ GL(M(D)(x)), this new family is still a family of q-Fuchsian systems in the sense of Definition 1.6.Moreover, this new family can be completed into a Lax pair via the matrix Indeed, from the q-Lax equation for the initial family, we get Note that B is given by In other words, the conjugated family σ q,x Y = A(x, t) Y is q-Schlesinger isomonodromic.To find this conjugated family, we had to solve a diagonal system of q-difference equations, which boils down to solving two scalar linear q-difference equations.Contrarily to the differential case, the resolution of q-difference equations even of such simple form is not trivial, and does not seem to be known in full generality.However, if some strong assumptions on the domain of definition D are satisfied, one can use for example the following lemma.
Note that the assumption on D of the above lemma is satisfied if for instance a ℓ q Z , for some a ℓ ∈ C * .
Proof.Let us define These are connected open sets satisfying . By [BHHW18, Lemma 4.4], there exist c 1 ∈ M(O 1 ), and c 2 ∈ M(O 2 ) such that c = c 1 c 2 .By construction, c 1 is a germ of meromorphic function at 0. By Remark 5.4, there exists 0 = m 1 that is meromorphic on a punctured neighborhood of 0 in C * such that σ q,t m 1 = c 1 m 1 .Using the functional equation and using the fact that D is stable by multiplication by q, we find that m 1 may be continued into a meromorphic function on qO 1 \ {0} where qO 1 = {qt, t ∈ O 1 }.Similarly, we construct a non-zero meromorphic solution of σ q,t m 2 = c 2 m 2 that is meromorphic on 2.3.Confluence.Let Θ(q) = (Θ 0 (q), Θ 1 (q), Θ t (q), Θ ∞ (q)) be a quadruple of rational functions in a complex variable q such that as q → 1, we have (39) Θ i (q) = 1 + (q − 1) with θ i ∈ C. We define Θ by Θ i = 1 Θ i .Recall from Section 1.3 that these requirements on (Θ, Θ) are a convenient setting for the discretization of sl 2 -Fuchsian systems with spectral data θ (if θ ∞ = 0).We shall now see under these requirements, the q-Schlesinger equations discretize the (differential) Schlesinger equations, and that the difference equation (36) generically characterizing q-isomonodromy discretizes the differential equation ( 27) generically characterizing isomonodromy.
2.3.1.The q-Schlesinger equations discretize the differential ones.The q-Schlesinger equations are obtained from the equations in point (2) of Proposition 2.10 by setting C = I 2 .With respect to the ∂ q,t -operator and the matrices , they read as follows: where B 0 = B 0 A 0 , A 1 , t, q is the function with values in defined, on the complement of some proper Zariski closed subset of GL 2 (C) × GL 2 (C) × C × C, as B 0 A 0 , A 1 , t, q being given by by this context, may write everything as a function of A 0 and A 1 by identifying With this notation, as q → 1, up to terms of order O(q − 1) 2 , we have Let f 0 , f 1 , f t be the functions with values in M 2 (C), defined on some obvious domain of definition inside GL 2 (C) × GL 2 (C) × C × (C \ {1}), that when evaluated in A 0 , A 1 , t, q , yield the right hand sides of the equations in (40).Then we have Using similar calculations, and the Taylor series expansion of B 0 until its second order term, one finds To summary, we have proved This proves that the q-Schlesinger equations (40) discretize the (differential) Schlesinger equations (26).
In order to complement this result, let us consider the function B with values in M 2 (C), given, on its obvious set of definition inside GL This function corresponds to the right hand side of the δ q,t -version of σ q,t Y = BY with B as in (38).It behaves, when q → 1, as By the above estimates, B can be continued analytically to {q = 1} and is there given by − At x−t .
2.3.2.The q-Lax pairs discretize the differential ones.Let be a connected subset with 1 in its closure.Let D ⊂ C * be an open connected subset.We shall assume that the pair (D, Q) satisfies the property that D is stable by multiplication by q ±1 , for every q ∈ Q.Note that unless D = C * , the subset Q cannot be too large.Two examples of a convenient pair (D, Q) with D ⊂ C * are the following: where q 0 ∈ C, with |q 0 | = 1.
• D is an open sector with infinite radius centered at 0 and Q =]1, +∞[.
In addition to our previous requirements on (Θ, Θ), let us now moreover assume that the (differential) spectral values θ satisfy the non-resonancy condition θ i ∈ Z * for i ∈ {0, 1, t, ∞} and θ ∞ = 0. Note that for values of q ∈ Q sufficiently close to 1, the (q-difference) non-resonancy condition (9) then is automatically is satisfied.
is a family of sl 2 -Fuchsian systems with spectral data θ as in Definition 1.1.Let A 0 , A 1 , A t be holomorphic functions in a neighborhood of D × Q ⊂ C 2 with values in GL 2 (C), such that for each q ∈ Q, the q-difference equation ∂ q,x Y = A(x, t, q)Y with A = A 0 x + A 1 x−1 + At x−t yields, via A = I 2 + x(q − 1) A, a family of q-Fuchsian systems with spectral data (Θ(q), Θ(q)) as in Definition 1.6.By Proposition 1.12 it is convenient to assume that ∀i ∈ {0, 1, t} , lim and that these limits are uniform on compact subsets of D, such that ∀i ∈ {0, 1, t} , Finally, let us assume that the family ∂ q,x Y = A(x, t, q)Y is q-isomonodromic for each q ∈ Q.By non-resonancy, §2.3.1 and the proof of Proposition 2.10, this means that this family can be completed into a q-Lax pair    σ q,x Y = I 2 + (q − 1)x A(x, t, q) Y σ q,t Y = C(t, q) I 2 + (q − 1)t B(x, t, q) Y .
Here f ∈ M(D × Q) \ {0} can be chosen arbitrarily and where X is some rational expression in terms of q, t, Θ 1 , Θ ∞ , A (1,1) 0 , A (1,2) 0 , A (1,1) 1 , A (1,2) 1 that can easily be made explicit.We assume these c i and X to be well-defined and finite.Using the Taylor series expansion of Θ 1 (q) and Θ ∞ (q), we readily compute that up to terms of order O(q − 1) 2 , we have .
Then, dividing by (q − 1) 2 xt and we obtain with ∂ q,x C = 0 that x−t + O(q − 1), this shows that we obtain the confluence of the q-Lax pair (41) to the differential Lax pair of
Of particular interest for the results in this paper is the discretization of the characterization of Schlesinger isomonodromy in terms of triples (λ(t), y(t), Z(t)) given in Proposition 2.6, namely the system of differential equations given by ( 27) and (28).Proposition 2.10 suggests that a convenient q-analogue of this differential equation is given by ( 42) where X(y, Z, t, q) and E(y, Z, q, t) are defined respectively as .
Here, as usual when considering confluence, we used our convention Θ i Θ 1 = 1.Let us now show that (42) discretizes the system of differential equations given by equations ( 28) and ( 27).Let f λ , f y , f Z be the rational functions in the variables y, Z, t, q forming the right hand sides of the equations in (42).Using the estimates (39), we may compute the Taylor series expansion of X(q) := X(y, Z, t, q) as q → 1.Up to terms of order O(q − 1) 3 , we have So in particular, we have X(q) = y−t y−1 + O(q − 1).Since moreover Θ ∞ = 1 + O(q − 1) and Θ ∞ − qΘ ∞ = (q − 1)(θ ∞ − 1) + O(q − 1) 2 , we conclude that In other words, the first difference equation in ( 42) discretizes (28).Similarly, up to order terms of order O(q − 1) 3 , we obtain Substituting the Taylor expansion of X(q) up to order O(q − 1) 3 yields .
Already from the Taylor expansion of E(q) up to order O(q − 1) 2 , we deduce that From the Taylor expansion of E(q) and X(q) up to order O(q − 1) 3 , we deduce by a series of tedious but straightforward calculations that . It follows that the second and third difference equation in (42) together discretize the system of differential equations (27). 4.We define the rational function H θ VI ∈ C(y, Z, t) in three variables given by

The sixth Painlevé equation
. Consider the non-autonomous Hamiltonian system defined by ( 43) Explicitly, it is given by (44) Recall from Corollary 2.7 that if for all i ∈ {0, 1, t, ∞}, we have θ i ∈ Z * and θ ∞ = 0, then this system of differential equations characterizes isomonodromy for families of sl 2 -Fuchsian systems.Substituting Z = t(t−1)y ′ (t) 2y(y−1)(y−t) − 1 2(y−t) (from the first equation in ( 44)) into the second, we obtain the sixth Painlevé equation associated to the spectral data θ: Conversely, given a meromorphic solution y of P VI (we will see in the sequel that it exists), and assuming it is not identically equal to 0, 1, t (which is a trivially satisfied if θ 0 θ 1 θ t = 0), then the substitution formula yields a meromorphic function Z such that the pair (y, Z) is a meromorphic solution of (44).
Let us briefly recall the well-known results concerning the existence of analytic solutions of P VI .By the Cauchy-Lipschitz theorem, for every t 0 ∈ C \ {0, 1} and every choice of (y 0 , y 1 ) ∈ (C \ {0, 1, t 0 }) × C, there exists a unique holomorphic function y(t) defined in a neighborhood of t 0 such that y(t 0 ) = y 0 and y ′ (t 0 ) = y 1 , and such that y is a solution of the sixth Painlevé equation.Equivalently, for every t 0 ∈ C\{0, 1} and every choice of (y 0 , Z 0 ) ∈ (C \ {0, 1, t 0 })×C, there exists a unique holomorphic solution (y(t), Z(t)) of the Hamiltonian system (44), defined in a neighborhood of t 0 , such that (y(t 0 ), Z(t 0 )) = (y 0 , Z 0 ).By the so-called Painlevé property, any such germ of holomorphic solution can be meromorphically continued along any path in C \ {0, 1}.In particular, on any simply connected subset U of P 1 \ {0, 1, ∞}, there exists a unique meromorphic solution satisfying some initial condition as above at t 0 ∈ U (see for instance [HL04,JK94], see also Section 4.1 for some details).

A discrete analogue.
Let us fix q ∈ C \ {0, 1} and let us consider the spectral data In [JS96] the q-Painlevé VI equation associated to such a spectral data was introduced.It is given by the following system of q-difference equations: (45) qP JS,VI (Θ, Θ) : The auxiliary parameters in [JS96] bear other names, but we have written the equation in a way that the dictionary between the auxiliary parameters in [JS96] and the above Θ i , Θ i is obvious, see (2).This system of difference equations has been derived in [JS96], for |q| = 1, from the pseudo-constancy condition of the Birkhoff connection matrix for q-Fuchsian systems with non-resonant spectral data (Θ, Θ).Note that the change of variable applied to (45) yields the q-difference system (36).In the case q ∈ e 2iπQ and non-resonant (Θ, Θ), solutions of the latter system have been shown in Proposition 2.10 to correspond (under some generic assumptions) to q-isomonodromic (in the sense of Definition 2.8) families of q-Fuchsian systems.Conversely, when starting with (36), the change of variable Z = (y−tΘt)(y−tΘt) q(y−1)(y−t)z − 1 (q − 1)y yields equation ( 45), which has a significantly shorter and more symmetric expression.Note that with this change of variable, one has σ q,t z = X, for X as in (36).
From now on, because we are ultimately interested in the behaviour under confluence, we will use our convention ∀i ∈ {0, 1, t, ∞}, Θ i Θ i = 1, by which equations ( 45) and (36) can obviously be simplified.In particular, from now on, the following system of q-difference equations will be referred to as the q-Painlevé VI equation associated to spectral data Θ ∈ (C * ) 4 : (46) qP VI (Θ) : Unfortunately, contrarily to the differential situation, the existence of a meromorphic solution having a prescribed value at a point t 0 ∈ C * is in general not known.Let us now focus on discrete solutions, i.e. the sequence of values on q Z t 0 for some t 0 ∈ C * \ q Z that a meromorphic solution defined on a domain containing the spiral q Z t 0 should interpolate.More precisely, a discrete solution of (46) is a sequence (y ℓ , z ℓ , t ℓ ) ℓ∈Z of points in P 1 × P 1 × C * , such that • the sequence (t ℓ ) ℓ∈Z is given by t ℓ = q ℓ t 0 for some t 0 ∈ C * 3.3.Confluence.As usual for matters of confluence, in this section we will only consider spectral data (Θ, Θ) related by Θ i Θ i = 1.We will first establish that the sixth Painlevé equation, up to the change of variable and spectral data that we previously found to be convenient for confluence, admits a q-analogue of Hamiltonian formulation.From this, we will deduce the confluence of discrete and meromorphic solutions.
The following result states that this modified q-Painlevé VI equation q P VI (Θ) is an appropriate q-analogue of a Hamiltonian system.First, let us introduce the Hamiltonian.To each datum θ = (θ 0 , θ 1 , θ t , θ ∞ ) ∈ C 4 , we associate the rational function H θ VI ∈ C(y, Z, t) in three variables given by (51) H θ VI (y, Z, t) := y(y−1)(y−t) . Note this is nothing else than the Hamiltonian for the differential case.In the following, we denote abusively 1 + (q − 1) θ 2 := 1 + (q − 1) ) be the (well-defined) rational functions in four variables such that the modified q-Painlevé VI equation (50) with spectral data given by 51) and the operator identity y,Z,t,q is not an irreducible component of the polar divisor of R.Moreover, the polar locus of the therefore well-defined rational function R| q=1 on C 3 y,Z,t is contained in the set Proof.Let us first say some words about the well-definedness of R θ 1 , R θ 2 .For quadrupels Θ, there are well defined rational functions f, g ∈ C(y, Z, t, q, Θ 0 , Θ 1 , Θ t , Θ ∞ ) such that q P VI (Θ) can be written as σ q,t y = f (y, Z, t, q, Θ) σ q,t Z = g(y, Z, t, q, Θ) .
Indeed, it suffices to substitute the first equation in (50) into the second, so that the right hand sides only depends on the variables y, Z, t, q, Θ.Note that ∂ q,Z H θ VI (y, Z, t) and ∂ q,y H θ VI (y, Z, t) can easily be calculated and are elements of C(y, Z, t, q).Then are indeed elements of C(y, Z, t, q) and are those required by the statement.Let us define Since ∂ ∂Z H θ VI and ∂ ∂y H θ VI are rational functions of (y, Z, t), these R θ 1 , R θ 2 are again rational functions in the variables y, Z, t, q.Denoting ∇ In order to compute these differences, note that H θ is rational with only simple poles independent of Z and for ⋆ ∈ {y, Z}, the operator ∇ ⋆ is C(t)-linear.So it suffices to compute ∇ x (x n ) for n ∈ N and ∇ x ( 1 x−a ) for a independent of x.We have ∇ x (1) = 0, and for n ∈ N * , we find where [k] q = q k −1 q−1 = k−1 i=0 q i with [0] q = 0.In particular, we find ∇ x (x) = 0, ∇ x (x 2 ) = x, ∇ x (x 3 ) = (2 + q)x 2 .For a independent of x, we find .
Obviously, these differences do not have {q = 1} as an irreducible component of their respective polar divisors, and their restrictions to q = 1 do not have poles outside P.This means that the statement holds for But for the latter, we have already done most of the work.Indeed, the calculations in §2.3.3 at the end of Section 2.3 show that in restriction to any line {(y, Z, t) = (y 0 , Z 0 , t 0 )} ⊂ C 4 y,Z,t,q with (y 0 , Z 0 , t 0 ) ∈ C 3 \ P, the two rational functions vanish both at q = 1.It follows that {q = 1} is an irreducible component of the zero divisor of both (q − 1) R θ 2 and (q − 1) R θ 2 .In particular, {q = 1} is not an irreducible component of the polar divisor of R θ 1 or R θ 2 .Moreover, even though we did not push the Taylor series expansions in Section 2.3 far enough as to have an explicit expression for R θ 1 | q=1 and R θ 2 | q=1 , it is still clear from the calculations that these functions cannot have poles outside P. The result follows.
3.3.2.Confluence of discrete solutions.We will now see that discrete solutions of the modified q-Painlevé VI equation yield holomorphic solutions of the differential Painlevé VI equation by some limit process.The idea is that the successive ∂ q,t -derivations should lead, when q → 1, to the coefficients of the Taylor series expansion of the limit functions.Let us consider the operator δ q,t := t∂ q,t = σ q,t − 1 q − 1 .
As one can easily check, for each n ∈ N, we have Here we use the convention δ 0 q,t = σ 0 q,t = 1.We will prove the following.
Then, there exists a family (y n , Z n ) n∈N of pairs of rational functions (y n , Z n ) ∈ C(q) × C(q) such that for generic values of q, the sequence (y n (q), Z n (q), q n t 0 ) n∈N is the positive part of the discrete solution of q P VI 1 + (q − 1) θ 2 with initial value (y 0 , Z 0 , t 0 ).Consider the sequence Then for each n ∈ N, neither a n (q) nor b n (q) has a pole at q = 1.Moreover, the power series both converge and yield the pair of functions q → (y(q • t 0 ), Z(q • t 0 )), where (y(t), Z(t)) is the unique solution of the Painlevé Hamiltonian system (43) with initial condition (y 0 , Z 0 ) at t 0 .
This result will be proven by the end of this section.As it turns out, rather than trying to calculate the limit for q → 1 for the (a n (q), b n (q)) directly, it is easier to first construct a particular sequence of rational functions that are finite at q = 1 and then show that this sequence is actually the one from the statement.First we will need some general remarks.
The δ q,t -operator on a field of functions with complex variable t is additive and satisfies the following algebraic properties: In particular, for any rational function F ∈ C(y, Z, t, q) we may define, by treating y and Z like functions of t, a rational function ∆ F with two additional variables such that δ q,t F (y, Z, t, q) = ∆ F (y, Z, t, q, δ q,t y, δ q,t Z) .
Let δ t = t∂ t that is the formal limit of δ q,t when q goes to 1.The q-analogue of the chain rule that we will need is the following.
Lemma 3.4.Let F ∈ C(y, Z, t, q) and ∆ F ∈ C(y, Z, t, q, δ q,t y, δ q,t Z) be as above.Define R F ∈ C(y, Z, t, q, δ q,t y, δ q,t Z) by If {q = 1} is not an irreducible component of the polar locus of F , then it is not an irreducible component of the polar locus of ∆ F and R F .Moreover, when F is seen as an element of C(y, Z, t, q, ∂ q,t y, ∂ q,t Z), then the affine parts of the polar locus of ∆ F | q=1 and R F | q=1 are contained in the polar locus of F | q=1 .
Proof.Let P, Q ∈ C[y, Z, t, q], 0 = Q, such that F = P/Q.We use (53) to compute successively Then (q − 1)R Q −1 is given by We have Finally, This proves that {q = 1} is not an irreducible component of the polar locus of ∆ F and R F .Note that Hence the affine parts of the polar loci of ∆ F | q=1 and R F | q=1 are contained in the zero locus of Q| q=1 .This concludes the proof.
Let H 1 , H 2 ∈ C(y, Z, t) and R 1 , R 2 ∈ C(y, Z, t, q) be rational functions in three and four complex variables respectively such that • the affine part of the polar locus of each of the functions H i with i ∈ {1, 2} is contained in the subset P ⊂ C 3 y,Z,t given by P := {y = 0} ∪ {y = 1} ∪ {y = t} ∪ {t = 0} ∪ {t = 1} .• for each i ∈ {1, 2}, the polar locus of R i does not contain {q = 1}, and the affine part of the polar locus of R i | {q=1} is contained in P. Consider the system of q-difference equations (54) δ q,t y = H 1 (y, Z, t) + (q − 1)R 1 (y, Z, t, q) δ q,t Z = H 2 (y, Z, t) + (q − 1)R 2 (y, Z, t, q) .
Applying the operator δ q,t on both sides and then substituting the values of δ q,t y, δ q,t Z imposed by this system yields a second order relation.There exist rational functions R (1) 2 ∈ C(y, Z, t, q) such that this second order system is of the form    δ 2 q,t y = H (1) 1 (y, Z, t) + (q − 1)R (1) 1 (y, Z, t, q) δ 2 q,t Z = H (1) 2 (y, Z, t) + (q − 1)R (1) 2 (y, Z, t, q) , where H (1) 1 , H 2 ∈ C(y, Z, t) and R (1) 2 ∈ C(y, Z, t, q) are given for i ∈ {1, 2} by We are therefore inclined to define the sequence (y n , Z n ) n∈N ∈ (C(q) × C(q)) N given by By construction, the elements of the sequence (y n , Z n , q n t 0 ) n∈N ∈ (C(q) × C(q) × C(q)) N are related to (y 0 , Z 0 , t 0 ) by the same rational relation as those of a discrete solution with initial value (y 0 , Z 0 , t 0 ) of the modified q-Painlevé VI equation (50) with spectral data Θ = 1+(q−1) θ 2 .Therefore, the sequence of rational functions (y n , Z n , q n t 0 ) n∈N is the (positive part of) the solution with initial value (y 0 , Z 0 , t 0 ), seen as a rational function of the variable q.
The process in (55) allowing to recover σ n q,t from 1, δ q,t , . . ., δ n q,t is inverse to the process in (52) allowing to recover δ n q,t from σ 0 q,t , . . ., σ n q,t .Therefore, the sequence ( a n , b n ) n∈N constructed above coincides with the (a n , b n ) n∈N defined in the statement of Theorem 3.3.This concludes the proof of Theorem 3.3.
Remark 3.5.Note that with respect to the notation in Theorem 3.3, for each n ∈ N, as q → 1, we have y n (q) − y(q n t 0 ) = O(q − 1) and Z n (q) − Z(q n t 0 ) = O(q − 1).
Proof.In the proof of Theorem 3.3, we may replace (y 0 , Z 0 ) by (y 0 (q), Z 0 (q)) in the definition of (a n (q), b n (q)) n∈N .Note that the latter then is a sequence of pairs of rational functions, each evaluated in a pair of continuous functions in q which, as q → 1, admit a finite limit which is not in the polar locus of the restriction to q = 1 of these rational functions.We conclude that for each n ∈ N, the pair (a n (q), b n (q)) may be continued to a continuous function on Q ∪ {1} with finite value at q = 1.Moreover, as before, we have the relation = (y 0 (q), Z 0 (q)) + O(q − 1) .
4. Okamoto's space of initial conditions 4.1.Differential case.Let us review the construction in [Oka86] of a convenient space of initial conditions for the Painlevé VI equation, and recall why it proves the Painlevé property.
We compactify this space of initial values to the second Hirzebruch surface F 2 , using the following coordinate charts of C 2 -spaces, endowed with their obvious rational transition maps: Here what we have added by the compactification is the union of the horizontal line and the four vertical lines given by Now the Hamiltonian system (43) defines a meromorphic vector field on F 2 × (C \ {0, 1}).Explicitly, it is given with respect to the coordinates (u, v, t) by + θ∞(θ∞−2)u(u−1)(u−t) 4t(t−1) .
One realizes that the vector field is infinite on the set given for each fixed t = t 0 by More precisely, it is infinite or undetermined (of the form " 0 0 ") precisely there.These indeterminacy points will be called base points in the following.If we assume that (57) then there are precisely eight such base points.With respect to the four charts of F 2 , these base points, each possibly visible in several charts, are precisely the following: In the following discussion, we assume (57).For fixed t, the Hirzebruch surface, as well as the configuration of particular lines and base points, are resumed in the following picture.Here "(n)" indicates "self-intersection number equal to n".
For any fixed t, let us denote by F t 2 the result of the above Hirzebruch surface F 2 after blow up of the the eight base points.For each i ∈ {0, 1, t, ∞}, we denote by D * * i the strict transform of D i after blow up of β ± i , i.e. the closure of D i \ {β ± i } in F t 2 .Note that each D * * i has self-intersection number −2.The Okamoto space of initial values at the time t for the sixth Painlevé equation with spectral data θ is by definition Oka t := F t 2 \ I t , where For example in order to blow up β − 0 , one replaces a neighborhood of β − 0 containing none of the other seven base points, by the corresponding neighborhood in the spaces C 2 u 01 ,v 01 and C 2 u 02 ,v 02 related to C 2 u,v according to the following transition maps: was the point β − 0 now corresponds to the exceptional line, isomorphic to P 1 , given by The complementary of E − 0 however is in biholomorphic correspondence with the corresponding open subset of C 2 u,v .The vector field in these new charts is given as follows: , Proof.We simply follow the proof in [Sak01], where generic t, Θ, were considered, and make sure that it goes through for fixed t, Θ as in the statement.In order to make the following argumentation shorter, let us first consider an example.Consider the rational map ϕ from P 1 × P 1 with standard coordinates (y, z) to P 1 × P 1 with standard coordinates (ŷ, ẑ) defined by (y, z) → (ŷ, ẑ) = (y, z(y − Θ 1 )) .This map is actually a so-called elementary transformation with respect to the ruling P 1 × P 1 → P 1 given by (y, z) → y.In the complement of the fiber {y = Θ 1 } of this ruling, its defines a biholomorphism onto its image.Moreover, this map has an indeterminacy point at (y, z) = (Θ 1 , ∞).With the exception of this indeterminacy point, every point in the fibre {y = Θ 1 } is mapped to the point (ŷ, ẑ) = (Θ 1 , 0).Conversely, the inverse rational map (ŷ, ẑ) → (y, z) = ŷ, ẑ ŷ−Θ 1 has an indeterminacy point at (ŷ, ẑ) = (Θ 1 , 0) and maps the rest of the fiber {ŷ = Θ 1 } to the point (y, z) = (Θ 1 , ∞).However, as one can easily check, the rational map obtained by considering the composition is biregular.So in summary, the elementary transformation ϕ blows up the point (y, z) = (Θ 1 , ∞) and contracts the strict transform of the line {y = Θ 1 }, and becomes biregular when pre-and post-composed with the blow-ups of (y, z) = (Θ 1 , ∞) and (ŷ, ẑ) = (Θ 1 , 0) respectively.
The key is now to use elementary transformations in order to decompose the map S t into a sequence of biregular isomorphisms.
• Consider the rational map from P 1 × P 1 with standard coordinates (y, z) to P 1 × P 1 with standard coordinates (y, z) defined by (y, z) → (y, z) = y, z (y−Θ 1 )(y−Θ 1 ) (y−tΘt)(y−tΘt) .This map can be seen as the composition of four (commuting) elementary transformations with respect to the ruling (y, z) → y.Note that by assumption, the set {Θ 1 , Θ 1 , tΘ t , tΘ t } has cardinality four, which implies that no two of these elementary transformations cancel each other out.Therefore, the considered rational map induces a biregular isomorphism from P t to the surface P (1) t , where P (1) t denotes the blow up of P 1 × P 1 with standard coordinates (y, z) at the eight points given, with respect to these coordinates, by • Consider the biregular map from P 1 ×P 1 with standard coordinates (y, z) to P 1 ×P 1 with standard coordinates (y, z) defined by (y, z) → (y, z) = y, 1 q z .This map induces a biregular isomorphism from P (1) t to the surface P (2) t , where P (2) t denotes the blow up of P 1 × P 1 with standard coordinates (y, z) at the eight points given, with respect to these coordinates, by 0, tΘ 0 , (0, tΘ • Consider the rational map from P 1 × P 1 with standard coordinates (y, z) to P 1 × P 1 with standard coordinates ( y, z) defined by (y, z) → ( y, z) = y ( z−Θ∞/q)( z−Θ∞) ( z−tΘ 0 )( z−tΘ 0 ) , z .This map can be seen as the composition of four (commuting) elementary transformations with respect to the ruling (y, z) → z.Note that by assumption, the set {tΘ 0 , tΘ 0 , Θ ∞ /q, Θ ∞ } has cardinality four, which implies that the considered rational map induces a biregular isomorphism from P (2) t to the surface P (3) t , where P (3) t denotes the blow up of P 1 × P 1 with standard coordinates ( y, z) at the eight points given, with Recall that P t is the blow up of F 0 in the eight points γ ± i with i ∈ {0, 1, t, ∞}.In P t , we denote by the strict transforms of the corresponding projective lines in F 0 .We denote Under the map S t : P t → P qt , the set J t is mapped to the set J qt .So in some sense, the set J t can be seen as invariant under the q-Painlevé map S t .Moreover, a discrete solution (y n , z n , q n t 0 ) n∈Z given by an initial condition in J t is not very interesting in the sense that both (y n ) n∈Z and (z n ) n∈Z simply oscillate between 0 and ∞.The Okamoto space qOka t for fixed t ∈ C * \ S q as introduced in [Sak01] is by definition the complement of J t in P t .So we define qOka t := P t \ J t .
Note that the strict transforms V ± * t of the vertical lines V t in F 0 are not contained in J t .They do however play a particular role in the relation to the construction of the modified Okamoto space q Oka t that we will now define.The letter will be better suited for the confluence problem.
Let us recall that Z = (y−tΘ t )(y−tΘ t ) q(y−1)(y−t)z −1 (q−1)y . Motivated by (56), we apply the change of variable to the modified q-Painlevé VI equation (50) with spectral data Θ.There are well-defined rational functions f , g ∈ C(u, v, t) such that with respect to these variables, (50) is of the form More precisely, we have .
We consider the second Hirzebruch surface 3 glued together along their C * × C * -subsets according to the transition maps given by (65) Here we will consider (u, v) as the standard coordinates, with respect to which we will define rational maps such as the following.For each fixed t = t 0 ∈ C * satisfying (59), we obtain a rational map (66) S t : Lemma 4.3.Let Θ ∈ (C * ) 4 such that (61) holds.Let t ∈ C * \ S q , where S q is defined in (63).
The indeterminacy points of the rational map S t defined in (66) are precisely the following eight points (in the source F 2 ).
Proof.The statement can easily be verified by direct computation.Note however that this lemma can also be deduced, with much less computation, from Proposition 4.4 below.
In addition to the eight indeterminacy points β ± i for i ∈ {0, 1, t, ∞}, we identify the following particular projective lines in F 2 : Moreover, we introduce the following curve (that corresponds to H 0 : {z = 0}): The configuration of these points, lines and the curve C in F 2 is illustrated in the following figure .Here the grey numbers indicate the self-intersection number of the corresponding curve.The use of colors will became clear later, see Remark 4.5.
Note that the points β ± 1 , respectively β ± t , are precisely the intersection between C and D ± 1 , respectively C and D ± t .Note further that C ∩ H = ∅, that β ± 0 ∈ H ∪ C because Θ 0 , Θ 0 = 0 and that β ± ∞ ∈ H ∪ C because Θ ∞ , Θ ∞ = 0. Now let us denote, for each t ∈ C * satisfying (59), by the blow up of F 2 at the eight points β ± i (t).Here we continue to assume (61).In P t , we denote by H, D * * 0 , D * * ∞ , D + * 1 , D − * 1 , D + * t , D − * t , C * * * * the strict transforms of the corresponding projective lines/curves in F 2 .We define q− Oka t := P t \ I t , where I t := H ∪ D * * 0 ∪ D * * ∞ ∪ C * * * * .As we shall see, this is an alternative q-Okamoto space of initial values of qP VI , and q Oka t is convenient for the study of confluence.Before formulating the equivalence of q-Oka t and q Oka t , let us give a name to the exceptional curves.We denote, for each i ∈ {0, 1, t, ∞}, by F ± i the exceptional lines in P t corresponding to blow up of γ ± t and by E ± i the exceptional lines in P t corresponding to blow up of β ± i .Proposition 4.4.Let q ∈ C \ {0, 1}.Let Θ ∈ (C * ) 4 such that (61) holds.Let t ∈ C * \ S q , where S q is defined in (63).Consider the birational map given, with respect to the standard coordinates and the above notation, by ϕ : (y, z) → (u, v) = y, (y−tΘt)(y−tΘt)−q(y−1)(y−t)z q(q−1)z .
This map is biregular and induces bijections Moreover, it induces a bijection J t ∼ → I t and therefore provides an isomorphism qOka t ∼ −→ q Oka t .
Proof.First, consider the rational map F 0 F 2 given, with respect to the standard coordinates, by the same formula as ϕ.We will abusively denote it again by ϕ.Note that ϕ preserves the fibers of the rulings F 2 → P 1 and F 0 → P 1 given by (u, v) → u and (y, z) → y.Hence we may restrict and corestrict ϕ to a map ϕ • : F 0 \ V ± t F 2 \ D ± t .It is however immediate to check that ϕ • is regular and a bijection, with inverse map given by ψ : (u−tΘt)(u−tΘt) (u−1)(u−t)+(q−1)v .Moreover, it is immediate to check that ϕ • maps the points γ ± i to the points β ± i for each i ∈ {0, 1, ∞}, and that it induces bijections To conclude, we use the same argument as in the proof of Proposition 4.1.Namely, in a neighborhood of the fibers V ± t and D ± t , ϕ is an elementary transformation blowing up the point γ ± t and contracting the strict transform of the fiber V ± t onto the point β ± t .Therefore, the induced map ϕ : Bl(F 0 ) γ ± t → Bl(F 2 ) β ± t is biregular.The result follows.
Remark 4.5.As shown in the above proof, the rational map ϕ : F 2 F 0 corresponding to the change of variable (64) composed with the change of variable (49) (which relates qP VI (Θ) to the modified q P VI (Θ)), respects the scheme of colors in the diagrams representing the particular lines in F 2 and F 0 .4.3.Confluence.In this section, we will see that the differential Okamoto space can be obtained from the second version of the q-difference one by a limit process.More precisely, we will show that they smoothly fit together into a family of Okamoto-spaces, parametrized by a neighborhood of q = 1 in C.
Remark 4.6.The behavior of the set S q when q goes to 1 may be wild.In order to fix this, analouglsy to [Sau00], we need to make q goes to 1 following a q-spiral.More precisely, let t ∈ C \ {0, 1} and fix |q 0 | > 1 with t / ∈ q R 0 .We have q ε 0 → 1, when ε > 0 is a real number going to 0 and for ε > 0 sufficiently close to 0, we find t / ∈ S q ε 0 .
5. Appendix: The relation between two notions of q-isomonodromy Some authors interpret the pseudo-constancy of the Birkhoff connection matrix as a suitable discrete analogue for the isomonodromy of families of Fuchsian systems.We will explain here how this is related to our notion of q-isomonodromy (see Section 2.2).This Birkhoff connection matrix is defined via certain fundamental solutions of the family of q-Fuchsian systems parameterized by t ∈ D. Therefore, we shall first recall from [Sau00] the construction of fundamental solutions (see also [Pra86,RSZ13,Dre14] for constructions in some more general settings).Note that these fundamental solutions will be meromorphic matrix functions on C * × D. In particular, they are uniform in t, which is one of the reasons why the definition of monodromy in the differential case should not be translated literally to the q-difference setting.
Let q be a complex number with |q| > 1.Let D be open connected subset of C * .Let (68) σ q,x Y (x, t) = A(x, t)Y (x, t), with A(x, t) = A 0 (t) + x A 1 (t) x − 1 + x A t (t) t(x − t) be a family of q-Fuchsian systems as in Definition 1.6 with non-resonant spectral data (Θ, Θ).Note that in particular, we assume that that for each i ∈ {0, 1, t}, we have A i ∈ M 2 (O(D)), i.e.
1 Strictly speaking, here one has to choose an order for the eight curves to be blown up in order to obtain a well-defined result in restriction to those q ∈ C * where the curves intersect.We will however neglect these values of q anyway afterwards, because they are not close to 1.