The holonomy group at infinity of the Painlevé VI Equation 2 1 0 2 Bassem Ben Hamed n Institut Supérieur des Sciences Appliquées et de Technologie de Gabès a J Département de Mathématiques 4 Rue Amor Ben El Khatab, 6029 Gabès, Tunisie 2 ] Lubomir Gavrilov A C Institut de Mathématiques de Toulouse, UMR 5219 h. Université de Toulouse, 31062 Toulouse, France t a m Martine Klughertz [ Institut de Mathématiques de Toulouse, UMR 5219 3 Université de Toulouse, 31062 Toulouse, France v 2 4 January 25, 2012 1 0 . 5 0 0 2000 MSC scheme numbers: 70H07, 34M55, 37J30 1 : v Abstract i X r We prove that the holonomy group at infinity of the Painlevé VI a equation is virtually commutative. 1 1 Introduction The sixth Painlevé equation (PVI) d2λ 1 (cid:18)1 1 1 (cid:19)(cid:18)dλ(cid:19)2 (cid:18)1 1 1 (cid:19) dλ = + + − + + dt2 2 λ λ−1 λ−t dt t t−1 λ−t dt (cid:20) (cid:21) λ(λ−1)(λ−t) t t−1 1 t(t−1) + α−β +γ +( −δ) (1) t2(t−1)2 λ2 (λ−1)2 2 (λ−t)2 is a family of differential equations parameterized by (α,β,γ,δ) ∈ C4. The purpose of the present paper is to show that the holonomy group of (PVI) at infinity is virtually commutative. The precise meaning is as follows. It is straightforward to check that (1) is equivalent to a non-autonomous Hamil- tonian system (the so called sixth Painlevé system)  dλ ∂H  = ,  dt ∂µ (2) dµ ∂H  = − dt ∂λ where 1 (cid:2) H := λ(λ−1)(λ−t)µ2 +{κ (λ−1)(λ−t) 0 t(t−1) +κ λ(λ−t)+(κ +1)λ(λ−1)}µ+κ(λ−t)] (3) 1 t and 1 1 1 1 1 α = κ2 , β = κ2, γ = κ2, δ = κ2,κ = (cid:2)(κ +κ +κ +1)2 −κ2 (cid:3). 2 ∞ 2 0 2 1 2 t 2 0 1 t ∞ (4) The phase space of the above system is {(λ,µ,t) ∈ C3 : t (cid:54)= 0,1} whichwepartiallycompactifytoM = P1×P1×{C\{0,1}}. Itisimmediately seen that the projective lines Γ = {µ = ∞,t = c} ⊂ M,c (cid:54)= 0,1 c are leaves of the one-dimensional foliation induced by (2) on M. On each leaf Γ the foliation has four singular points defined by λ = 0,1,c,∞. Let c 2 P (cid:54)= 0,1,c,∞ be a point on Γ and consider a germ of a cross-section (C2,0) c to Γ at P. The holonomy group G at infinity is then the image of the c holonomy representation π (Γ \{0,1,t,∞},P) → Diff(C2,0). (5) 1 c It is defined up to a conjugation by a diffeomorphism, depending on the germ of cross-section and the initial point P. Our main result is Theorem 1 The holonomy group at infinity of the sixth Painlevé equation is virtually commutative. Recall that a group G is said to be virtually commutative, provided that there is a normal commutative subgroup G0 ⊂ G, such that G/G0 is finite. The isomorphism class of the holonomy group G along the leaf Γ has in fact c a canonical meaning. As we shall see in section 2, the leaf Γ coincides with c the divisor D (c) in the Okamoto compactification[15] of the phase space of 0 PVI, see fig.1. In particular, the holonomy group along Γ is isomorphic c to the holonomy group along the Okamoto divisor D (c). The remaining 0 divisors shown on fig.1 are topological cylinders, the associated holonomy has therefore one generator and is commutative. The proof of Theorem 1 is based on Lemma 2 which claims that the local holonomies near the singular points of the leaf Γ are involutions, as well on c the algebraic Lemma 1. Lemma 2 and Lemma 1 suggest that Theorem 1 is related to the fact that the vertical divisor shown on fig.1 belongs to the Kodaira list of degenerate elliptic curves. Let E be the k-th order variational equation along Γ and G the as- k c k sociated differential Galois group. E defines a connection on the Riemann k sphere Γ = P\{0,1,c,∞} with four regular singular points at the punctures c {0,1,c,∞}. The monodromy group of E represents the k-th order jet of the k holonomy group along D (c). We describe these monodromy groups in the 0 simplest cases k = 1,2 in section 3. It follows, for instance, that the mon- odromy group of E is isomorphic to a semi direct product Z2 (cid:111) Z , while 1 2 G = C2(cid:111)Z . In particular G , as well G , are virtually commutative. This 1 2 1 2 isaparticularcaseofageneralfact. AccordingtoTheorem1themonodromy group of E is virtually commutative for all k. As its Zarisky closure is G , k k then we also have Theorem 2 For every k the differential Galois group G is virtually com- k mutative. 3 The present paper was motivated by the study of the Liouville non- integrability of the PVI system through the Ziglin-Morales-Ramis-Simo the- ory of non-integrability [14, 11, 10, 13]. This theory asserts that integrability in a Liouville sense along a particular solution Γ implies that the variational c equation E , as well all higher order variational equations E along this solu- 1 k tion, have virtually commutative differential Galois groups. Indeed, in such a way the "semi-local" non-integrability in a neighborhood of some particular solutions and parameter values of the PVI system has been recently proved by Horozov and Stoyanova [7, 16], see also Morales-Ruiz [12]. To prove the non-integrability for all parameters we need, however, an explicitly known particular solution which exists for all parameter values. The only such ap- propriate solution is the vertical divisor Γ = D (c), defined in Theorem 1. c 0 The result of Theorem 2 shows that, contrary to what we expected, one can not prove the absence of a first integral of the PVI equation, by making use of the Ziglin-Morales-Ramis-Simo theory. It is an open question, whether the PVI equation has a first integral, meromorphic along the divisor Γ "at c infinity". This question, but in a more general setting, has been raised in [11, section 7]. Non-integrability or transcendency of solutions is one of the central sub- jects in the study of the PVI equation. The fact that its general solution can not be reduced to a solution of a first order differential equation has been claimed already by Painlevé, and proved more recently by Watanabe [18] and others. A different approach to the transcendency, going back to Drach and Vessiot, is to interpret it as an irreducibility of the Galois groupoid defined by Malgrange, see [9, 3, 2, 4]. The irreducibility of the PVI equation in the sense of Drach-Vessiot-Malgrange has been shown by Cantat and Loray [1, Theorem 7.1]. It follows from these results that the PVI equation does not allow an additional rational first integral. The relation between the irre- ducibility of the Galois groupoid of a Hamiltonian system and the differential Galois group along a given algebraic solution is studied recently by Casale [5]. In this context, our Theorem 1 comes at a first sight as a surprise. The solution Γ which we use is however rather special : it is an irreducible com- c ponent of the anti-canonical divisor of the space of initial conditions, and hence it is invariant under the action of the Galois groupoid. This leads to special properties of the Galois groupoid along Γ too. c The paper is organized as follows. In section 2, we resume briefly the OkamotocompactificationofthephasespaceofPVIequation[15]. Insection 3 we describe the monodromy group of the first and the second variational 4 equation along Γ , in terms of complete elliptic integrals of first and second c kind. These groups provide an approximation of the holonomy group along Γ . Our main result, Theorem 1, is proved in section 4. c 2 The Okamoto compactification Let (E,π,B) be a complex-analytic fibration with base B, total space E and projection π : E → B. Consider a foliation F on E of dimension equal to the dimension of B. Following [15] we say that F is P-uniform, if for every leaf Γ ⊂ E the induced map π : Γ → B is an analytic covering. Thus, for every initial point e ∈ E, and every con- tinuous path γ ⊂ B starting at b = π(e), there is a unique continuous path γ˜ ⊂ E starting at e, which is a lift of γ with respect to π (the "Painlevé property" of the foliation). The analyticity of π implies moreover that at each point e ∈ E the leaf of the foliation is transversal to the corresponding fiber of the fibration. From now on we put E = {(λ,µ,t) ∈ C3 : t (cid:54)= 0,1},B = C\{0,1} π : E → B : (λ,µ,t) (cid:55)→ t being the natural projection. The system (2) defines a one-dimensional foli- ation F on the total space E which is not P-uniform, but can be completed ¯ to a P-uniform foliation after an appropriate partial compactification E of E. The main result of [15] may be formulated as follows. Theorem 3 Thereexistsacanonicalcompactcomplex-analyticfibration(E¯,π¯,B), such that • E ⊂ E¯, π¯| = π E • Each fiber E¯ = π¯−1(t) is compact t • E¯ \E is a union of nine transversal projective lines, as it is shown on t t fig.1. The intersection points of the lines depend analytically on t. 5 ¯ Figure 1: The divisor E \E t t • Let D be the union of five solid lines shown on fig.1.The foliation t induced by (2) on E˜ = E¯ \ ∪ D is P-uniform with respect to the t∈B t induced projection. A similar result holds true for the remaining Painlevé equations [15]. ¯ Remark. E \D is the so called "space of initial conditions" of the Painlevé t t VI equation which we describe next. Sketch of the proof of Theorem 3. Following [15, Okamoto], define first the Hirzebruch surface Σ(2), ε ∈ C, using four charts W = C2, with local ((cid:15)) i coordinates (λ ,µ ), i = 1,...,4, where i i  1  λ = λ , µ = in W ∩W ,  2 1 2 µ1 1 2  1 λ = , µ = (cid:15)λ −λ2µ in W ∩W , (6) 3 λ 3 1 1 1 1 3  1  1  λ = λ , µ = in W ∩W .  4 3 4 3 4 µ 3 6 If (cid:15) (cid:54)= 0, then the Hirzebruch surface Σ(2) is isomorphic to P1×P1; otherwise ((cid:15)) it is isomorphic to the tangent projective bundle of P1 with projection Σ(2) → P1 ((cid:15)) (λ ,µ ) (cid:55)→ λ . i i i The vector field (2) extends on the total space of the trivial bundle Σ(2) ×B →π B,B = P1 \{0,1,∞} (7) ((cid:15)) where (cid:15) = −(κ +κ +κ +κ +1). For instance, in the chart W it takes 0 1 t ∞ 2 the form  1  µ λ(cid:48) = [2E(t,λ )+F(t,λ )µ ],  2 2 t(t−1) 2 2 2 (8) 1  µ(cid:48) = [E (t,λ )+F (t,λ )µ +Gµ2],  2 t(t−1) λ 2 λ 2 2 2 where  E(t,λ) = λ(λ−1)(λ−t),    F(t,λ) = κ (λ−1)(λ−t)+κ λ(λ−t)+(κ +1)λ(λ−1),  0 1 t  ∂E ∂F (9) E = , F = , λ λ  ∂λ ∂λ   1   G = − (cid:15)(κ +κ +κ −κ +1) = κ,  0 1 t ∞ 2 The above meromorphic vector field induces a singular foliation on Σ(2) hav- ((cid:15)) ing four one-parameter families of singular points Sθ, θ = 0,1,∞,t defined by Sθ ∩π−1(t) = aθ(t), a0(t) = {(λ ,µ ) = (0,0)}, 2 2 a1(t) = {(λ ,µ ) = (1,0) or (λ ,µ ) = (1,0)}, 2 2 4 4 (cid:26) (cid:27) 1 at(t) = (λ ,µ ) = (t,0) or (λ ,µ ) = ( ,0) , 2 2 4 4 t a∞(t) = {(λ ,µ ) = (0,0)}. 4 4 7 ReplacetheHirzebruchsurfaceΣ(2) byΣ(2) blownupataθ(t)foreveryt ∈ B. ((cid:15)) ((cid:15)) This replaces each aθ(t) by a projective line denoted Dθ(t). The induced fo- 1 liation has still four one-parameter families of singular points which belong to Dθ(t). We blow up once again the surfaces at these singular points to 1 ¯ obtain the fibers E of the fibration described in Theorem 3, see fig.1. The t remaining claims of the Theorem follow by computation.(cid:3) 3 Higher order variational equations and their monodromy groups In this section, we consider the foliation F defined by the vector field (2) on ¯ the total space of the fibration (E,π¯,B), see Theorem 3. This foliation has in each fiber π−1(t) a vertical leaf D (t), which in the chart W takes the 0 2 form D (t) : µ = 0. 0 2 According to (8) the foliation F in the local chart W is defined by 2  [E (t,λ)+F (t,λ)µ+Gµ2]µ  dµ = λ λ dλ  2E(t,λ)+F(t,λ)µ (10) t(t−1)µ  dt = dλ  2E(t,λ)+F(t,λ)µ where E, F and G are given by (9). Here, as well until the end of the paper, we replace for simplicity µ ,λ by µ,λ. 2 2 In this section we compute the first and the second variational equations of (10) along D (c) and study the corresponding monodromy groups. For 0 this purpose we put, following [7, 11], ε2 ε2 t = c+εη + η +..., µ = εξ + ξ +..., ε ∼ 0 1 2 1 2 2 2 where η = η (λ),ξ = ξ (λ) are unknown functions, and substitute these k k k k expressions in (10). Equating the coefficients of εk we get a recursive system of linear non-homogeneus equations on (η ,ξ ) - the higher order variational k k equations. We note that these equations, except in the case k = 1, are non- linear. In order to obtain a linear system we add suitable monomials in η ,ξ , i j 8 e.g. [11, 14]. The fundamental matrices of solutions of these equations are then explicitly computed by the Picard method in terms of iterated integrals. This implies also a description of the corresponding monodromy matrices. In the next two subsections we carry out this procedure in the particular case of the first and the second variational equation. 3.1 The first variational equation The first variational equation E along D is the linear system 1 0 (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) η˙ 0 b(λ) η 1 = 1 , (11) ξ˙ 0 a(λ) ξ 1 1 where E (c,λ) λ a(λ) = , 2E(c,λ) c(c−1) b(λ) = . 2E(c,λ) The general solution of the system (11) is given by (cid:90) λ c(c−1)dλ η (λ) = c +c , 1 1 (cid:112) 2 2 λ(λ−1)(λ−c) p (cid:112) ξ (λ) = c λ(λ−1)(λ−c). 1 1 where (c ,c ) ∈ C2 and p ∈ C is a fixed initial point. The fundamental 1 2 matrix of solutions  c(c−1)dλ  (cid:82)λ 1  p 2(cid:112)λ(λ−1)(λ−c)  X(λ) =     (cid:112) λ(λ−1)(λ−c) 0 is multivalued, and the result of the analytic continuation of X(.) along small loops making one turn around λ = 0,1,c respectively is X → XT ,X → XT ,X → XT . 0 1 c 9 The matrices T ,T ,T generate the monodromy group of (11) and can be 0 1 c computed as follows. Let S be the compact elliptic Riemann surface of the c (cid:112) algebraic function λ(λ−1)(λ−c). It has an affine equation {(λ,y) : y2 = λ(λ−1)(λ−c)}. (12) The one-form dλ (cid:112) λ(λ−1)(λ−c) is holomorphic on S and hence X(.) can be seen as a globally multivalued, c but locally meromorphic matrix function on S . This implies that c (cid:18) (cid:19) 1 0 T2 = T2 = T2 = 0 1 c 0 1 and hence (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) −1 0 −1 0 −1 0 T = ,T = ,T = . (13) 0 α 1 1 α 1 c α 1 0 1 c The constants α ,α ,α depend on the initial point p and can be determined 0 1 c as follows. The matrix (cid:18) (cid:19) 1 0 T T = 0 1 α −α 1 1 0 represents the monodromy of X(.) along a closed loop on the λ-plane, which (cid:112) lifts, on on the Riemann surface of λ(λ−1)(λ−c) to a closed loop too, which we denote γ. The monodromy of the fundamental matrix X along this loop is T T and we have 0 1 (cid:18) (cid:19) Π 0 X → XT T = X + 0 1 0 0 where (cid:90) c(c−1)dλ (cid:90) 1 c(c−1)dλ Π = = (cid:112) (cid:112) 2 λ(λ−1)(λ−c) λ(λ−1)(λ−c) γ 0 is a period of the holomorphic one-form on S . Therefore c (cid:90) 1 c(c−1)dλ α −α = 1 0 (cid:112) λ(λ−1)(λ−c) 0 10