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Finite branch solutions to Painleve VI around a fixed singular point

Every finite branch solutions to the sixth Painleve equation around a fixed singular point is an algebraic branch solution. In particular a global solution is an algebraic solution if and only if it is finitely many-valued globally. The proof of this result relies on algebraic geometry of Painleve VI, Riemann-Hilbert correspondence, geometry and dynamics on cubic surfaces, resolutions of Kleinian singularities, and power geometry of algebraic differential equations. In the course of the proof we are also able to classify all finite branch solutions up to Backlund transformations.


Introduction
We are interested in a finite branch local solution to the sixth Painlevé equation around a fixed singular point. We show that every such solution is in fact an algebraic branch solution (see Definition 1.1 for the terminology). In particular a global solution is an algebraic solution if and only if it is finitely many-valued globally. Although the problem under study is local in nature, our solution to it relies on an effective combination of some global technologies and some local tools. The former includes the algebraic geometry of the sixth Painlevé equation, Riemann-Hilbert correspondence, geometry and dynamics on cubic surfaces, Kleinian singularities and their minimal resolutions [15,16,17,18,20], while the latter includes the power geometry of algebraic differential equation [5,6,7], which is a method of constructing formal solutions by means of Newton polygons, and the theory of nonlinear differential equations of "regular singular type" [10,11], which discusses the convergence of formal solutions.
It is well known that equation (1) has the analytic Painlevé property, that is, any meromorphic solution germ at a base point z ∈ Z can be continued meromorphically along any path in Z emanating from z. Thus a solution can branch only around a fixed singular point. We are interested in finite branch solutions around it, by which we mean the following. Definition 1.1 A finite branch solution to equation (1), say, around z = 0 is a local solution (q(z), p(z)) on a punctured disk D × = D − {0} centered at z = 0 such that its lift (q(z),p(z)) along some finite branched covering ϕ : (D,0) → (D, 0),z → z =z n around z = 0 is a singlevalued meromorphic function onD × =D − {0}. Such a solution is said to be an algebraic branch solution if it can be represented by a convergent Puiseux-Laurent expansion with a i = b i = 0 for all sufficiently small i ≪ 0, namely, if the lift (q(z),p(z)) is a single-valued meromorphic function onD with at most pole at the originz =0.

Problem 1.2 Is any finite branch solution to P VI (κ) an algebraic branch solution ?
In this article we settle this problem in the affirmative as is stated in the following. It is an interesting problem to consider algebraic solutions to Painlevé VI. Many algebraic solutions have been constructed in [1,2,3,8,13,14,24,25], but a complete classification seems to be outstanding. We hope that Theorem 1.3 will play an important part in discussing this issue. The following remark explains what Theorem 1.3 signifies and why it is remarkable. Remark 1.4 Logically, according to Definition 1.1, a finite branch solution (q(z), p(z)) around z = 0 may have a very transcendental singularity at z = 0, to the effect that its lift (q(z),p(z)) may have infinitely many poles inD × accumulating to the originz =0, or even if such an accumulation phenomenon does not occur, it may have an essential singularity atz =0. Rather surprisingly, however, Theorem 1.3 excludes the possibility for a finite branch solution to admit such transcendental phenomena. This result becomes more intriguing if we recall that wild behaviors of a generic solution to Painlevé VI have been observed in [9,12,22,31,32] and examples of solutions with infintely many poles accumulating toz = 0 are given in [12,31]; such a distribution of poles may be expected for a generic solution, though it is not rigorously verified yet to the author's knowledge. Thus we can think that a finite branch solution is quite distinguished from generic solutions, necessarily being an algebraic branch solution. The main idea for the proof of Theorem 1.3 is presented in Figure 1. We have natural inclusions i and j in the top line of Figure 1 and we wish to show that the injection i is in fact a surjection. Our strategy consists of the "upper bound part" and the "lower bound part".
(1) Upper bound part: In this part we investigate the inclusion j : II ֒→ III in Figure 1, considering how the locus of finite branch solutions is included in the moduli space of all solutions. In other words, we make a confinement of the locus II in the entire space III. What we shall really do is not an upper bound estimation of this locus but rather a pinpoint identification of it. This is the main part of the article and we use the algebraic geometry of Painlevé VI, Riemann-Hilbert correspondence, geometry and dynamics on cubic surfaces, and minimal resolutions of Kleinian singularities [15,16,17,18,19,20].
(2) Lower bound part: In this part we fill in the diagram of Figure 1 by adding the bottom line to the top one. We try to construct as many algebraic branch solutions as possible in order to make the set I ′ as large as possible. The construction is based on the power geometry technique developed in [5,6,7] and the convergence arguments in [10,11]. We are done if the set I ′ is large enough to show that the injection i ′ : I ′ ֒→ II is in fact a surjection. This does not mean that we verify the equality I ′ = II directly. (If such a direct approach were feasible, then our problem would not be difficult from the beginning!) Instead, we prove it very indirectly based on the following idea.
(3) Key trick: Suppose that a component A of I ′ injects into a component B of II. If the cardinalities of A and B are finite and the same, then the injection i ′ : A ֒→ B is in fact a surjection. If A and B are biholomorphic to C and the injection i ′ : A ֒→ B is holomorphic, then it must be a surjection because any holomorphic injection C ֒→ C is a surjection (use Casorati-Weierstrass or Picard's little theorem). The same argument holds true if C is replaced by C × , since any holomorphic injection C × ֒→ C × is a surjection (lift it to the universal covering C ֒→ C). These tricks enable us to identify the component A ⊂ I ′ with the component B ⊂ II. We show that each component involved is either of the three types mentioned above. Then we make this kind of argument componentwise to get an identification I ′ = II, which leads to the desired coincidence I ′ = I = II.
In view of the way in which Theorem 1.3 is established, the power geometry technique provides us with an efficient method of identifying all finite branch solutions (up to Bäcklund transformations), which have now turned out to be algebraic branch solutions, by determining the leading terms of their Puiseux-Laurent expansions.
In some sense this article is a counterpart of the previous paper [20] where an ergodic study of Painlevé VI is developed (see also the survey [21]). Put z 1 = 0, z 2 = 1, z 3 = ∞. {i, j, k} = {1, 2, 3}, let γ i be a loop in Z surrounding z i once anti-clockwise and leaving z j and z k outside as in Figure 2. Then the fundamental group π 1 (Z, z) is represented as A loop γ ∈ π 1 (Z, z) is said to be elementary if it is conjugate to γ m i for some i ∈ {1, 2, 3} and m ∈ Z; otherwise, it is said to be non-elementary. The main theme of [20] is the dynamics of the nonlinear monodromy of P VI (κ) along a given loop γ. It is shown there that, along every non-elementary loop, the nonlinear monodromy is chaotic and the number of its periodic points grows exponentially as the period tends to infinity. On the other hand, it is Liouville integrable along an elementary loop, in the sense that it preserves a Lagrangian fibration. Now we notice that from the dynamical point of view the main problem of this article is nothing other than discussing the periodic points of the nonlinear monodromy along the basic loop γ i , which is of course an elementary loop. In view of its integrable character, one may doubt if there is something very deep with this issue. As Theorem 1.3 and Remark 1.4 show, however, this issue is actually quite interesting from the function-theoretical point of view.
The plan of this article is as follows. In §2 the phase space of Painlevé VI is introduced as a moduli space of stable parabolic connections. In §3 the Riemann-Hilbert correspondence from the moduli space to an affine cubic surface is formulated and its character as an analytic minimal resolution of Kleininan singularities is stated. In §4 the dynamical system on the cubic surface representing the nonlinear monodromy of Painlevé VI is formulated and some preliminary properties of it are given. In §5 we briefly review Bäcklund transformations and their relation to the Riemann-Hilbert correspondence. In §6 fixed points and periodic points of the dynamical system are discussed. A stratification of the parameter space K is also introduced in order to describe the singularities of the cubic surfaces. In §7 a case-by-case study of fixed points and periodic points is made according to the stratification, thereby a pinpoint identification of finite branch solutions is made on each stratum. In §8 power geometry of algebraic differential equations is applied to Painlevé VI in order to construct as many algebraic branch solutions as possible. In §9 we consider the inclusion of those solutions constructed in §8 into the moduli space of all finite branch solutions. After some preliminaries on Riccati solutions, we show that this inclusion is in fact a surjection, thereby complete the proof of Theorem 1.3. singularities

Phase Space
Equation (1) is only a fragmentary appearance of a more intrinsic object constructed algebrogeometrically [16,17,18]. We review this construction following the expositions of [20,21]. The sixth Painlevé dynamical system P VI (κ) is formulated as a holomorphic, uniform, transversal foliation on a fibration of certain smooth quasi-projective rational surfaces whose fiber M z (κ) := π −1 κ (z) over z ∈ Z, called the space of initial conditions at time z, is realized as a moduli space of stable parabolic connections. The total space M(κ) is called the phase space of P VI (κ). In this formulation, the uniformity of the Painlevé foliation, in other words, the geometric Painlevé property of it is a natural consequence of a solution to the Riemann-Hilbert problem (see Theorem 3.5), especially of the properness of the Riemann-Hilbert correspondence [16]. Then equation (1) is just a coordinate expression of the foliation on an affine open subset of M(κ) and the analytic Painlevé property for equation (1) is an immediate consequence of the geometric Painlevé property for the foliation and the algebraicity of the phase space M(κ). Moreover there exists a natural compactification M z (κ) ֒→ M z (κ) of the moduli space M z (κ) into a moduli space M z (κ) of stable parabolic phi-connections.
Here we include a very sketchy explanation of the terminology used in the last paragraph. A stable parabolic connection is a Fuchsian connection equipped with a parabolic structure on a (rank 2) vector bundle over P 1 having a Riemann scheme as in Table 1, where the parabolic structure corresponds to the first exponents, which satisfies a sort of stability condition in geometric invariant theory. Here the parameter κ i stands for the difference of the second exponent from the first one at the regular singular point t i . On the other hand, a stable parabolic phi-connection is a variant of stable parabolic connection allowing a "matrix-valued Planck constant" called a phi-operator φ such that the generalized Leibniz rule is satisfied, where the key point here is that the field φ may be degenerate or simi-classical. Then the moduli space M z (κ) can be compactified by adding some semi-classical objects, that is, some stable parabolic phi-connections with degenerate phi-operator φ. There is the following characterization of our moduli spaces (see Figure 3). (1) The compactified moduli space M z (κ) is isomorphic to an 8-point blow-up of the Hirzebruch surface Σ 2 → P 1 of degree 2. Figure 3: Nonlinear monodromy γ * : M z (κ) along a loop γ ∈ π 1 (Z, z) (2) M z (κ) has a unique effective anti-canonical divisor Y z (κ), which is given by where E 0 is the strict transform of the section at infinity and E i (i = 1, 2, 3, 4) is the strict transform of the fiber over the point t i ∈ P 1 of the Hirzebruch surface Σ 2 → P 1 .
(3) The support of the divisor Y z (κ) is exactly the locus where the phi-operator φ is degenerate, with the coefficients of formula (5) being the ranks of degeneracy of φ. In particular, This theorem implies that M z (κ) is a moduli-theoretical realization of the space of initial conditions for P VI (κ) constructed "by hands" in [26], M z (κ) is a generalized Halphen surface of type D 4 in [30] and (M z (κ), Y z (κ)) is an Okamoto-Painlevé pair of type D 4 in [28]. Since the Painlevé foliation has the geometric Painlevé property [16], each loop γ ∈ π 1 (Z, z) admits global horizontal lifts along the foliation and induces an automorphism called the nonlinear monodromy along the loop γ (see Figure 3). Note that a fixed point or a periodic point of the map γ * : M z (κ) can be identified with a solution germ at z which is single-valued or finitely many-valued along the loop γ, respectively.

Riemann-Hilbert Correspondence
Generally speaking, a Riemann-Hilbert correspondence is the map from a moduli space of flat connections to a moduli space of monodromy representations, sending a connection to its Figure 4: Four loops in P 1 − {0, z, 1, ∞} monodromy. In our situation an appropriate Riemann-Hilbert correspondence is formulated in [16,17,18]. For each a = (a 1 , a 2 , a 3 , a 4 ) ∈ A := C 4 , let R z (a) denote the moduli space of Jordan equivalence classes of linear monodromy representations with the prescribed local monodromy data Tr ρ(C i ) = a i (i = 1, 2, 3, 4), where C i is a loop as in Figure 4. Any stable parabolic connection Q ∈ M z (κ), restricted to P 1 − {0, z, 1, ∞}, induces a flat connection and determines the Jordan equivalence class ρ ∈ R z (a) of its monodromy representations, where the correspondence of parameters κ → a is described as follows. If The Riemann scheme in Table 1 then implies that the monodromy matrix ρ(C i ) has an eigenvalue b i for each i = 1, 2, 3, 4. Since ρ(C i ) ∈ SL 2 (C), its trace a i = Tr ρ(C i ) is given by Given any θ = (θ 1 , θ 2 , θ 3 , θ 4 ) ∈ Θ := C 4 θ , consider the affine cubic surface Then there exists an isomorphism of affine algebraic surfaces for {i, j, k} = {1, 2, 3}, where the correspondence of parameters a → θ is given by a 1 a 2 a 3 a 4 + a 2 1 + a 2 2 + a 2 3 + a 2 4 − 4 (i = 4). The composition of the sequence κ → b → a → θ of the three maps (8), (9) and (10) is referred to as the Riemann-Hilbert correspondence in the parameter level [16] and is denoted by Then the Riemann-Hilbert correspondence (7) is reformulated as a holomorphic map The map (11) admits a remarkable affine Weyl group structure [16,19], from which the Bäcklund transformations of Painlevé VI emerge [15]. In view of formula (2) the affine space K can be identified with the linear space C 4 by the forgetful isomorphism K → C 4 , κ = (κ 0 , κ 1 , κ 2 , κ 3 , κ 4 ) → (κ 1 , κ 2 , κ 3 , κ 4 ), where the latter space C 4 is equipped with the standard (complex) Euclidean inner product. For each i ∈ {0, 1, 2, 3, 4}, let w i : K → K, κ → κ ′ , be the orthogonal reflection in the hyperplane { κ ∈ K : κ i = 0}, which is explicitly represented as where C = (c ij ) is the Cartan matrix of type D (1) 4 given in Figure 5. Then the group generated by w 0 , w 1 , w 2 , w 3 , w 4 is an affine Weyl group of type D where the signs ± may be chosen arbitrarily. Let Wall be the union of all these hyperplanes. Then the affine Weyl group structure on (11) is stated as follows [16] (see Figure 6).
where we put The Riemann-Hilbert correspondence in the parameter level (11) is a branched W (D The singularity structure of the cubic surfaces S(θ) can be described in terms of the stratification of K by proper Dynkin subdiagrams, which we now define.  Let K I be the set obtained from K I by removing the sets K J with #J = #I + 1. Then it turns out that we have either K I = K I ′ or K I ∩ K I ′ = ∅ for any distinct subsets I, I ′ ∈ I (see Remark 3.3). So we can think of the stratification of K by the subsets K I (I ∈ I), called the W (D  Table 2.
There is a mistake in the definition of K I in [16,Definition 9.3] and [21], which is now corrected in Definition 3.2. (As for [16], correction may be possible before it is published.)   (2) There are six W (D 4 )-strata of abstract type A ⊕2 1 or A 3 .
With this stratification, we have a very neat solution to the Riemann-Hilbert problem. (1) if κ ∈ K I then S(θ) has Kleinian singularities of Dynkin type D I , (2) the Riemann-Hilbert correspondence (12) is a proper surjective map that is an analytic minimal resolution of Kleinian singularities.
If κ ∈ K − Wall then the surface S(θ) is smooth and RH z,κ is a biholomorphism, while if κ ∈ Wall, it is not a biholomorphism but only gives a resolution of singularities (proper and surjective, but not injective). For example, see Figure 8 for the case κ = (0, 0, 0, 0, 1) where a singularity of type D 4 occurs. In the latter case, however, if we take a standard algebraic minimal resolution of Kleinian singularities as constructed by Brieskorn [4] and others, The lifted Riemann-Hilbert correspondence RH z,κ is a biholomorphism and hence gives a strict conjugacy between the nonlinear monodromy (6) of P VI (κ) and a certain automorphism This latter map will be described explicitly in Section 4 (see Theorem 4.1). The singularity structure of the affine cubic surface S(θ) is closely related to the Riccati solutions to P VI (κ) [16], where a Riccati solution is a particular solution that arises from the Riccati equation associated to a Gauss hypergeometric equation. Let E z (κ) ⊂ M z (κ) be the exceptional set of the resolution of singularities by the Riemann-Hilbert correspondence (12). Similarly, let E(θ) ⊂ S(θ) be the exceptional set of the algebraic resolution of singularities (15). All Riccati solution germs at time z ∈ Z are parametrized by the exceptional set E z (κ) ⊂ M z (κ), which precisely corresponds to the exceptional set E(θ) ⊂ S(θ) through the lifted Riemann-Hilbert correspondence (16).
Fot this reason we may refer to E z (κ) and M • z (κ) := M z (κ) − E z (κ) as the Riccati locus and non-Riccati locus of M z (κ) respectively. They are invariant under the action of the nonlinear monodromy (6). Corresponding to them, let Sing(θ) and S • (θ) := S(θ)−Sing(θ) be the singular locus and the smooth locus of the cubic surface S(θ) respectively. (1) By Theorem 3.5 the Riemann-Hilbert correspondence (12) restricts to a biholomorphism between the non-Riccati locus of M z (κ) and the smooth locus of S(θ), while it collapses the Riccati locus E z (κ) to the singular locus Sing(θ). In order to resolve this degeneracy and obtain an isomorphism, we had to take the lifted Riemann-Hibert correspondence (16), which induces an isomorphism between the exceptional sets E z (κ) and E(θ).
(2) For the Riccati solutions the main problem of this article is trivial; if a Riccati solution is a finite branch solution around a fixed singular point, then it is an algebraic branch solution, because the Riccati solution is (essentially) the logarithmic derivative of a Gauss hypergeometric function. Thus we may restrict our attention to the non-Riccati locus.

Dynamics on Cubic Surface
We shall describe the strict conjugacy (17) of the nonlinear monodromy (6). For a cyclic permutation (i, j, k) of (1, 2, 3) we define an isomorphism g i : Through the resolution of singularities (15), the map g i is uniquely lifted to an isomorphism We remark that the square g 2 i is an automorphism of S(θ) withg 2 i being its lift to S(θ).
Let Fix j (θ) be the set of all fixed points of the transformationg 2 j : S(θ) . Moreover, for any integer n > 1, let Per j (θ; n) be the set of all periodic points of prime period n of the transformationg 2 j : S(θ) . Theorem 4.1 then implies that all single-valued solution germs and all n-branch solution germs to P VI (κ) around the fixed singular point z j are parametrized by the sets Fix j (θ) and Per j (θ; n) respectively. By Remark 3.7, considering Fix j (θ) and Per j (θ; n) upstairs is the same thing as considering Fix j (θ) and Per j (θ; n) downstairs, except for the exceptional locus upstairs and the singular locus downstairs. Here Fix j (θ) and Per j (θ; n) denote the set of all fixed points and the set of all periodic points of prime period n of the transformation g 2 j : S(θ) downstairs. In order to make the situation more transparent, we begin by investigating simultaneous fixed points of g 2 1 , g 2 2 , g 2 3 downstairs.
is the set of all simultaneous fixed points of g 2 1 , g 2 2 , g 2 3 : S(θ) , then Proof. A point x ∈ S(θ) is a singular point of the surface S(θ) if and only if its gradient vector field y(x, θ) = (y 1 (x, θ), y 2 (x, θ), y 3 (x, θ)) vanishes at the point x, where On the other hand, an inspection of formula (19) readily shows that x ∈ S(θ) is a simultaneous fixed point of g 2 1 , g 2 2 , g 2 3 if and only if x is a common root of equations Then the equality (20) immediately follows from these observations. 2 As is announced in [16], this theorem yields a characterization of the rational solutions. Proof. If a single-valued solution Q ∈ M z (κ) belongs to the non-Riccati locus Q ∈ M • z (κ), then the Riemann-Hilbert correspondence (18) sends Q to a smooth point x ∈ S • (θ). Since the single-valued solution Q is a simultaneous fixed point of the nonlinear monodromies γ 1 * , γ 2 * , γ 3 * , the corresponding point x must lie in Fix(θ). Then Theorem 4.2 implies that x ∈ Sing(θ), which contradicts the fact that x ∈ S • (θ). Hence any single-valued solution is a Riccati solution.
Since any Riccati solution is (essentially) the logarithmic derivative of a Gauss hypergeometric function, any single-valued Riccati solution must be a rational solution. 2 All the rational solutions to Painlevé VI are classified in [25]. We come back to our discussion downstairs and give a simple characterization of the sets Fix j (θ) and Per j (θ; n).
be any point and let n be any integer > 1.
(2) x ∈ Per j (θ; n) if and only if there exists an integer 0 < m < n coprime to n such that Proof. We put (x ′ , θ ′ ) = g j (x, θ) and y ′ = y(x ′ , θ ′ ). Then formula (19) yields For each integer n ∈ Z we write (x (n) , θ (n) ) = g n i (x, θ) and y (n) = y(x (n) , θ (n) ). From formulas (19) and (25), we can easily obtain three recurrence relations The characteristic equation of the recurrence relation (26) is the quadratic equation the roots of which are denoted by α and β = α −1 . Since αβ = 1, we may and shall assume that |α| ≥ 1 ≥ |β| > 0 in the sequel. The discussion is divided into two cases. Case x i ∈ C − {±2}: In this case, the roots α and β are distinct and different from ±1 and the recurrence relation (26) is settled as Then it follows from (27) and (28) that the sequences x where the constants p, q, r 1 and r 2 are given by , (23). Indeed, the condition p = q = 0 is equivalent to αy j + y k = βy j + y k = 0, which is equivalent to the condition y j = y k = 0, because the roots α and β are distinct. Now we assume that x is a root of equations (23). Then (30) implies that the sequence x (n) is periodic of period two, that is, x is a fixed point of g 2 j . Next we assume that x is not a root of equations (23). If x is a periodic point of g 2 j of prime period n ≥ 1, then (30) yields Here it cannot happen that p − qβ 2n = pα − qβ 2n+1 = 0. Indeed, otherwise, we have p = qβ 2n and q(1 − β 2 ) = 0. Since at least one of p and q is nonzero, we have β ∈ {±1} and hence x i ∈ {±2}, which contradicts the assumption that x i ∈ {±2}. Therefore, α 2n = 1, that is, α is a primitive 2n-th root of unity. Note that n ≥ 2 since α ∈ {±1}. Thus there is an integer 0 < m < n comprime to n such that α = exp(πim/n) and so x i = α + α −1 = 2 cos(πm/n), which leads to condition (24). Conversely, if condition (24) is satisfied, then it is easy to see that x is a periodic point of g 2 j of prime period n. Case x i ∈ {±2}: In this case we have x i = −2ε for some sign ε ∈ {±1} and hence equation (29) has a double root α = β = ε. Then the recurrence equation (26) is settled as y j . This is the case if and only if y j + εy k = 0. Conversely, if this condition is satisfied, then we have y Hence the sequence x (2n) is periodic if and only if y j = y k = 0, namely, if and only if x is a root of (23). In this case x is a fixed point of g 2 j . 2 In order to give the relation between the fixed points upstairs and those downstairs, we put For the periodic points of prime period n > 1, we define Per • j (θ; n), Per • j (θ; n) and Per e j (θ; n) in a similar manner. Then there exist direct sum decompositions where the exceptional components Fix For any n > 1 we have Per(θ; n) ∩Sing(θ) = ∅, that is, Per • j (θ; n) = Per j (θ; n), and the minimal resolution (15) induces an isomorphism Proof. The isomorphism (32) is trivial from the definition. The assertion Per(θ; n)∩Sing(θ) = ∅ follows from (20). Then the isomorphism (33) is again trivial from the definition. 2 The fixed point set and the periodic point set, upstairs or downstairs, will be investigated more closely in §6. For this purpose it is convenient to consider the symmetric group S 4 of degree 4 acting on K by permuting the entries κ 1 , κ 2 , κ 3 , κ 4 of κ ∈ K and fixing κ 0 . Through the Riemann-Hilbert correspondence in the parameter level, rh : K → Θ, the action S 4 K induces an action of S 3 ⋉ Kl on Θ, where Kl is Klein's 4-group realized as the group of even triple signs, Kl = {ε = (ε 1 , ε 2 , ε 3 ) ∈ {±1} 3 : ε 1 ε 2 ε 3 = 1}, acting on Θ by the sign changes (θ 1 , θ 2 , θ 3 , θ 4 ) → (ε 1 θ 1 , ε 2 θ 2 , ε 3 θ 3 , θ 4 ), while S 3 acts on Θ by permuting the entries θ 1 , θ 2 , θ 3 of θ ∈ Θ and fixing θ 4 . This construction defines an isomorphism of groups with respect to which the map rh : K → Θ becomes S 4 -equivariant. Viewed as a subgroup of Then the polynomial f (x, θ) is σ-invariant and hence σ induces an isomorphism of algebraic surfaces, σ : S(θ) → S(σ(θ)). As for the action g 2 j : S(θ) , we have the commutative diagram for any element σ ∈ S 4 with τ ∈ S 3 determined by (34). It induces isomorphisms which, via the minimal resolution (15), lift up to isomorphisms The action of the symmetric group S 4 on K mentioned above is just induced from its action on the index set {0, 1, 2, 3, 4} fixing the element 0, namely, from the realization of S 4 as the automorphism group of the Dynkin diagram D    Table 2, there is a unique W (F (1) 4 )-stratum of type * . As for the W (D (1) 4 )-strata, we have the following classification (see also Figure 9).
4 )-stratum of abstract type * and this unique stratum is denoted by the same symbol * . Figure 10: Adjacency relations among W (D If something about the transformation g 2 j is discussed for a fixed index j, the relevant stratification is the W (D

Bäcklund Transformations
In this section we briefly discuss Bäcklund transformations, especially the characterization of them in terms of Riemann-Hilbert correspondence [15,16]. This topic is included here in order to confirm that our problem may be treated modulo Bäcklund transformations. For each σ ∈ S 4 we define the isomorphism of affine cubic surfaces Since the Riemann-Hilbert correspondence (12) is an analytic minimal resolution of singularities, for each w ∈ W (F (1) 4 ), there exists an analytic isomorphism w : M z (κ) → M z (w(κ)) such that the diagram is commutative, for any fixed κ ∈ K with θ = rh(κ) ∈ Θ. The commutative diagram (36) characterizes the Bäcklund transformations of Painlevé VI. Namely the map w : M z (κ) → M z (w(κ)) turns out to be algebraic and there are suitable affine coordinates on M z (κ) and M z (w(κ)) in terms of which the map w can be represented by the usual formula for Bäcklund transformations known as birational canonical transforamtions [27] (see [15,16] for the precise statement). In other words the Riemann-Hilbert correspondence is equivariant under the Bäcklund transformations and so is our main problem.
4 ) or more strictly the S 3 -factor of S 4 = S 3 ⋉ Kl permutes the three fixed singular points 0, 1 and ∞, while they are fixed by W (D 4 ). Hence we may consider our problem only around the origin z = 0 and, upon restricting our attention to z = 0, we may discuss it modulo the Bäcklund action of W (D (1) 4 ).

Fixed Points and Periodic Points
We shall more closely investigate the fixed point set Fix j (θ), or rather its subset Fix • j (θ) ∼ = Fix • j (θ) of smooth fixed points, by solving the system of equations (23). In view of (21) the last two equations in (23) are expressed as a linear system for the unknowns (x j , x k ), If its determinant 4 − x 2 i is nonzero, then system (37) is uniquely settled as Substituting (38) into equation f (x, θ) = 0 yields a quartic equation for the unknown x i , Conversely, if x i is a root of equation (39) with nonzero x 2 i − 4, then subsituting this into formula (38) yields a root of system (23). The four roots of quartic equation (39) are given by We pick up the root If this is the case, then substituting gives a root of system (23) with nonzero x 2 i − 4 provided that b 2 i b 2 4 = 1. If x is at this root, then y i (x, θ) admits the following nice factorization label fixed point existence smoothness condition is a simple root of equation (39). In terms of κ ∈ K, the existence and smoothness conditions for P (b i , b 4 ; b j , b k ) are given by κ i + κ 4 ∈ Z and κ i ∈ Z, κ 4 ∈ Z, κ i + κ 4 ± κ j ± κ k ∈ 2Z + 1, respectively.
Lemma 6.1 The smooth fixed points x ∈ Fix • j (θ) with nonzero x 2 i − 4 are precisely those points in Table 3 which satisfy the existence and smoothness conditions mentioned there. Table 3 is closely related to the configuration of lines on the affine cubic surface S(θ) or on its compactification S(θ) by the standard embedding

The fixed points in
where the projective cubic surface S(θ) is defined by the homogeneous equation It is obtained from the affine surface S(θ) by adding three lines at infinity , whose union L = L 1 ∪ L 2 ∪ L 3 is called the tritangent lines at infinity. It is well known that a smooth projective cubic surface has exactly 27 lines on it. We describe them in the current situation [20]. Let L i (b i , b 4 ; b j , b k ) be the line in P 3 defined by   Table 4 are the only lines on S(θ) that intersect the i-th line L i at infinity, but they do not intersect the remaining two lines L j and L k at infinity. These lines are divided into four pairs as in Table 4. The surface S(θ) is always smooth at infinity [20] and hence, if κ ∈ K − Wall, then S(θ) is smooth everywhere. In this case, the two lines in the same pair intersect, while two lines from different pairs do not. The intersection point of the i-th pair is exactly the i-th fixed point in Table 3. See Figure 11 for a total image of these situations. Caution: for a pair of distinct indices i and j, the intersection relations between L ± iµ and L ± jν are not depicted in the Figure 11. We also remark that in some degenerate cases the lines L + iµ and L − iµ may meet in a point on the line L i at infinity. Next we consider the case where the determinant 4 − x 2 i of system (37) vanishes. In other words we ask when the fixed point set Fix j (θ) contains points x such that x i ∈ {±2}. Lemma 6.2 Fix j (θ) contains a point x such that x i = 2δ with δ ∈ {±1} if and only if either If this is the case, then θ k = δθ j and all such poins x are exactly those points on the line In particular ℓ δ j ⊂ Fix j (θ) precisely when x i = 2δ is a multiple root of the quartic equation (39).  Proof. If x i = 2δ with δ ∈ {±1} then system (37) is linearly dependent, so that θ k − δθ j = 0.
Taking the equation ; the former case falls into case (1) while the latter falls into case (3). In a similar manner the other case b j b ε k k = δ falls into case (2) or case (3). Next, if Fix j (θ) contains the line ℓ δ j , then what we have just proved implies that Hence x i = 2δ is a multiple root of the quartic equation (39). Conversely, if x i = 2δ is a multiple root of (39), then we can trace the argument backwards to conclude that the system (23) admits the line solution ℓ δ j , that is, Fix j (θ) contains ℓ δ j . 2 Summarizing the arguments so far yields a classification of the irreducible components of the algebraic set Fix j (θ) in terms of certain roots of quartic equation (39). (1) A simple root with x i ∈ {±2} corresponds to a point component that is a smooth point of the surface S(θ) and is given in Table 3.
(2) A multiple root with x i ∈ {±2} corresponds to a point component that is a singular point of the surface S(θ) and is associated with Riccati solutions.
(3) A multiple root with x i ∈ {±2} corresponds to a line component; either ℓ + j or ℓ − j .
(4) A simple root with x i ∈ {±2} corresponds to no component of Fix j (θ).
A summary of Theorem 6.3 is given in Table 5 and the following remark may be helpful.
Remark 6.4 The assertions (3) and (4) of Theorem 6.3 may be well understood through the degeneration of line configration on the projective surface S(θ) as the parameter θ = rh(κ) tends to a special position. For a generic value of θ the lines L ± iµ intersect in a single (smooth) point on the affine part S(θ) of S(θ). If the parameter θ tends to a special position so that a corresponding root x i of quartic equation (39) approaches {±2}, then the two line L ± iµ are getting "parallel" and eventually either coincide completely or meet in a point at infinity. The former case falls into assertion (3) and the latter case falls into assertion (4) respectively.
(1) Fix j (θ) contains either ℓ + j or ℓ − j but not both of them if and only if κ lies in a W (D 4 )stratum appearing in the following adjacency diagram (see also Figure 10) : (2) Fix j (θ) contains both ℓ + j and ℓ − j if and only if θ = (0, 0, 0, −4), that is, precisely when κ is in the W (D which reduces condition (a) to condition (b). Next, formula (13) implies that the reflection w i induces the multiplicative transformation w i : Applying w 4 or w k if necessary, we may assume from the beginning that ε 4 = 1 and ε k = −1 in condition (c). Then using w 0 there yields b 2 , that is, condition (b). Note that condition (b) means κ j , κ k ∈ Z. On the other hand, the extended affine Weyl group W (D Repeated applications of these operations and their inverses can shift κ j and κ k independently by arbitrary integers. Thus the condition κ j , κ k ∈ Z can further be reduced to κ j = κ k = 0. Thus we have shown that if Fix j (θ) contains at least one of ℓ + j and ℓ − j , then κ must lie in the W (D (1) 4 )-stratum of type (A ⊕2 1 ) i or on its boundary strata of types (A 3 ) i , A ⊕3 1 , D 4 , A ⊕4 1 . Moreover, it is easy to see that the converse is also true.
For a sign δ ∈ {±1} the conditions (1), (2), (3) in Lemma 6.2 are denoted by (1 δ ), (2 δ ), (3 δ ), respectively. Now we assume that Fix j (θ) contains both ℓ + j and ℓ − j . Then there exists a pair of conditions, one from {(1 + ), (2 + ), (3 + )} and the other from {(1 − ), (2 − ), (3 − )}, that are valid at the same time. Such a pair can be consistent only if it is either (1 − ) + (2 − ); or (2 + ) + (1 − ); or (3 + ) + (3 − ) where if the sign for (3 + ) is (ε k , ε 4 ) then the sign for (3 − ) must be its antipode (−ε k , −ε 4 ). The first and second pairs lead to b 2 These are nothing but the conditions (a) and (b) in Example 3.4. (1). Therefore κ must lie in the stratum of type A ⊕4 1 . Combining this with the discussion in the last paragraph establishes the assertion (1), as well as a large part of the assertion (2). The only thing yet to be proved is the assertion that if Fix j (θ) contains both ℓ + j and ℓ − j , then Fix j (θ) = ℓ + j ∐ ℓ − j . For this, the last part of Lemma 6.2 implies that both x i = 2 and x i = −2 are multiple roots of the quartic equation (39), so that there are no other roots of the equation (39). Thus Fix j (θ) has no elements other than those in ℓ + j ∐ ℓ − j . 2 Now we turn our attention to periodic points and investigate the set Per • j (θ; n) of periodic points of prime period n > 1 on the non-Riccati locus. Lemma 6.6 For any integer n > 1 the set Per • j (θ; n) is biholomorphic to the disjoint union of ϕ(n) copies of C × , where ϕ(n) denotes the number of integers 0 < m < n coprime to n.

Proof. By Lemma 4.5 we can identify Per
• j (θ; n) with Per j (θ; n) and hence may work downstairs. For any integer 0 < m < n coprime to n, we consider the projective curve C m in P 3 defined by where (47) is obtained from F (X, θ) = 0 by substituting (48) and factoring X 0 out of it. It follows from −2 < 2 cos(πm/n) < 2 that C m is an irreducible smooth conic curve. By equations (24) of Lemma 4.4 the closure Per j (θ; n) of Per j (θ; n) in S(θ) is the union of these ϕ(n) curves C m . The curve C m intersects the lines L = L i ∪ L j ∪ L k at infinity in the two points is the disjoint union of the ϕ(n) curves C m with 0 < m < n, (m, n) = 1, and hence biholomorphic to the disjoint union of ϕ(n) copies of C × .

Case-by-Case Study
We make case-by-case studies of Fix j (θ) and Per e j (θ; n) according to the adjacency diagram in Figure 10. Now we need to introduce some notation. Recall that we have the resolution of singularities (15) which restricts to an isomorphism ϕ : S • (θ) → S • (θ) and that the smooth fixed points Fix • j (θ) in S • (θ) are listed in Table 3. For each P ∈ Fix • j (θ) let P ∈ Fix • j (θ) denote its lift through the isomorphism ϕ. For example, , then we denote by {{· · · }} its subset obtained by discarding those expressions which do not satisfy either the existence condition or the smoothness condition of Table 3. An example is given in (49) below.
Here we have only to care the existence condition, as we are in the big open where the smoothness condition is fulfilled by hypothesis. If a finer stratification of K attached to the W (F 4 )action on K is introduced, then a more precise description of (49) is feasible, detecting how many and which elements are there in (49), but the details are omitted. We only remark that Fix j (θ) consists of four distinct points in the most generic case where none of κ i ±κ 4 and κ j ±κ k are integers. As for the periodic points, since there is no Riccati locus, we have Per j (θ; n) = Per 4 )-stratum of type A 1 . We may assume that κ 0 = 0 so Applying a shift as in (46) to κ repeatedly, we may assume that κ j + κ k = 0 while keeping the condition κ 0 = 0. Then the transformation w 0 w j sends κ to κ ′ with κ ′ j = 0 and κ ′ k = κ j + κ k = 0, so that one has κ ∈ K {j,k} , namely, κ lies in the closure of the stratum of type (A ⊕2 1 ) i . This contradicts the assumption that we are in the stratum of type A 1 . In this case, the surface S(θ) has a unique singular point of type A 1 at Blow up S(θ) at this point to obtain a minimal resolution (15). Write the blowing-up as in terms of coordinates (u i , u j , u k ). The exceptional set e is the irreducible quadratic curve which can be paramatrized as u j = 0 and In terms of this parametrization, the lifted transformationg 2 j acts on the exceptional curve e ≃ P 1 by the multiplication t → b 2 j b 2 k t. Since b 2 j b 2 k = 1, the set Fix e j (θ) consists of the two e + e − P 1 Figure 12: Surface of types A 1 (left) and A 2 (right) points, say p and q, corresponding to t = 0 and t = ∞ (see Figure 12, left). On the other hand, the possible candidates for the smooth fixed points Fix  Table 3, since those of labels 1 and 3 do not satisfy the smoothness condition. Thus, As for the Riccati periodic points Per e j (θ; n), the discussion above implies that for any n > 1, We may assume that κ 0 = κ i = 0 so that b 0 = b i = 1. Then the surface S(θ) has a unique singular point of type A 2 at Blow up S(θ) at this point to obtain a minimal resolution (15). Write the blowing-up as in terms of coordinates (u i , u j , u k ). The exceptional set e is the union of two lines intersecting in a point. These lines are parametrized as with the intersection point corresponding to s = t = 0. In terms of these parametrizations, the lifted transformationg 2 j acts on e + and e − by the multiplications Since we are assuming that κ i = κ j + κ k + κ 4 = 1, the points of labels 1, 2, 3 in Table 3 do not satisfy the smoothness condition and that of label 4 is the only smooth fixed point. Thus Fix In the remaining cases presented below, Fix j (θ) contains at least one line component. 1 ) i . We may assume that κ j = κ k = 0 so that b j = b k = 1. Since our stratum is not of type In this case Fix j (θ) contains the line ℓ + j but does not the line ℓ − j and the surface S(θ) has two singular points of type We denote the former singularity by q i and the latter by q 4 respectively; both singularities lie on the line ℓ + j . Blow up S(θ) at these points to obtain a minimal resolution as in (15). Let ℓ + j be the strict transform of ℓ + j , and let e i and e 4 be the exceptional curves over q i and q 4 respectively. Moreover let p i be the intersection point of ℓ + j and e i . Similarly let p 4 be the intersection point of ℓ + j and e 4 (see Figure 13). Then the blowing-up at the point q i is represented as 0, 0, 0). The strict transform ℓ + j and the exceptional curve e i are given by u i = u k + 1 = 0 and The exceptional curve e i admits a parametrization where the intersection point p i has coordinates (u i , u j , u k ) = (0, 0, −1), which corresponds to t = ∞. The lifted transformationg 2 j acts on e i as a Möbius transformation fixing p i . Some computations show that in terms of the variable t this transformation is just the shift and hence a parabolic transformation. Thusg 2 j has no periodic points on e i other than the fixed point p i . By symmetry,g 2 j also acts on e 4 as a parabolic Möbius transformation fixing p 4 only. Summarizing the arguments, we conclude that on the W (D Next we consider Fix i (θ) on the W (D  Table 3 with (i, j, k) replaced by (k, i, j) is not satisfied for any labels there. Thus we have Fix i (θ) = Fix e i (θ) = {p ii , q ii , p i4 , q i4 } and Fix 4 )-stratum of type (A 3 ) i . We may assume that κ 0 = κ j = κ k = 0 and κ i + κ 4 = 1 so that b j = b k = 1 and b i b 4 = 1. But we have b i b −1 4 ∈ {±1}, since our stratum is not of type D 4 . In this case Fix j (θ) contains the line ℓ + j but does not the line ℓ − j . The surface S(θ) has only one singular point of type , which lies on the line ℓ + j . Blow up the singular point. This blowing-up is expressed as ( 0, 0, 0). The strict transform of the surface S(θ) is given by which has yet one singular point, say q. The exceptional curve consists of two line components u j = u i +b 4 u k +b 4 = 0 and u j = b 4 u i +u k +1 = 0, whose intersection point (u i , u j , u k ) = (0, 0, −1) is exactly the singular point q. The strict transform of ℓ + j is now given by u i = u k +1 = 0, which e 0 ℓ + j e j e k P 1 Figure 14: Surface of type (A 3 ) i also passes through q. Blow up again the singular point q. Let e 0 be the exceptional curve and let e j , e k ,l + j be the strict transforms of the lines u j = u i + b 4 u k + b 4 = 0, u j = b 4 u i + u k + 1 = 0, u i = u k + 1 = 0 respectively. If we express this blowing-up as ( , then the exceptional curve e 0 is given by j acts on e k as the shift t → t + b 4 − b i . Henceg 2 j acts on e j and e k as parabolic Möbius transformations fixing only p j and q j . Theng 2 j acts on e 0 as the identity, because it also fixes the intersection point p 0 of e 0 andl + j . Summarizing the arguments we see that on the stratum of type ( where ℓ + j ∪ p 0 e 0 indicates that the curves ℓ + j and e 0 meet in the point p 0 . Next we consider Fix i (θ) and Per e i (θ; n) on the W (D (1) 4 )-stratum of type (A 3 ) i . If we take a parametrization of e 0 such that t = 0 and t = ∞ correspond to the points p j and p k respectively, then a simple check shows that the transformationg 2 i on e j is expressed as t → b −2 4 t. There is a parametrization of e j such that t = 0 corresponds to p j andg 2 i is given by t → b 2 4 t. Since b 2 4 = 1, the transformationg 2 i has exactly two fixed points on e j , one of which is just p j and the other is denoted by p ij . Similarly, there is a parametrization of e k such that t = 0 corresponds to p k andg 2 i is given by t → b −2 4 t, and henceg 2 i has exactly two fixed points on e k , one of which is just p k and the other is denoted by p ik . There are no smooth fixed points Fix • i (θ), because the smoothness condition of Table 3 with (i, j, k) replaced by (k, i, j) is not satisfied for any labels there. So we have Fix i (θ) = Fix e i (θ) = {p j , p ij , p k , p ik } and Fix 1 . In this case the surface S(θ) has three singular points of type , which are called q i , q j , q k respectively. Note that the two points q j and q k lie on the line ℓ + j but q i does not lie on the union ℓ + j ∐ ℓ − j . The minimal resolution (15) is obtained by blowing up these three points (see Figure 15). First, consider the blowing-up at q k and represent it by (x i , x j , x k ) = (u i u j + 2, u j + 2, u k u j + b 4 + b −1 4 ). Then the strict transform ℓ + j of the line ℓ + j is given by u i = u k + 1 = 0, while the exceptional curve e k is given by b 4 (u i + u k + 1) 2 + (b 4 − 1) 2 u i = 0. The curves ℓ + j and e k intersect in the point (u i , u j , u k ) = (0, 0, −1); this point is called p k . If we parametrize the curve e k as where t = ∞ corresponds to the point p k , then the lifted transformation g 2 j induces the shift t → t + 1 and hence acts on e k as a parabolic Möbius transformation fixing p k only. In a similar manner g 2 j acts on the exceptional curve e j over q j as a parabolic Möbius transformation fixing only the intersection point p j of ℓ + j and e j . Next we consider the blowing-up at q i and represent it by (x i , x j , x k ) = (u i u j + b 4 + b −1 4 , u j + 2, u k u j + 2). Then the exceptional curve e i is given by b 4 (u i + u k + 1) 2 + (b 4 − 1) 2 u k = 0, which can be parametrized as Figure 16: Surface of type D 4 In terms of this parametrization, the transformation g 2 j restricts to the map t → b 2 4 t on the exceptional curve e i . Let r 0 and r ∞ be the points on e i corresponding to t = 0 and t = ∞ respectively. Since b 2 4 = 1, the map g 2 j acts on e i as a Möbius transformation with exactly two fixed points r 0 and r ∞ . Hence the set Fix j (θ) contains the line component ℓ + j and the Riccati component {r 0 , r ∞ }, but has no smooth-point component, since is a double root of the quartic equation (39) (see Theorem 6.3). Thus we have As for the Riccati periodic points, since g 2 j acts on e i as t → b 2 4 t, we have for any n > 0, 4 )-stratum with value θ = (8,8,8,28). In this case the surface S(θ) has only one singular point of type D 4 at (x i , x j , x k ) = (2, 2, 2). The minimal resolution (15) is obtained by successive blowing-ups: Blow up the singular point. If we express the blowing-up as (x i , x j , x k ) = (u i u j + 2, u j + 2, u k u j + 2) in terms of coordinates (u i , u j , u k ), then the strict transform of the surface S(θ) is represented as u i u j u k + (u i + u k + 1) 2 = 0. The exceptional curve e is given by u j = u i + u k + 1 = 0. The strict transforms of ℓ + i and ℓ + j are given by u i + 1 = u k = 0 and u i = u k + 1 = 0, while the strict transform of ℓ + k is at infinity and not expressible in terms of the coordinates (u i , u j , u k ). The blow-up surface has three singularities, all of which are of type A 1 and located at the points in which the exceptional curve e intersects the strict transforms of ℓ + i , ℓ + j , ℓ + k . The lifts of the transformations g 2 i , g 2 j , g 2 k fix the curve e pointwise, since they fix the three singular points on it. Again blow up these points. Then we obtain a minimal resolution (15) of the surface S(θ) as depicted in Figure 16, where e i , e j , e k are the exceptional curves over the singular points and Figure 17: Surface of type A ⊕4 1 e 0 , ℓ + i , ℓ + j , ℓ + k are the strict transforms of e, ℓ + i , ℓ + j , ℓ + k , respectively. Being the strict transform of e, the exceptional curve e 0 is fixed pointwise by the liftsg 2 i ,g 2 j ,g 2 k of g 2 i , g 2 j , g 2 k , and hence carries rational solutions. Moreover the liftg 2 j fixes e j pointwise. This can be seen without computation. Sinceg 2 j is area-preserving and fixes ℓ + j pointwise, it has derivative 1 at p j along the curve e j . So the Möbius transformation on e j induced byg 2 j is either identity or a map of parabolic type. But the latter is impossible because it has at least two fixed points at p j and q j (see Figure 16). Hence g 2 j acts on e j as the identity. Next we shall observe that g 2 j acts on e i as a parabolic Möbius transformation fixing q i only. If we express the blowing-up Then g 2 j acts on e i by the shift t → t + 1. Similarly g 2 j acts on e k as a parabolic transformation fixing q k only. By symmetry, g 2 i and g 2 k act on e j as parabolic transformations fixing q j only. Notice that the exceptional curve e 0 carries rational Riccati solutions, while e j − {q j } carries Riccati solutions of infinte period. Thus we have 4 )-stratum of type A ⊕4 1 , where θ = (0, 0, 0, −4) and Fix j (θ) = ℓ + j ∐ ℓ − j . In this case the surface S(θ) has four singularities of type Blow up at these points to obtain a minimal resolution as in (15). Let e ε i ε j ε k be the exceptional line over (x i , x j , x k ) = (2ε i , 2ε j , 2ε k ) andl ε i j be the strict transform of ℓ ε i j . Moreover let p ε i ε j ε k denote the intersection point of the lines e ε i ε j ε k and ℓ ε i j (see Figure 17). Then the lifted transformationg 2 j : S(θ) acts on the exceptional line e ε i ε j ε k ∼ = P 1 as a Möbius transformation. It is a parabolic transformation with the only fixed point p ε i ε j ε k . Let us check this for (ε i , ε j , ε k ) = (−1, −1, −1). The blowing-up of C 3 at (x i , x j , x k ) = (−2, −2, −2) is described by x i = u i u j − 2, x j = u j − 2, x k = u j u k − 2, in terms of coordinates (u i , u j , u k ) around (0, 0, 0). Then the exceptional line e −−− is represented by the equations u j = 0 and (u i − u k ) 2 − 2(u i + u k ) + 1 = 0 and hence it is parametrized as where the fixed point p −−− corresponds to t = ∞. Then we can check thatg 2 j acts on the line e −−− as the translation t → t+4, as desired. Thus the only fixed points ofg 2 j on the exceptional set E(θ) are the four points p ε i ε j ε k with ε i ε j ε k = −1 and there are no periodic points, so that

Power Geometry
We apply the method of power geometry [5,6,7] to construct as many algebraic branch solutions to P VI (κ) as possible around each fixed singular point. Basically we can follow the arguments of [7]. However, while the attention of [7] is restricted to generic parameters, we require a thorough treatment of all parameters, where much ampler varieties of patterns are present. Moreover, the way in [7] of representing the parameters of Painlevé VI is not convenient for our purpose. So we have to redevelop the necessary arguments on power geometry from scratch. In view of Remark 5.1, it is sufficient to work around the origin z = 0. In order to apply the method in [5,6,7], we reduce the system (1) into a single second-order equation. If (q, p) = (q(z), p(z)) is a solution to system (1) such that q ≡ 0, 1, z, ∞, then we solve the first equation of system (1) with respect to p = p(z) to obtain Substituting this into the second equation yields the single second-order equation Multiply equation (61) by 2z 2 (z − 1) 2 q 0 q 1 q z and move its right-hand side to the left to obtain where P (z, q) is a polynomial of (z, q, q ′ , q ′′ ), that is, a differential sum of (z, q), whose explicit formula is omitted here but can be found in [7]. Therefore system (1) is equivalent to equation (62) together with (60) except for the possible solutions such that q ≡ 0, 1, z, ∞. A simple check shows that the Newton polygon of equation (62) is given as in Figure 18, where there are four patterns according as the parameters κ 1 and κ 4 are zero or not. First we search for a holomorphic solution germ q = q(z) to equation (62) around z = 0. We have only to construct formal power series solutions of the form q = cz r + (higher order terms),   (62) is convergent [10,11]. Then it follows from (60) that the associated formal Laurent series for p = p(z) is also convergent.
Proof. We only derive (73), as (72) is derived in a similar manner. Substituting q = t −1 + t −1 (t−1)Q into (62) yields P (z, t −1 +t −1 (t−1)Q) = t −4 (t−1) 2 z −1 p(z, Q; t), where p(z, Q; t) is a differential sum of (z, Q) with coefficients in C[t] whose Newton polygon is given as in Figure 22 (right). Consider the edge connecting (0, 1) and (1, 0). The vertex (0, 1) carries the differential sum LQ = −2(zQ ′ + z 2 Q ′′ ), whose characteristic polynomial is −2k 2 , while the vertex (1, 0) carries the monomial z. Since the critical value k = 0 is smaller than 1, the coefficients c k (t), k ≥ 1, in (73) are determined uniquely and recursively, where the leading term is c 1 (t) = 1/2. Substituting the resulting series q = q(z) into (60) we have p = p(z) as in (73). 2 So far we have considered the edges Γ 1 and Γ 0 of the Newton polygon in Figure 18 and constructed meromorphic solutions around the origin z = 0. Now let us consider the vertex (0, 3) of the polygon, which gives rise to algebraic branch solutions around z = 0. The truncation of the differential sum P = P (z, q) at the vertex (0, 3) is given by Its charactersistic polynomial χ(r) is defined by substituting q = z r into P 3 (z, q) and dividing the result by q 3 = z 3r . In the present situation we see that χ(r) is identically zero; χ(r) ≡ 0. The normal cone of the vertex (0, 3) is U 3 = { (p 1 , p 2 ) ∈ R 2 : p 1 < 0, 0 < r = p 2 /p 1 < 1 }. Thus the truncated solutions at the vertex (0, 3) are q = tz r for an arbitrary 0 < r < 1 and t ∈ C × . The Fréchet derivative with respect to q at the truncated solution q = tz r is given by The corresponding characteristic equation is given by v 3 (k) := −2t 2 (k − r) 2 = 0, which has the only root k = r. Let n be any integer greater than 1. In order to search for an algebraic n-branch solution around z = 0, we take r = m/n for any integer 0 < m < n coprime to n, and consider a formal Puiseux series solution of the form Since the characteristic equation v 3 (k) = 0 has no roots such that k > m/n, the coefficients a ν = a ν (t) can be determined uniquely and recursively for any given initial coefficient t ∈ C × . The convergence of the formal solution and its holomorphic dependence on parameters follow easily if we rewrite the equation (62) in terms of the new independent variable ζ = z 1/n and apply the convergence arguments in [10,11]. Thus we have established the following lemma.
Lemma 8.7 For any integer n > 1, there exist ϕ(n) mutually disjoint 1-parameter families of n-branch solution germs to P VI (κ) around the origin z = 0, q(z) = tz m/n + ∞ ν=m+1 a ν (n, m, t; κ) z ν/n , where the discrete parameter m ranges over all integers 0 < m < n coprime to n and the continuous parameter t takes any value of the punctured complex line C × .

Remark 8.8
The family (74) contains no Riccati solutions, even if κ ∈ Wall. This will be shown in the proof of Lemma 9.14.
Example 9.6 (∅) We combine the results of Example 7.1, Lemmas 8.1 and 8.2. A key observation is that Lemmas 8.1 and 8.2 give us as many meromorphic solutions around z = 0 as the cardinality of the set Fix j (θ) in (49). For example, if P (b i , b 4 ; b j , b k ) ∈ Fix j (θ), then the existing and smoothness conditions for it (see Table 3) makes it possible to apply Lemma 8.2 to conclude that the meromorphic solution (66) exists corresponding to the fixed point P (b i , b 4 ; b j , b k ). Thus the set Fix j (θ) is exhausted by meromorphic solutions around z = 0. Example 9.7 (A 1 ) We combine the results of Example 7.2, Lemmas 8.1, 8.2 and 9.1. First we notice that the two single-valued Riccati solutions in Lemma 9.1 correspond to the two Riccati fixed points Fix e j (θ) = {p, q} in (50). On the other hand, for the same reason as in Example 9.6, formulas (65) and (67) in Lemmas 8.1 and 8.2 give us as many meromorphic solutions around z = 0 as the cardinality of smooth fixed points Fix 4 )}} in (50). Thus the set Fix j (θ) is exhausted by meromorphic solutions around z = 0. Example 9.8 (A 2 ) We combine the results of Example 7.3, Lemmas 8.2 and 9.2. As (51) shows, the set Fix j (θ) consists of the four points p 0 , p + , p − and P (b j , b −1 k ; b i , b 4 ). On the other hand, we have the three single-valued Riccati solutions of Lemma 9.2 and one non-Riccati holomorphic solution (67). Clearly, the three Riccati solutions correspond to the points p 0 , p + and p − , while the non-Riccati solution corresponds to the remaining point P (b j , b −1 k ; b i , b 4 ). Thus any single-valued solution around z = 0 is a meromorphic solution.  (53), so that we have an injection C ֒→ C. Since this injection is holomorphic, it must be a surjection. Thus the line ℓ + j is exhausted by the family (68). Moreover we have the two holomorphic solutions (66) and (67), which do not lie in the family (68). Thus they must correspond to the points P (b i , b 4 ; b j , b k ) and P (b i , b −1 4 ; b j , b k ) in (53). So on the stratum of type (A ⊕2 1 ) i the set Fix j (θ) is exhausted by meromorphic solutions around z = 0. Next we consider the W (D  1 ) i . The P 1 -family (77) of single-valued Riccati solutions exactly corresponds to the exceptional curve e 0 ≃ P 1 in (55). So the C-family (70) of holomorphic solutions must inject into the line ℓ + j ≃ C in (55). Since this injection C ֒→ C is holomorphic, it must be a surjection. Hence ℓ + j is exhausted by the family (70). Moreover there is the holomorphic solution (67), which must correspond to the point P (b i , b −1 4 ; b j , b k ) in (55). Thus on the stratum of type (A ⊕2 1 ) i the set Fix j (θ) is exhausted by meromorphic solutions around z = 0. Next we consider the W (D (1) 4 )-strata of types (A ⊕2 1 ) j and (A ⊕2 1 ) k . On these strata the equality Fix j (θ) = Fix e j (θ) in (56) implies that any single-valued solution around z = 0 is a Riccati and hence meromorphic solution.
Example 9.11 (A ⊕3 1 ) We combine the results of Example 7.6 and Lemma 8. 3. As (57) shows, Fix j (θ) has only one line component ℓ + j ≃ C. Hence the C-family (68) of holomorphic solutions must inject into this line, so that we have an inclusion C ֒→ C. Since this injection is holomorphic, it must be a surjection. Thus ℓ + j is exhausted by the family (68). there. Hence C-family (71) of holomorphic solutions, which is different from (78) and (79), must inject into the line ℓ + j ≃ C in (58). Since this injection C ֒→ C is holomorphic, it must be a surjection. Thus ℓ + j is exhausted by the family (71).
Example 9.13 (A ⊕4 1 ) We combine the results of Example 7.8 and Lemma 8.6. In view of Fix j (θ) = ℓ + j ∐ ℓ − j , the C-family (72) of holomorphic solutions injects into the line ℓ ε j ≃ C for some sign ε ∈ {±1}. So we have an injection C ֒→ C. Since this injection is holomorphic, it must be a surjection, so that ℓ ε j is exhausted by the family (72). Then the other C-family (73) of holomorphic solutions injects into the remaining line ℓ −ε j ≃ C. So we have another injection C ֒→ C. Since this injection is holomorphic, it must be a surjection. Hence Fix j (θ) = ℓ + j ∐ ℓ − j is exhausted by the families (72) and (73). The proof of Lemma 9.5 is now complete. 2 Finally we argue the periodic point case using the "injection-implies-surjection" principle.
Lemma 9.14 For any n > 1 the set Per j (θ; n) is exhausted by algebraic n-branch solutions around z = 0.
Proof. We combine Lemmas 6.6 and 8.7. First we consider the generic case where κ ∈ K−Wall, namely, where θ = rh(κ) is such that ∆(θ) = 0. In this case there is no Riccati locus and hence Per j (θ; n) = Per • j (θ; n), which is biholomorphic to the disjoint union of ϕ(n) copies of C × by Lemma 6.6. On the other hand, by Lemma 8.7, there are ϕ(n) mutually disjoint C × -parameter families of algebraic n-branch solutions around z = 0 as in (74). Number these families from 1 to ϕ(n). The first family injects into a (unique) connected component (≃ C × ) of Per j (θ; n), which we call the first component, and we have an injection C × ֒→ C × . Since this injection is holomorphic, it must be a surjection and hence the first component is exhausted by the first family. Consider the second family of solutions and the corresponding second component of Per j (θ; n). Notice that the second component is different from the first one, because the first component is already occupied by the first family and so it cannot contain the second family. For the same reason as above, the second component is exhausted by the second family. Since the families and the components have the same cardinality ϕ(n), we can repeat this argument to conclude that Per j (θ; n) is exhausted by the ϕ(n) families of algebraic n-branch solutions.
Next we consider the case where κ ∈ Wall, namely, where the Riccati part Per e j (θ; n) may appear. Since the lemma is trivial for the Riccati part, we have only to consider the non-Riccati part Per • j (θ; n). The argument proceeds just in the same manner as in the last paragraph, once we show that the family of solutions in (74) contains no Riccati solutions (see Remark 8.8). To see this, we consider the family S → Θ of surfaces S(θ) parametrized by θ ∈ Θ and put where E(θ) is the exceptional set in S(θ). (Precisely speaking, the parameter space Θ should be replaced by a finite covering of it to get a simultaneous minimal resolution.) Then Per • j (n) and E are closed subsets of S which are disjoint by Lemma 4.5. Now we look at the family of solutions in (74). It depends continuously on κ ∈ K. Take any point κ * ∈ Wall and let K − Wall ∋ κ → κ * . For any κ ∈ K − Wall, the family at κ is contained in Per • j (θ; n) with θ = rh(κ) and hence in Per • j (n). Taking the limit κ → κ * , we see that the family at κ * is contained in Per • j (n), hence in Per • j (θ * ; n) with θ * = rh(κ * ). Since Per • j (θ * ; n) is disjoint from E(θ * ), the family at κ * contains no Riccati solutions. Therefore the proof is complete. 2 Now the local statement of Theorem 1.3 around a fixed singular point, say z = 0, is an immediate consequence of Lemmas 9.5 and 9.14. At the same time all the finite branch solutions around z = 0 have been classified up to Bäcklund transformations. The global statement about algebraic solutions follows readily from the local statements around z = 0, 1, ∞, together with the analytic Painlevé property on Z = P 1 − {0, 1, ∞}. The proof of Theorem 1.3 is complete.