Geometric Aspects of the Painlevé Equations PIII(D₆) and PIII(D₇)

The Riemann-Hilbert approach for the equations PIII(D6) and PIII(D7) is studied in detail, involving moduli spaces for connections and monodromy data, Okamoto-Painlevé varieties, the Painlevé property, special solutions and explicit Bäcklund transformations.

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Опубліковано в: :	Symmetry, Integrability and Geometry: Methods and Applications
Дата:	2014
Автори:	Marius van der Put, Top, J.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2014
Онлайн доступ:	https://nasplib.isofts.kiev.ua/handle/123456789/146693
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Цитувати:	Geometric Aspects of the Painlevé Equations PIII(D₆) and PIII(D₇) / Marius van der Put, J. Top // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 13 назв. — англ.

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spelling	Marius van der Put Top, J. 2019-02-10T19:01:20Z 2019-02-10T19:01:20Z 2014 Geometric Aspects of the Painlevé Equations PIII(D₆) and PIII(D₇) / Marius van der Put, J. Top // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 13 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 14D20; 14D22; 34M55 DOI:10.3842/SIGMA.2014.050 https://nasplib.isofts.kiev.ua/handle/123456789/146693 The Riemann-Hilbert approach for the equations PIII(D6) and PIII(D7) is studied in detail, involving moduli spaces for connections and monodromy data, Okamoto-Painlevé varieties, the Painlevé property, special solutions and explicit Bäcklund transformations. The authors thank Yousuke Ohyama for his helpful answers to our questions and his remarks concerning the tau-divisor (see Section 3.3.4). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Geometric Aspects of the Painlevé Equations PIII(D₆) and PIII(D₇) Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Geometric Aspects of the Painlevé Equations PIII(D₆) and PIII(D₇)
spellingShingle	Geometric Aspects of the Painlevé Equations PIII(D₆) and PIII(D₇) Marius van der Put Top, J.
title_short	Geometric Aspects of the Painlevé Equations PIII(D₆) and PIII(D₇)
title_full	Geometric Aspects of the Painlevé Equations PIII(D₆) and PIII(D₇)
title_fullStr	Geometric Aspects of the Painlevé Equations PIII(D₆) and PIII(D₇)
title_full_unstemmed	Geometric Aspects of the Painlevé Equations PIII(D₆) and PIII(D₇)
title_sort	geometric aspects of the painlevé equations piii(d₆) and piii(d₇)
author	Marius van der Put Top, J.
author_facet	Marius van der Put Top, J.
publishDate	2014
language	English
container_title	Symmetry, Integrability and Geometry: Methods and Applications
publisher	Інститут математики НАН України
format	Article
description	The Riemann-Hilbert approach for the equations PIII(D6) and PIII(D7) is studied in detail, involving moduli spaces for connections and monodromy data, Okamoto-Painlevé varieties, the Painlevé property, special solutions and explicit Bäcklund transformations.
issn	1815-0659
url	https://nasplib.isofts.kiev.ua/handle/123456789/146693
citation_txt	Geometric Aspects of the Painlevé Equations PIII(D₆) and PIII(D₇) / Marius van der Put, J. Top // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 13 назв. — англ.
work_keys_str_mv	AT mariusvanderput geometricaspectsofthepainleveequationspiiid6andpiiid7 AT topj geometricaspectsofthepainleveequationspiiid6andpiiid7
first_indexed	2025-11-26T17:52:25Z
last_indexed	2025-11-26T17:52:25Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 10 (2014), 050, 24 pages Geometric Aspects of the Painlevé Equations PIII(D6) and PIII(D7) Marius VAN DER PUT and Jaap TOP Johann Bernoulli Institute, University of Groningen, P.O. Box 407, 9700 AK Groningen, The Netherlands E-mail: mvdput@math.rug.nl, j.top@rug.nl URL: http://www.math.rug.nl/~top/ Received October 15, 2013, in final form April 10, 2014; Published online April 23, 2014 http://dx.doi.org/10.3842/SIGMA.2014.050 Abstract. The Riemann–Hilbert approach for the equations PIII(D6) and PIII(D7) is studied in detail, involving moduli spaces for connections and monodromy data, Okamoto– Painlevé varieties, the Painlevé property, special solutions and explicit Bäcklund transfor- mations. Key words: moduli space for linear connections; irregular singularities; Stokes matrices; monodromy spaces; isomonodromic deformations; Painlevé equations 2010 Mathematics Subject Classification: 14D20; 14D22; 34M55 1 Introduction The aim of this paper is a study of the Painlevé equations PIII(D6) and PIII(D7) by means of isomonodromic families in a moduli space of connections of rank two on the projective line. This contrasts the work of Okamoto et al. on these third Painlevé equations, where the Hamiltonians are the main tool. However, there is a close relation between the two points of view, since the moduli spaces turn out to be the Okamoto–Painlevé varieties. We refer only to a few items of the extensive literature on Okamoto–Painlevé varieties. More details on Stokes matrices and the analytic classification of singularities can be found in [11]. A rough sketch of this Riemann–Hilbert method is as follows (see for some background [10] and see for details concerning PI, PII, PIV [8, 9, 12]). The starting point is a family S of differential modules M of dimension 2 over C(z) with prescribed singularities at fixed points of P1. The type of singularities gives rise to a monodromy setR built out of ordinary monodromy, Stokes matrices and ‘links’. The map S→ R associates to each module M ∈ S its monodromy data in R. The fibers of S → R are parametrized by T ∼= C∗ and there results a bijection S→ R×T . The set S has a priori no structure of algebraic variety. A moduli spaceM over C, whose set of closed points consists of certain connections of rank two on the projective line, is constructed such that S coincides with M(C). There results an analytic Riemann–Hilbert morphism RH : M → R. The fibers of RH are the isomonodromic families which give rise to solutions of the corresponding Painlevé equation. The extended Riemann–Hilbert morphism RH+ :M→R×T is an analytic isomorphism. From these constructions the Painlevé property for the corresponding Painlevé equation follows and the moduli space M is identified with an Okamoto–Painlevé space. Special properties of solutions of the Painlevé equations, such as special solutions, Bäcklund transformations etc., are derived from special points of R and the natural automorphisms of S. The above sketch needs many subtle refinements. One has to construct a geometric mon- odromy space R̃ (depending on the parameters of the Painlevé equation) which is a geometric quotient of the monodromy data. The ‘link’ involves (multi)summation and in order to avoid mailto:mvdput@math.rug.nl mailto:j.top@rug.nl http://www.math.rug.nl/~top/ http://dx.doi.org/10.3842/SIGMA.2014.050 2 M. van der Put and J. Top the singular directions one has to replace T by its universal covering T̃ ∼= C. In the construction of M, one represents a differential module M ∈ S by a connection on a fixed vector bundle of rank two on P1. This works well for the case (1,−, 1/2), described in Section 2. In case (1,−, 1), described in Section 3, this excludes a certain set of reducible modules M ∈ S. The moduli space of connections M is replaced by the topological covering M̃ = M×T T̃ . Now the main result is that the extended Riemann–Hilbert morphism RH+ : M̃ → R̃ × T̃ is a well defined analytic isomorphism. Each family of linear differential modules and each Painlevé equation has its own story. For the family (1,−, 1/2), corresponding to PIII(D7), the computations of R, M and the Bäcklund transformations present no problems. For the resonant case (i.e., α = ±1 and θ ∈ Z, see Sections 2.1, 2.2 and 2.5 for notation and the statement) there are algebraic solutions of PIII(D7). The spaces R(α) with α = ±i have a special point corresponding to trivial Stokes matrices. This leads to a special solution q for PIII(D7) and θ ∈ 1 2 +Z, which is transcendental according to [5]. According to [2], q is a univalent function of t and is a meromorphic at t = 0. For the general case (i.e., α 6= β±1) of the family (1,−, 1), corresponding to PIII(D6), the computations of R, M present no problems. The formulas for the Bäcklund transformations, derived from the automorphisms of S, have denominators. These originate from the complicated cases α = β±1 and/or α = ±1 where reducible connections and/or resonance occur. Isomon- odromy for reducible connections produces Riccati solutions and resonance is related to algebraic solutions. 2 The family (1,−, 1/2) and PIII(D7) In this section the set S consists of the (equivalence classes of the) pairs (M, t) of type (1,−, 1/2) (see [10] for the terminology), corresponding to the Painlevé equation PIII(D7). The differential module M is given by dimM = 2, the second exterior power of M is trivial, M has two singular points 0 and∞. The Katz invariant r(0) = 1 and the generalized eigenvalues at 0 are normalized to ± t 2z −1 with t ∈ T = C∗. The singular point∞ has Katz invariant r(∞) = 1/2 and generalized eigenvalues ±z1/2. Further (M, t) is equivalent to (M ′, t′) if M is isomorphic to M ′ and t = t′. The Riemann–Hilbert approach to PIII(D7) in this section differs from [10] in several ways. The choice (1,−, 1/2) is made to obtain the classical formula for the Painlevé equation. Further we consider pairs (M, t) rather than modules M . This is needed in order to distinguish the two generalized eigenvalues at z = 0 and to obtain a good monodromy space R. Finally, the definitions of the topological monodromy and the ‘link’ need special attention. 2.1 The construction of the monodromy space R→ P This is rather subtle and we provide here the details. Given is some (M, t) ∈ S and we write δM for the differential operator on M . First we fix an isomorphism φ : Λ2M → (C(z), z d dz ). 1. C((z))⊗M has a basis F1, F2 such that the operator δM has the matrix ( −ω 0 0 ω ) with ω = tz−1+θ 2 . We require that φ(F1 ∧ F2) = 1. Then F1, F2 is unique up to a transformation (F1, F2) 7→ (λF1, λ −1F2). The solution space V (0) at z = 0 has basis f1 = e− t 2 z−1 zθ/2F1, f2 = e t 2 z−1 z−θ/2F2. The formal monodromy and the two Stokes matrices have on the basis f1, f2 the matrices ( α 0 0 α−1 ) , ( 1 0 c1 1 ) , ( 1 c2 0 1 ) , where α = eπiθ. Their product (in this order) is the topological monodromy top0 at z = 0. 2. C((z−1))⊗M has a basis E1, E2 such that the operator δM has the matrix ( 1 4 1 z −1 4 ) with respect to this basis. Since this differential operator is irreducible, the basis E1, E2 is unique Geometric Aspects of the Painlevé Equations PIII(D6) and PIII(D7) 3 up to a transformation (E1, E2) 7→ (µE1, µE2) with µ ∈ C∗. We require that φ(E1 ∧ E2) = 1 and then µ ∈ {1,−1}. The solution space V (∞) at z =∞ has basis e1 = 1√ −2 z−1/4e2z 1/2 ( E1 + ( −1 2 + z1/2 ) E2 ) , e2 = 1√ −2 z−1/4e−2z 1/2 ( E1 + ( −1 2 − z1/2 ) E2 ) . The formal monodromy and the Stokes matrix have on the basis e1, e2 the matrices ( 0 −i −i 0 ) ,( 1 0 e 1 ) . Their product (in this order) is the topological monodromy top∞ at z =∞. 3. The link L : V (0) → V (∞) is a linear map obtained from multisummation at z = 0, analytic continuation along a path from 0 to ∞ and the inverse of multisummation at z = ∞. The matrix ( `1 `2 `3 `4 ) of L with respect to the bases e1, e2 and f1, f2 has determinant 1, due to the isomorphism φ : Λ2M → (C(z), z d dz ) and the careful choices of the bases. The relation( α αc2 c1 α 1+c1c2 α ) = top0 = L−1 ◦ top∞ ◦L yields α = −i`14e + i`12 − i`34 6= 0 where `ij := `i`j . One observes that α, c1, c2 are determined by L and top∞. Thus the affine space, given by the above data and relations, has coordinate ring C [ e, `1, `2, `3, `4, 1 −`14e+ `12 − `34 ] /(`14 − `23 − 1). 4. The group G generated by the base changes (e1, e2)7→(−e1,−e2) and (f1, f2)7→(λf1, λ −1f2), acts on this affine space. The monodromy space R is the categorical quotient by G and has coordinate ring C [ e, `12, `14, `23, `34, 1 −`14e+ `12 − `34 ] /(`14 − `23 − 1, `12`34 − `14`23). This is in fact a geometric quotient. The morphismR → P = C∗ is given by (e, `12, `14, `23, `34) 7→ α := −i`14e + i`12 − i`34 6= 0. For a suitable linear change of the variables, the fibers R(α) of R → P are nonsingular, affine cubic surfaces with three lines at infinity, given by the equation x1x2x3 + x21 + x22 + αx1 + x2 = 0. A detailed calculation resulting in this equation is presented in [10, Section 3.5]. Lemma 2.1. The space R(α) is simply connected. Proof. We remove from R(α) the line L := {(0, 0, x3) \|x3 ∈ C} and project onto C2 \ S by (x1, x2, x3) 7→ (x1, x2). Here S is the union of {(0, x2) \|x2 6= −1} and {(x1, 0) \|x1 6= −α}. If x1x2 6= 0, then the fiber is one point. If x1x2 = 0, then the fiber is an affine line. Since C2 \ S is simply connected, R(α) \ L is simply connected. Then R(α) is simply connected, too. � Remark on the differential Galois group. The differential Galois group of a module M , with (M, t) ∈ S, can be considered as algebraic subgroup of GL(V (0)). It is the smallest algebraic subgroup containing the local differential Galois group G0 ⊂ GL(V (0)) at z = 0 and L−1G∞L, where G∞ ⊂ GL(V (∞)) is the local differential Galois group at z = ∞. Now G0 is generated (as algebraic group) by the exponential torus {( s1 0 0 s−11 ) ∣∣∣∣∣s1 ∈ C∗ } , the formal monodromy( α 0 0 α−1 ) and the Stokes maps ( 1 0 c1 1 ) , ( 1 c2 0 1 ) . The group G∞ is (as algebraic group) 4 M. van der Put and J. Top generated by the exponential torus {( s2 0 0 s−12 ) ∣∣∣∣∣s2 ∈ C∗ } , the formal monodromy ( 0 −i −i 0 ) and the Stokes map ( 1 0 e 1 ) . This easily implies that the differential Galois group is SL(2,C). In particular, M is irreducible and the same holds for the differential module C( m √ z)⊗M over C( m √ z) for any m ≥ 2. The construction needed to define the topological monodromy and the link. For the definition of the link and the topological monodromies we have to choose nonsingular directions for the two multisummations and a path from 0 to ∞. At z = ∞ the singular direction does not depend on t ∈ T and we can take a fixed nonsingular direction. However, at z = 0, the singular directions for t ∈ T = C∗, t = \|t\|eiφ are φ and π + φ and they vary with t. Thus we cannot use a fixed path from 0 to ∞. In order to obtain a globally defined map L : V (0) → V (∞) we replace T = C∗ by its universal covering T̃ = C → T , t̃ 7→ et̃. The elements of T̃ ∼= R>0 × R are written as t̃ = \|t\|eiφ. Consider the path z̃ = reid(r), 0 < r < ∞, with d(r) = (φ − π 2 ) 1 1+r + π 2 r 1+r on the universal covering of P1 \ {0,∞}. Now L is defined by summation at z = 0 in the direction φ − π 2 , followed by analytic continuation along the above path and finally the inverse of the summation at z =∞ in the direction π 2 . Write S̃ = S×T T̃ . The elements of S̃ are the pairs (M, t̃) with (M, et̃) ∈ S. For the elements in S̃ the link and the monodromy at z = 0 are defined as above. Since R is a geometric quotient, [10, Theorem 1.9] implies: The above map S̃→ R× T̃ is bijective. Fix α ∈ P = C∗. Let S(α), S̃(α) be the subsets of S and S̃, consisting of the pairs (M, t) and (M, t̃) which have ( α 0 0 1 α ) as formal monodromy at z = 0. The map S̃(α)→ R(α)× T̃ is bijective, since S̃→ R× T̃ is bijective. 2.2 The construction of the moduli space M(θ) Fix θ with eπiθ = α. The moduli space M(θ) is obtained by replacing each (M, t) ∈ S(α) by a certain connection (V,∇) on P1. This connection is uniquely determined by the data: Its generic fiber is M ; ∇z d dz is formally equivalent at z = 0 to z d dz + ( ω 0 0 −ω ) with ω = tz−1+θ 2 and is formally equivalent at z =∞ to z d dz + ( −3 4 1 z −1 4 ) . It follows that Λ2(V,∇) is isomorphic to (O(−1), d). Since (V,∇) is irreducible one has that V ∼= O ⊕O(−1) and the vector bundle V is identified with Oe1 +O(−[∞])e2. Then D := ∇z d dz : V → O([0] + [∞])⊗V has with respect to e1, e2 the matrix ( a b c −a ) with a = a−1z −1 + a0 + a1z, c = c−1z −1 + c0 and b = b−1z −1 + b0 + b1z + b2z 2. The condition at z = 0 is a2 + bc ∈ ( tz−1+θ 2 )2 + C[[z]], equivalently a2−1 + b−1c−1 = t2 4 , 2a−1a0 + b−1c0 + b0c−1 = tθ 2 . The condition at z =∞ is a2 + a+ bc = z + C[[z−1]], equivalently a21 + b2c0 = 0, 2a1a0 + a1 + b2c−1 + b1c0 = 1. The space, given by the above variables and relations has to be divided by the action of the group {e1 7→ e1, e2 7→ λe2 + (x0 + x1z)e1} (with λ ∈ C∗, x0, x1 ∈ C) of automorphisms of the vector bundle. Using the standard forms below one sees that this is a good geometric quotient. Geometric Aspects of the Painlevé Equations PIII(D6) and PIII(D7) 5 A standard form for c−1 6= 0 is z d dz + ( a1z b z−1 + c0 −a1z ) with b = b−1z −1 + · · · + b2z 2 and equations a21 + b2c0 = 0, a1 + b2 + b1c0 = 1, b−1 = t2 4 , t2 4 c0 + b0 = tθ 2 . A standard form for c0 6= 0 is z d dz + ( a−1z −1 b c−1z −1 + 1 −a−1z−1 ) with equations b2 = 0, b1 = 1, a2−1 + b−1c−1 = t2 4 , b−1 + b0c−1 = tθ 2 . By gluing the two standard forms, one obtains the nonsingular moduli space M(θ). The map M(θ)→ S(α), where α = eiθ, is a bijection. Observation. After scaling some variables one sees that M(θ) is the union of two open affine spaces U1 × T and U2 × T , where U1 is given by the variables a1, b1, c0 and the relation a21 + (1 − a1 − b1c0)c0 = 0, and U2 is given by the variables a−1, b0, c−1 and the relation a2−1 + ( θ2 − b0c−1)c−1 − 1 4 = 0. Let U12 ⊂ U1 be defined by c0 6= 0 and U21 ⊂ U2 by c−1 6= 0. The gluing of U1 × T and U2 × T is defined by the isomorphism U12 × T → U21 × T obtained by suitable base changes in the group {e1 7→ e1, e2 7→ λe2 + (x0 + x1z)e1}. Using the two projections U1 → C, (a1, b1, c0) 7→ c0 and U2 → C, (a−1, b0, c−1) 7→ c−1 one finds that U1 and U2 are simply connected. Define M(θ) := f−1(1), where f :M(θ)→ T is the canonical morphism. The space M(θ) is simply connected since it is the union of the two simply connected spaces U1, U2. The universal covering of M(θ) is M̃(θ) = M(θ) ×T T̃ . Indeed, it is the union of the two simply connected spaces U1 × T̃ and U2 × T̃ . Using the explicitly defined link L one obtains a globally defined analytic morphism M̃(θ)→ R(α)× T̃ which is bijective (and thus an analytic isomorphism, see [3]). Indeed, M(θ)→ S(α) and S̃(α)→ R(α)× T̃ are bijections. Theorem 2.2. Let θ ∈ C, α = eπiθ. The extended Riemann–Hilbert map M̃(θ) → R(α) × T̃ , with T̃ ∼= C, is a well defined analytic isomorphism. Comment. The existence of an analytic isomorphism as in Theorem 2.2 is called the “geo- metric Painlevé property” in [1]. They prove this property for a number of Painlevé equations under a restriction on the parameters (loc. cit., Theorem 6.3). We prove it here for PIII(D7) and in Sections 3.3.2 and 3.4.4 below for PIII(D6) without any restriction. 2.3 Isomonodromy and the Okamoto-Painlevé space The calculation is done on the ‘chart’ c0 6= 0 and q is supposed to be invertible. The data for the operator z d dz +A are c−1 = −q, b−1 = q−1 ( a2−1 − t2 4 ) , b0 = − tθ 2 q−1 + q−2 ( a2−1 − t2 4 ) , b1 = 1, b2 = 0. Now q and a−1 are functions of t and the family is isomonodromic if there is an operator d dt +B commuting with z d dz +A. Equivalently A′ = o B + [A,B], where ′ denotes d dt and o denotes z d dz . The Lie algebra sl2 has standard basis H = ( 1 0 0 −1 ) , E1 = ( 0 1 0 0 ) , E2 = ( 0 0 1 0 ) . One writes A = a−1z −1H+bE1+(−qz−1+1)E2 and B = BHH+B1E1+B2E2 with B∗ = 2∑ i=−1 B∗,iz i 6 M. van der Put and J. Top for ∗ = H, 1, 2 and B∗,i only depending on t. Using the Lie algebra structure one obtains the equations: a′−1z −1 = o BH +B2(z + b0 + b−1z −1)−B1(−qz−1 + 1), (H) b′0 + b′−1z −1 = o B1 + 2B1a−1z −1 − 2BH(z + b0 + b1z −1), (E1) − q′z−1 = o B2 − 2B2a−1z −1 + 2BH(−qz−1 + 1). (E2) By Maple one obtains the system q′ = q + 2a−1 t , a′−1 = −t2 − θtq + 4a2−1 + 2qa−1 + 2q3 2tq and finally q′′ = (q′)2 q − q′ t − θ t + 2q2 t2 − 1 q . We note that the change q = −Q, t = −T brings this equation in the form ∗∗ Q = ( ∗ Q)2 Q − ∗ Q T − θ T − 2Q2 T 2 − 1 Q with notation ∗ − = d− dT . This is the standard form for PIII′(D7) (see [6]). As in [8, 9, 12] one obtains: Theorem 2.3. The equation PIII(D7) has the Painlevé property. The analytic fibration t̃ : M̃(θ) → T̃ = C with its foliation {RH−1(r) \| r ∈ R(α)}, where RH : M̃(θ) → R(α) is the Riemann–Hilbert map, is isomorphic to the Okamoto–Painlevé space for the equation PIII(D7) with parameter θ. Moreover, M(θ) ∼= R(α) is the space of initial values. 2.4 Automorphisms of S and Bäcklund transforms The automorphism s1 of S is defined by s1(M, t) = (M,−t). The induced action on R leaves all data invariant except for interchanging the basis vectors f1, f2 of V (0). As a consequence α is mapped to α−1. The automorphism s2 of S is defined by s2(M, t) = (N ⊗M,−t). Here N = C(z)b is the differential module given by δ(b) = 1 2b. Starting with a local presentation z d dz + ( ω 0 0 −ω ) with ω = tz−1+θ 2 of M at z = 0 one obtains, after conjugation with ( z 0 0 1 ) , the local presentation z d dz + ( −τ 0 0 τ ) , with τ = tz−1−θ+1 2 , of N ⊗M at z = 0. Starting with a local presentation D := z d dz + ( 1 4 1 z −1 4 ) of M at z = ∞ (say on the basis E1, E2, described in Section 2.1, part 2), one obtains the local presentation z d dz + ( 3 4 1 z 1 4 ) of N ⊗M at z = ∞. This is the matrix of D with respect to the basis zE2, E1. The induced action of s2 on R maps α to −α−1, the formal monodromy at ∞ is multiplied by −1 and the Stokes data are essentially unchanged. The group of automorphisms of S, generated by s1, s2, has order 4. The Bäcklund transfor- mations are the lifts of the elements of this group to isomorphisms (preserving isomonodromy) between various moduli spaces M̃(θ). Geometric Aspects of the Painlevé Equations PIII(D6) and PIII(D7) 7 s+1 : M̃(θ) → M̃(−θ) is the obvious lift of s1, given by t̃ 7→ t̃ + πi, θ 7→ −θ. Further any solution q(t̃) of PIII(D7) for the parameter θ is mapped to the solution q(t̃+πi) for the parameter −θ. s+2 : M̃(θ) → M̃(1 − θ) is the obvious lift of s2 with t̃ 7→ t̃ + πi. The formula for s+2 is not obvious and its computation is given below. Put B := (s+1 )2 = (s+2 )2. Then B is the automorphism of M̃(θ) = M(θ) × T̃ , which is the identity on M(θ) and B : t̃ 7→ t̃+ 2πi. The group 〈s+1 , s + 2 〉 generated by s+1 , s+2 (for their action on θ, t̃) has 〈B〉 ∼= Z as normal subgroup and 〈s+1 , s + 2 〉/〈B〉 is the affine Weyl group of type A1. Computation of the Bäcklund transformation s+2 . A point ξ ∈ M(θ), lying in the affine open subset defined by c0 6= 0 and c1 6= 0, is represented by the operator in standard form z d dz + ( az−1 b 1− qz−1 −az−1 ) , where b = z − tθ 2q + a2− t 2 4 q2 + a2− t 2 4 q z−1. The map s+2 changes this operator into z d dz + A, where A is obtained from the above matrix by t 7→ −t and adding( 1 2 0 0 1 2 ) . The point s+2 (ξ) ∈ M(1 − θ) is supposed to be represented by the operator z d dz + Ã, where Ã = ( ãz−1 b̃ 1− q̃z−1 −ãz−1 ) with b̃ = z − t(1−θ) 2q̃ + ã2− t 2 4 q̃2 + ã2− t 2 4 q̃ z−1. Since the two matrix differential operators represent the same irreducible differential module over C(z), there is a T ∈ GL(2,C(z)) 6= 0, unique up to multiplication by a constant, such that (z d dz + A)T = T (z d dz + Ã). A local computation shows that T has the form T−2z −2 + T−1z −1 + T0 6= 0 with constant matrices T∗. The ã, q̃ and the entries of the T∗ are the unknows in the identity (z d dz + A)(T−2z −2 + T−1z −1 + T0) = (T−2z −2 + T−1z −1 + T0)(z d dz + Ã). A Maple computation yields q̃ = − t(θq + 2a− t) 2q2 and ã = t(4a2 − 4at+ 2aq + 2θa q + q2θ + t2 − tqθ − qt− 2q3) 4q3 . The isomorphism s+2 respects the foliations. For a leaf one has q′ = q+2a t and substitution in the first formula produces q̃ = − t(θt+tq′−q−t) 2q2 for this Bäcklund transformation on solutions of PIII(D7). The Bäcklund transformation s+2 s + 1 maps a solution q for the parameter θ to the solution t(θq−2a+t) 2q2 , with a = tq′−q 2 , with parameter 1 + θ. 2.5 Remarks 1. One considers for (M, t) ∈ S the connection (V0,∇) with generic fibre M and the local data z d dz + ( 1 4 z 1 −1 4 ) at z =∞ and z d dz + ( ω 0 0 −ω ) with ω = tz−1+θ 2 at z = 0. The second exterior power of (V0,∇) is (O, d) and thus V0 has degree 0. Since (V0,∇) is irreducible, there are two possibilities for V0 namely O ⊕ O and O(1)⊕ O(−1). Suppose that V0 ∼= O(1)⊕ O(−1). Then one can identify V0 with O([∞])e1 + O(−[∞])e2 and for a good choice of e1, e2 one obtains the operator ∇z d dz = z d dz + ( 0 b z−1 0 ) with b = z2 + · · · . One concludes that the locus of the modules M with V0 = O(1) ⊕ O(−1) is the set of the closed points of q−1(∞) for the map q := − c0 c1 :M(θ)→ P1 (see also Section 3.3.4). 8 M. van der Put and J. Top 2. Algebraic solutions of PIII(D7). One easily finds the algebraic solution(s) q with q3 = t2 2 for PIII(D7) with θ = 0. Using the Bäcklund transformations one finds an algebraic solution for PIII(D7) for every θ ∈ Z. According to [5, 6], these are all the algebraic solutions of PIII(D7). More precisely, qj(t̃) = e2πij/3 e 2t̃/3 3√2 , j = 0, 1, 2 are algebraic solution for θ = 0. We note that q1(t̃) = q0(t̃ + 4πi) and q2(t̃) = q0(t̃ + 2πi). Since α = 1 and top3 = 1, these solutions are mapped to a single point of R(1) corresponding to c1c2 = −3, e = −i and certain values for the invariants `12, `14, `23, `34 (which we cannot make explicit). The isomonodromic family for this solution q is z d dz + ( − q 6z −1 b −qz−1 + 1 q 6z −1 ) with b = z + ( 1 36 − q 2 ) + q ( 1 36 − q 2 ) z−1 and q3 = t2 2 . It is not clear what makes this family and the corresponding point of R(1) so special. 3. Special solutions of PIII(D7). Consider an isomonodromy family for which the Stokes matrices are trivial, i.e., c1 = c2 = e = 0. Then α = i or α = −i. In the first case one computes that `12 = `14 = 1 2 , `23 = `34 = −1 2 and one finds a unique point of R(i) and a special solu- tion q(t̃) of PIII(D7) for θ = 1 2 . Using Bäcklund transformations one obtains a similar special solution for any θ ∈ 1 2 + Z. Y. Ohyama informed us that the condition c1 = c2 = 0 implies that the corresponding solution q of PIII(D7) is a univalent function of t and is meromorphic at t = 0. Further θ ∈ 1 2 + Z is equivalent to e = 0. See [2] for details. 3 The family (1,−, 1) 3.1 Definition of the family The set S consists of the equivalence classes of pairs (M, t), where M is a differential module M over C(z) and t ∈ C∗ such that: dimM = 2, Λ2M is the trivial module, M has two singularities 0 and ∞, both singularities have Katz invariant 1, the (generalized) eigenvalues are normalized to ± t 2z −1 at 0 and ± t 2z at ∞. Further, two pairs (M1, t1) and (M2, t2) are called equivalent if there exists an isomorphism M1 →M2 and t1 = t2. As in Section 2, we will have to replace T by its universal covering T̃ = C→ T , t̃ 7→ et̃. Write S̃ = S×T T̃. Define for α, β ∈ C∗ the subset S(α, β) of S consisting of the pairs (M, t) such that C((z)) ⊗M is represented by z d dz + ( ω 0 0 −ω ) with ω = tz−1+θ0 2 , α = eπiθ0 and C((z−1)) ⊗M is represented by z d dz + ( τ 0 0 −τ ) with τ = tz+θ∞ 2 , β = eπiθ∞ . Further S̃(α, β) = S(α, β)×T T̃. 3.2 The monodromy space For (M, t) ∈ S, the monodromy data are given by (compare [10]): the symbolic solutions spaces V (0) and V (∞) at z = 0 and z =∞ (including formal monodromies and Stokes matrices) and the link L : V (0)→ V (∞). We make this more explicit. The module C((z)) ⊗M has a basis E1, E2 with δ(E1 ∧ E2) = 0 and δE1 = − tz−1+θ0 2 E1, δE2 = tz−1+θ0 2 E2. We note that t is used to distinguish between E1 and E2. This basis is unique up to a transformation E1 7→ c1z mE1, E2 7→ c2z −mE2 with c1, c2 ∈ C∗, m ∈ Z. After fixing θ0, the E1, E2 are unique up to multiplication by constants. The symbolic solution space V (0) at z = 0 is Ce1 + Ce2, with e1 = e− t 2 z−1+ θ0 2 log zE1 and e2 = e+ t 2 z−1− θ0 2 log zE2. Geometric Aspects of the Painlevé Equations PIII(D6) and PIII(D7) 9 Now α = eπiθ0 is well defined and does not depend on the choices for E1, E2. Similarly, C((z−1)) ⊗M = C((z−1)F1 ⊕ C((z−1))F2 with δ(F1 ∧ F2) = 0, δF1 = − tz+θ∞ 2 F1 and δF2 = tz+θ∞ 2 F2. The space V (∞) has basis f1 = e tz 2 + θ∞ 2 log zF1 and f2 = e− tz 2 − θ∞ 2 log zF2 over C. Moreover, β = eπiθ∞ . For the basis e1, e2 of V (0), the formal monodromy and the Stokes matrices are:( α 0 0 1 α ) , ( 1 0 a1 1 ) , ( 1 a2 0 1 ) with product ( α αa2 a1 α 1+a1a2 α ) . This product is the topological monodromy top0 at z = 0. For the basis f1, f2 of V (∞), the formal monodromy and the Stokes matrices are:( β 0 0 1 β ) , ( 1 0 b1 1 ) , ( 1 b2 0 1 ) with product ( β βb2 b1 β 1+b1b2 β ) . This is the topological monodromy top∞ at z =∞. The link L : V (0)→ V (∞) with matrix ( `1 `2 `3 `4 ) has determinant 1. The relations are given by the matrix equality( β βb2 b1 β 1+b1b2 β ) = L ◦ ( α αa2 a1 α 1+a1a2 α ) ◦ L−1. In particular, β = `1`4α + `2`4 a1 α − `1`3αa2 − `2`3 1+a1a2 α . This defines a variety T , given by the variables α, a1, a2, `1, . . . , `4 with the only restrictions `1`4 − `2`3 = 1, α 6= 0 and `1`4α + `2`4 a1 α − `1`3αa2 − `2`3 1+a1a2 α 6= 0. For fixed values of α, β ∈ C∗ we obtains a variety T (α, β) defined by the variables a1, a2, `1, . . . , `4 and the relations: `1`4 − `2`3 = 1 and β = `1`4α+ `2`4 a1 α − `1`3αa2 − `2`3 1 + a1a2 α . The group Gm×Gm acts on T and T (α, β), by base change (e1, e2, f1, f2) 7→ (γe1, γ −1e2, δf1, δ−1f2). The categorical quotient of T by Gm×Gm is R → P with parameter space P = C∗×C∗ given by (α, β). This is a family of affine cubic surfaces R(α, β) (this is the categorical quotient of T (α, β)) given by the equation x1x2x3 + x21 + x22 + (1 + αβ)x1 + (α+ β)x2 + αβ = 0, where x1 = `1`4 − 1, x2 = αa2`1`3 − α`1`4, x3 = 1 + a1a2 α + α. Observation: R(α, β) is simply connected if α 6= β±1. Define U by removing the two lines {(0,−β, ∗)}, {(−1, 0, ∗)} from R(α, β). The image of the projection (x1, x2, x3) ∈ U 7→ (x1, x2) ∈ C2 is C∗ × C∗ ∪ {(0,−α), (−αβ, 0)}. This image is simply connected. For x1x2 6= 0, the fiber is one point. For x1x2 = 0, the fiber is an affine line. It follows that U is simply connected and thus R(α, β) is simply connected, too. Definition of the link and the topological monodromies. A construction similar to the one in Section 2 is needed for the definition of the link. For t̃ = \|t\|eiφ ∈ T̃ , φ ∈ R the singular directions at z = 0 and z = ∞ are φ, φ − π and −φ, π − φ. On the universal covering of P1 \ {0,∞} one considers the path z̃ = reid(r), 0 < r < ∞ with d(r) = 1 1+r (φ − π 2 ) + r 1+r (π2 − φ). The link L : V (0) → V (∞) is defined by (multi)summation at zero in the direction φ − π 2 , followed by 10 M. van der Put and J. Top analytic continuation along the above path and finally the inverse of (multi)summation in the direction π 2 − φ at infinity. Now the map S̃(α, β)→ R(α, β)× T̃ is well defined. In the general case, i.e., α 6= β±1, the space R(α, β) is the geometric quotient of T (α, β) and this space is nonsingular. Therefore the natural map S̃(α, β)→ R(α, β)× T̃ is a bijection. Let (M, t̃) ∈ S̃(α, β). Then M is reducible if and only its monodromy data (in R(α, β)) is reducible. Further R(α, β) contains reducible monodromy data if and only if α = β±1. Thus S(α, β) contains reducible modules if and only if α = β±1. We will first investigate the general case α 6= β±1. The special case α = β±1 presents many difficulties and will be handled later on. 3.3 The general case α 6= β±1 3.3.1 The moduli space M(θ0, θ∞) Fix θ0, θ∞ with α = eiπθ0 , β = eiπθ∞ . We will construct a moduli spaceM(θ0, θ∞) of connections on the bundle Oe1 ⊕ O(−[0])e2 on P1 such that the map, which associates to a connection in this space its generic fiber, is a bijection M(θ0, θ∞)→ S(α, β). The elements of the set M(θ0, θ∞) are the connections ∇ : V → Ω(2[0] + 2[∞])⊗ V defined by: the generic fiber M satisfies (M, t) ∈ S(α, β); the invariant lattices at z = 0 and z = ∞ are given by the local matrix differential operators z d dz + ( tz−1+θ0 2 0 0 − tz−1+θ0 2 + 1 ) and z d dz + ( tz+θ∞ 2 0 0 − tz+θ∞ 2 ) . Since the second exterior power of M is trivial, the degree of V is −1. By assumption M is irreducible and therefore the type of V is O⊕O(−1) and one can identify V with Oe1 ⊕O(−[0])e2. By construction the map M(θ0, θ∞)→ S(α, β) is bijective. The operator ∇z d dz has, with respect to the basis {e1, e2}, the form z d dz + ( a b c −a ) with a = a−1z −1 + a0 + a1z, b = b−2z −2 + · · · + b1z, c = c0 + c1z. The lattice condition at z = 0 is equivalent to a(a− 1) + bc ∈ ( tz −1+θ0 2 )2 − ( tz −1+θ0 2 ) + C[[z]]. This leads to the equations a2−1 + b−2c0 = t2 4 , 2a−1a0 − a−1 + b−2c1 + b−1c0 = t ( θ0 2 − 1 2 ) . The lattice condition at z =∞ is equivalent to a2 + bc ∈ ( tz+θ∞2 )2 + C[[z−1]] and one finds the equations a21 + b1c1 = t2 4 , 2a0a1 + b0c1 + b1c0 = tθ∞ 2 . Define the (quasi-)affine space A(θ0, θ∞) by the variables a∗, b∗, c∗, t, the four equations and the open condition (c0, c1) 6= (0, 0). For the general case α 6= β±1, the module M and the corresponding connection are irreducible and thus (c0, c1) 6= (0, 0) holds (see Remarks 3.2.3). The space A(θ0, θ∞) is divided out by the group of the automorphisms of the bundle Oe1 ⊕ O(−[0])e2. This action amounts to dividing A(θ0, θ∞) by the group G of transformations e1 7→ e1, e2 7→ λe2 + (γz−1 + δ)e1. Proposition 3.1. The quotient of A(θ0, θ∞) by G is geometric and has no singularities. Proof. The open subset A(θ0, θ∞)1, defined by c1 6= 0, contains the closed ‘standard subset’ ST1, given by the connections z d dz + ( a−1z −1 b z + c0 −a−1z−1 ) with b = b−2z −2 + · · ·+ b1z, b1 = t2 4 , b0 = − t 2 4 c0 + tθ∞ 2 , b−2 = a−1 − b−1c0 + t ( θ0 2 − 1 2 ) . Geometric Aspects of the Painlevé Equations PIII(D6) and PIII(D7) 11 and the equation a2−1 + c0 ( a−1 − b−1c0 + t ( θ0 2 − 1 2 )) − t2 4 = 0. The natural morphism G× ST1 →M(θ0, θ∞)1 is an isomorphism. Similarly, let ST0 denote the closed subset of the open subset A(θ0, θ∞)0, defined by c0 6= 0, be given by the connections z d dz + ( a1z b c1z + 1 −a1z ) with b = b−2z −2 + · · ·+ b1z, b−2 = t2 4 , b−1 = t ( θ0 2 − 1 2 ) − t2 4 c1, b1 = tθ∞ 2 − b0c1 and the equation a21 + ( tθ∞ 2 − b0c1 ) c1 − t2 4 = 0. Again G × ST0 → A(θ0, θ∞)0 is an isomorphism. The quotient of A(θ0, θ∞) by G is obtained by gluing the two non singular spaces ST1 and ST0. � NowM(θ0, θ∞) is, as algebraic variety, defined as the quotient of A(θ0, θ∞) by G. Since this is a geometric quotient, the map M(θ0, θ∞)(C)→ S(α, β) is bijective. Remarks 3.2. 1. As in Section 2, the Observation, one sees that, after scaling some of the variables, the two charts ST1, ST0 of M(θ0, θ∞), have the form U1 × T , U2 × T with simply connected spaces U1, U2. Define M(θ0, θ∞) = f−1(1), where f : M(θ0, θ∞) → T is the canonical morphism. Then M(θ0, θ∞) is simply connected. Further M̃(θ0, θ∞) :=M(θ0, θ∞)×T T̃ is the universal covering of M(θ0, θ∞). 2. Let q :M(θ0, θ∞)→ P1 denote the morphism given by q = − c0 c1 . 3. In the cases α = β±1, the spaceA(θ0, θ∞) is defined as before, and including the assumption (c0, c1) 6= (0, 0). The space M(θ0, θ∞) is again the geometric and non singular quotient of A(θ0, θ∞) by G. The canonical map M(θ0, θ∞)(C) → S(α, β) is injective and, in general, not surjective. Indeed, the open condition (c0, c1) 6= (0, 0) is valid for (M, t) ∈ S(α, β) such that M is irreducible but may exclude certain reducible modules in S(α, β). We note that c0 = c1 = 0 implies ± θ∞ 2 = θ0 2 − ε with ε ∈ {0, 1}. 3.3.2 The Okamoto–Painlevé space M̃(θ0, θ∞) for θ0 2 ± θ∞ 2 6∈ Z The moduli space M(θ0, θ∞) is replaced by M̃(θ0, θ∞) := M(θ0, θ∞) ×T T̃ . The bijections S̃(α, β)→ R(α, β)×T̃ andM(θ0, θ∞)→ S(α, β) imply that the analytic morphism M̃(θ0, θ∞)→ R(α, β)× T̃ is bijective and hence an analytic isomorphism. As in Theorem 2.3, using arguments presented in [8, 9, 12] this implies the following result. Theorem 3.3. Suppose that θ0 2 ± θ∞ 2 6∈ Z (equivalently α 6= β±1). Then M̃(θ0, θ∞) → T̃ , provided with the foliation given by the fibers of M̃(θ0, θ∞) → R(α, β) (i.e., the isomonodromy families), is the Okamoto–Painlevé space corresponding to the equation PIII(D6), namely q′′ = (q′)2 q − q′ t − 4(θ0 − 1) t + 4θ∞q 2 t + 4q3 − 4 q . Moreover, this equation satisfies the Painlevé property. 12 M. van der Put and J. Top Observations 3.4. 1. The above formula differs slightly from the one given in [10, Section 4.5]. This is due to different choices of the standard matrix differential operator. 2. The transformation t 7→ −t, θ∞ 7→ −θ∞ and θ0 7→ −θ0 + 2 leaves the family of matrix differential operators invariant. This has the consequence that a solution q(t) of PIII(D6) with parameters θ0 and θ∞, yields the solution q(−t) of PIII(D6) with parameters −θ0 + 2 and −θ∞. This can also be seen directly from the differential equation. 3. The solutions q(t) of PIII(D6) are in fact meromorphic functions in t̃ ∈ T̃ = C. Thus Q(t̃) := q(et̃) is well defined and satisfies the equation Q′′ = (Q′)2 Q − 4(θ0 − 1)et̃ + 4θ∞Q 2et̃ + 4Q3e2t̃ − 4e2t̃ Q , where ′ = d dt̃ . 4. The space of initial conditions is analytically isomorphic to R(α, β) and can also be identified with M(θ0, θ∞). Indeed, the extended Riemann–Hilbert isomorphism M̃(θ0, θ∞) → R(α, β)× T̃ induces an analytic isomorphism M(θ0, θ∞)→ R(α, β). 3.3.3 Verif ication of the formula in Theorem 2.3 On the chart ST1 of M(θ0, θ∞) the matrix differential operator has the form z d dz + A = z d dz + a−1z −1H + bE1 + (z − q)E2, where H = ( 1 0 0 −1 ) , E1 = ( 0 1 0 0 ) , E2 = ( 0 0 1 0 ) . We will use [H,E1] = 2E2, [H,E2] = −2E2, [E1, E2] = H. Further q := −c0, b = b−2z −2 + b−1z −1 + b0 + b1z, where b−2 6= 0 (by assumption), b1 = t2 4 , b0 = q t2 4 + tθ∞ 2 , b−2 = a−1 + qb−1 + t ( θ0 2 − 1 2 ) , a2−1 − q ( a−1 + qb−1 + t ( θ0 2 − 1 2 )) − t2 4 = 0. Now q, a−1 are considered as (meromorphic) functions of t and A is a matrix depending on z and t. The family z d dz + A is isomonodromic if and only if there is an operator of the form d dt + B = d dt + BHH + B1E1 + B2E2 which commutes with z d dz + A. This is equivalent to d dt(A) = z d dz (B) + [B,A]. FurtherB∗ = BH , B1, B2 are functions of t, z and are supposed to have the formB∗,−2(t)z −2+ B∗,−1(t)z −1+B∗,0(t)+B∗,1(t)z. The two operators commute if and only if a′−1z −1H+b′E1−q′E2 is equal to o BHH + o B1E1 + o B2E2 − [ BHH +B1E1 +B2E2, a−1z −1H + bE1 + (z − q)E2 ] , where o X stands for z d dz (X) and X ′ := d dt(X). On obtains the equations (H) a′−1z −1 = o BH −B1(z − q) +B2b, (E1) b′ = o B1 − 2BHb+ 2B1a−1z −1, (E2) − q′ = o B2 + 2BH(z − q)− 2B2a−1z −1. A Maple computation shows that this system of differential equations for q, a−1 is equivalent to q′ = 4a−1 − q t , a′−1 = 4a2−1 − t2 + q(t− a−1 − tθ0) + q3tθ∞ + q4t2 tq . Geometric Aspects of the Painlevé Equations PIII(D6) and PIII(D7) 13 The equation q′′ = (q′)2 q − q′ t − 4(θ0 − 1) t + 4θ∞q 2 t + 4q3 − 4 q follows by substitution. Using the transformation z 7→ z−1 one finds that Q = 1 q satisfies the PIII(D6) equation with θ0 − 1 and θ∞ interchanged: Q′′ = (Q′)2 Q − Q′ t − 4θ∞ t + 4(θ0 − 1)Q2 t + 4Q3 − 4 Q . 3.3.4 After a remark by Yousuke Ohyama Let (M, t) ∈ S(α, β) with α = eπiθ0 , β = eπiθ∞ have the property that M is irreducible. Consider the connection (W,∇) with generic fiber M and locally represented by z d dz + ( tz−1+θ0 2 0 0 − tz−1+θ0 2 ) at z = 0 and z d dz + ( tz+θ∞ 2 0 0 − tz+θ∞ 2 ) at z =∞. The second exterior product of (W,∇) is trivial and thus Λ2W has degree 0. Since M is irreducible one has W ∼= O(k)⊕O(−k) with k ∈ {0, 1}. Suppose that the connection W has type O(1) ⊕ O(−1). Then we can identify W with O([0])B1⊕O(−[0])B2. Put D := ∇z d dz . Now DB1 is not a multiple of B1 since M is irreducible. After multiplying B1 with a scalar, the matrix of D with respect to the basis B1, B2 has the form ( α β z −α ) with α = α−1z −1 + α0 + α1z, β = β−3z −3 + · · ·+ β1z. The base vector B2 can be replaced by B2 + hB1 with h = h0 + h−1z −1 + h−2z −2. The result is a new representation of D, namely( 1 h 0 1 )−1{ z d dz + ( α β z −α )}( 1 h 0 1 ) = z d dz + ( α− hz 2αh− h2z + z dhdz + β z −α+ hz ) . For unique h0, h−1 and at most two values of h−2, the last operator is z d dz + ( a−1z −1b z − a−1z−1 ) , where b = b−2z −2 + b−1z −1 + b0 + b1z. Let e1, e2 denote the new basis. Then V = Oe1 ⊕ O(−[0])e2 and the corresponding point ξ ∈ M(θ0, θ∞) satisfies q(ξ) = 0. The converse holds, too. One observes that q−1(0) ⊂ M(θ0, θ∞) has two connected components, each one isomorphic to A1 × T . We note that the map Q := 1 q can also be used in this context, since for a monodromic family Q satisfies a PIII(D6) equation (see the end of Section 3.3.3). According to Malgrange, the locus where the bundleW is not free is the tau-divisor. Thus we find that the tau-divisor coincides with the locus Q−1(∞) ⊂ M(θ0, θ∞). The statement: ‘the tau-divisor coincides with q−1(∞)’ holds for PI, PII, PIII(D7), PIII(D8), PIV, too (see [8, 9, 12]). 3.4 The cases α = β±1 3.4.1 Geometric quotients of the monodromy data We use here the notation of Section 3.2. 1. If α = β 6= ±1, then R(α, α) has a singular point, namely (x1, x2, x3) = (0,−α, α+ α−1). The preimage in T (α, α) of this singular point consists of the tuples (a1, a2, `1, . . . , `4) such that 14 M. van der Put and J. Top the matrices L, top0 have the form ( `10 `3`4 ) , ( α 0 a1 α 1 α ) or ( `1 `2 0 `4 ) , ( α αa2 0 1 α ) . In particular, R(α, α) is not a geometric quotient. The remedy consists of replacing T (α, α) by T (α, α)∗ which is the complement of the closed subset of T (α, α) given by the equations `2 = `3 = a1 = a2 = 0. We claim that T (α, α)∗ has a nonsingular geometric quotient by the action of Gm × Gm. A proof is obtained by writing T (α, α)∗ as the union of the four affine open subsets `2 6= 0, `3 6= 0, a1 6= 0 and a2 6= 0. On each of these subsets one explicitly computes the quotient by Gm × Gm, which turns out to be nonsingular and geometric. Gluing these four quotients produces the required geometric quotient which will be denoted by R(α, α)∗. Let S(α, α)∗ the complement in S(α, α) of the set of the modules which are direct sums and S̃(α, α)∗ = S(α, α)∗ ×T T̃ . Then the canonical map S̃(α, α)∗ → R(α, α)∗ × T̃ is bijective. Define the closed space T (α, α)∗red of T (α, α)∗ by the condition that the data is reducible. This space has two irreducible components, given in terms of the matrices L, top0 by: (a) ( `1 0 `3 `4 ) , ( α 0 a1 α 1 α ) with `1`4 = 1 and (`3, a1) 6= 0. One easily verifies that the map which sends (L, top0) to (`3 : a1) ∈ P1 is the geometric quotient. (b) ( `1 `2 0 `4 ) , ( α αa2 0 1 α ) with `1`4 = 1 and (`2, a2) 6= 0. One easily verifies that the map which sends the (L, top0) to (`2 : a2) ∈ P1 is the geometric quotient. Therefore the ‘reducible locus’ R(α, α)∗red (i.e., corresponding to reducible monodromy data) is the union of two, not intersecting, projective lines. 2. The case α = β−1 6= ±1 can be handled as in (1). One finds (with a similar notation) a geometric quotient R(α, α−1)∗ of T (α, α−1)∗ and a bijection S̃(α, α−1)∗ → R(α, α−1)∗ × T̃ . Further R(α, α−1)∗red is the union of two, non intersecting, projective lines. 3. α = β = 1. The categorical quotient R(1, 1) of T (1, 1) has two singular points, namely (x1, x2, x3) = (0,−1, 2) and (x1, x2, x3) = (−1, 0, 2). The preimage of the first singular point con- sists of the pairs (L, top0) equal to (( `1 `2 0 `4 ) , ( 1 a2 0 1 )) or to (( `1 0 `3 `4 ) , ( 1 0 a1 1 )) . The pre- image of the second singular point consists of the pairs (L, top0) equal to (( 0 `2 `3 `4 ) , ( 1 a2 0 1 )) or to (( `1 `2 0 `4 ) , ( 1 0 a1 1 )) . Clearly, R(1, 1) is not a geometric quotient. The locus of the points in T (1, 1) which describe the monodromy data for modules in S(1, 1) which are direct sums is the union of the two closed sets a1 = a2 = `2 = `3 = 0 and a1 = a2 = `1 = `4 = 0. Let T (1, 1)∗ ⊂ T (1, 1) denote the complement of this locus. This set is the union of the six open subsets given by the inequalities a1 6= 0, a2 6= 0, `12 6= 0, `13 6= 0, `24 6= 0 and `34 6= 0. The group Gm×Gm acts on each of these open affine sets and the categorical quotient is a geometric quotient and is nonsingular. Therefore the quotient R(1, 1)∗ of T (1, 1)∗, obtained by gluing the six quotients, is a geometric quotient and nonsingular. Example. The open subset `12 6= 0 is defined by the variables `1, . . . , `4, a1, a2 and relations 0 = −`23a12 + `24a1 − `13a2, `14 − `13 − 1 = 0, `12 6= 0. Division by Gm × Gm is equivalent to the normalisation `1 = `2 = 1. Elimination of `4 by `4 = `3 + 1 yields the equation `3(a1 − a2 + a1a2) + a1 = 0. This is a nonsingular surface. Using the projection (`3, a1, a2) 7→ (a1, a2) one finds that this surface is simply connected. Similar computations lead to the statements: R(1, 1)∗ is a nonsingular geometric quotient and is simply connected. The natural map S̃(1, 1)∗ → R(1, 1)∗ × T̃ is a well defined bijection. The reducible locus R(1, 1)∗red is the union of four, non intersecting, projective lines. 4. α = β = −1. The categorical quotient R(−1,−1) has two singular points, namely (x1, x2, x3) = (0, 1,−2) and (x1, x2, x3) = (−1, 0,−2). As in (3), one defines T (−1,−1)∗ and its Geometric Aspects of the Painlevé Equations PIII(D6) and PIII(D7) 15 geometric non singular quotient R(−1,−1)∗. The space R(−1,−1)∗ contains four, non inter- secting, projective lines. These lines correspond to the reducible locus of R(−1,−1)∗. As in (3) one defines S̃(−1,−1)∗ and concludes: R(−1,−1)∗ is a nonsingular geometric quotient and is simply connected. The natural map S̃(−1,−1)∗ → R(−1,−1)∗ × T̃ is a bijection. 3.4.2 Reducible modules in S We use here the notation of Sections 3.2 and 3.4.1. Observations 3.5. Let N ⊂ M be a 1-dimensional submodule, then C((z)) ⊗ N = C((z))Ei and C((z−1)) ⊗ N = C((z−1))Fj with i, j ∈ {1, 2}. Since N has no other singularities than 0, ∞ one has N = C(z)n with δ(n) = (±tz −1±tz 2 + d)n, where d ∈ C is unique modulo Z. Indeed, δ(n) = (±tz−1±tz 2 + f ) n, where f ∈ C(z) has no poles at 0 and ∞. Using that N has only singularities at 0 and ∞, one can change the generator n of N such that f is a constant d. Any other base vector of N with this property has the form zkn with k ∈ Z. Further d ∈ ±θ0/2 + Z and d ∈ ±θ∞/2 + Z and hence α = β±1. Proposition 3.6 (Reducible modules). 1. A module (M, t) ∈ S is reducible if and only if there are i, j ∈ {1, 2} such that ai = 0, bj = 0 and L(Cei) = Cfj. 2. Let (M, t) ∈ S be reducible, but not a direct sum of two submodules of dimension one. Then there are unique elements ε1, ε2 ∈ {−1, 1}, a complex number d, unique modulo Z, and a polynomial c = c1z+c0 6= 0, unique up to multiplication by a scalar, such that M is represented by the matrix differential operator z d dz + ( − ε1tz−1+ε2tz 2 − d 0 c1z + c0 ε1tz−1+ε2tz 2 + d ) . 3. For α 6= ±1, the reducible locus S(α, α)∗red of S(α, α)∗ is represented by the union of the two families in (2) given by e2πid = α, ε1 = ε2 = 1 and ε1 = ε2 = −1. Each of the two families is isomorphic to P1 × T , by sending the matrix differential operator to ((c1 : c0), t) ∈ P1 × T . The isomorphism S̃(α, α)∗red → R(α, α)∗red × T̃ yields two isomorphism P1 × T̃ → P1 × T̃ . These have the form (p, t̃) 7→ (A(t̃)p, t̃) where A(t̃) is an automorphism of P1, depending on t̃. 4. A similar result holds for α 6= ±1 and α = β−1. 5. S(1, 1)∗red is represented by the families d ∈ Z and the four possibilities of ε1, ε2. Then S̃(1, 1)∗red identif ies with the disjoint union of four copies of P1 × T̃ . The same holds for R(1, 1)∗red× T̃ . The isomorphism S̃(1, 1)∗red → R(1, 1)∗red× T̃ yields four isomorphism P1× T̃ → P1 × T̃ . These have the form described in (3). 6. The case α = β = −1 is similar to case (5). Proof. 1. This follows from Observations 3.5 and the statement that a differential module over C(z) is determined by its monodromy data (i.e., ordinary monodromy, Stokes matrices and links) and the formal classification of the singular points (see [10, Theorem 1.7]). 2. For convenience we consider the case i = j = 1 of (1). Using the above Observation, one finds that M has a basis m1, m2 such that z∂(m2) = am2 and z∂(m1) = −am1 + fm2 with a := tz−1+tz 2 + d and f ∈ C(z). If we fix d, then m2 is unique up to multiplication by a scalar. Further, m1 is unique up to a transformation m1 7→ λm1 + hm2 with λ ∈ C∗, h ∈ C(z). This transformation changes f into λf + 2ah+ zh′. We start considering the subgroup of transformations with λ = 1 and h ∈ C(z). For a suitab- le h the term f1 := f + 2ah+ zh′ has in C∗ at most poles of order one. A pole of order one of f1 in C∗ cannot disappear by a transformation of the form under consideration. Since M has only 16 M. van der Put and J. Top singularities at 0 and ∞ we conclude that f1 ∈ C[z, z−1]. For suitable h ∈ C[z, z−1] the term c := f1 + 2ah+ zh′ is a polynomial of degree ≤1 and is 6=0, by assumption. For any h ∈ C(z), h 6= 0 the term c + 2ah + zh′ is not a polynomial of degree ≤1. This yields a unique c for this subgroup of transformations. Finally, the transformation m1 7→ λm1 shows that c is unique up to multiplication by a scalar. Cases (3)–(6) are consequences of the above computation of the R(α, β)∗red. � 3.4.3 The reducible connections in M(θ0, θ∞) The image of the injective map M(θ0, θ∞) → S(α, β)∗, where α = eπiθ0 , β = eπiθ∞ , will be denoted by S(θ0, θ∞). By Remarks 3.2 part (3), this image contains the irreducible modules (M, t). For α 6= β±1, S(θ0, θ∞) is equal to S(α, β) = S(α, β)∗. Now we consider the other cases. Proposition 3.7. Suppose α = β 6= ±1. Then S(θ0, θ∞) consists of: (a) the irreducible elements (M, t); (b) the (ε1, ε2) = (1, 1) family of reducible modules ⇔ θ0 − θ∞ ≥ 2; (c) the (ε1, ε2) = (−1,−1) family of reducible modules ⇔ θ0 − θ∞ ≤ 0. Proof. Part (a) is known. Consider an element (M, t) in the (ε1, ε2) = (1, 1) family of reducible modules. Let the connection (V,∇), corresponding to (M, t), be defined as in Section 3.3.1. Then V can be identified with the vector bundle O(k[0])e1 +O((−k − 1)[0])e2 for some integer k ≥ 0. Then (M, t) lies in S(θ0, θ∞) ⇔ k = 0. Consider the case k > 0. Then ∇z d dz e1 = (a−1z −1+a0+a1z)e1 and ∇z d dz (z−ke1) = (a−1z −1+ a0 + a1z − k)(z−ke1). Comparing with the prescribed local operator at z = 0 yields the possibilities: a−1z −1 + a0 equals (i) t 2z −1 + θ0 2 or (ii) − t 2z −1 − θ0 2 + 1. Comparing with the prescribed local operator at z = ∞ yields the possibilities: a0 + a1z equals (A) t 2z + θ∞ 2 or (B) − t 2z − θ∞ 2 . Combining one obtains (i), (A) a−1 = t 2 , a1 = t 2 , θ0 = θ∞ − 2k, (ε1, ε2) = (1, 1), (i), (B) a−1 = t 2 , a1 = − t 2 , θ0 = −θ∞ − 2k, (ε1, ε2) = (1,−1), (ii), (A) a−1 = − t 2 , a1 = t 2 , θ0 = −θ∞ + 2k + 2, (ε1, ε2) = (−1, 1), (ii), (B) a−1 = − t 2 , a1 = − t 2 , θ0 = θ∞ + 2k + 2, (ε1, ε2) = (−1,−1). Since α = β 6= ±1, reducible modules of types (1,−1) and (−1, 1) are not present in S(α, α)∗. Now we consider the presence of reducible modules of type (1, 1) in S(θ0, θ∞).The condition θ0 − θ∞ ≥ 0 is necessary because of (i), A. Consider the case θ0 = θ∞ = 2d. A reducible module of type (1, 1) yields a connection on W = Of1 ⊕ Of2 with the local data z d dz + ( −( t2z −1 + d) 0 ∗ t 2z −1 + d ) at z = 0 and z d dz +( −( t2z + d) 0 ∗ t 2z + d ) at z = ∞. Now V = O(−[0])f1 ⊕ Of2 = Oe1 ⊕ O(−[0])e2, with e1 = f2, e2 = f1, has the required local data and the matrix of ∇z d dz with respect to the basis e1, e2 is( ω ∗ 0 −ω ) . Thus c0 = c1 = 0 and the reducible modules of type (1, 1) are not present according to the construction of M(θ0, θ∞). Geometric Aspects of the Painlevé Equations PIII(D6) and PIII(D7) 17 Consider the case θ0 = θ∞+2. The standard form of Proposition 3.6.2 for type (1, 1) belongs to M(2d + 2, 2d). Further, see Remarks 3.2.3, (c0, c1) 6= (0, 0) holds for θ0 − θ∞ > 2. This proves (b). The proof of (c) is similar. � Similarly one shows: α = β−1 6= ±1. Then S(θ0, θ∞) consists of: (a) the irreducible elements (M, t); (b) the (ε1, ε2) = (1,−1) family of reducible modules ⇔ θ0 + θ∞ ≥ 2; (c) the (ε1, ε2) = (−1, 1) family of reducible modules ⇔ θ0 + θ∞ ≤ 0. α = β = ±1. Then S(θ0, θ∞) consists of: (a) the irreducible elements (M, t); (b) the (ε1, ε2) = (1, 1) family of reducible modules ⇔ θ0 − θ∞ ≥ 2; (c) the (ε1, ε2) = (1,−1) family of reducible modules ⇔ θ0 + θ∞ ≥ 2; (d) the (ε1, ε2) = (−1, 1) family of reducible modules ⇔ θ0 + θ∞ ≤ 0; (e) the (ε1, ε2) = (−1,−1) family of reducible modules ⇔ θ0 − θ∞ ≤ 0. 3.4.4 M(θ0, θ∞) for α = β±1 Let R(θ0.θ∞) denote the open subspace of R(α, β)∗ which corresponds to the subset S(θ0, θ∞) of S∗(α, β), defined in Section 3.4.3. By Sections 3.4.1–3.4.3, the extended Riemann–Hilbert morphism M̃(θ0, θ∞) → R(θ0, θ∞) × T̃ is a well defined analytic isomorphism. This has as consequence: Theorem 3.3 holds for the cases θ0 2 ± θ∞ 2 ∈ Z with R(α, β) replaced by R(θ0, θ∞). 3.4.5 Isomonodromy for reducible connections The fibers of the locally defined map S(α, α±1)∗red → R(α, α±1)∗red are the isomonodromy families of reducible modules. As a start, we consider the reducible familly of type (ε1, ε2) = (1, 1) lying in M(2d + 2, 2d). This family is represented by z d dz + ( − tz−1+tz 2 − d 0 z − q tz−1+tz 2 + d ) . For an isomonodromy subfamily of this, q is a function of t and the Stokes data at 0 and ∞ and the link are fixed. Isomonodromy is equivalent to the statement that the above matrix differential operator commutes with an operator of the form d dt +B−1z+B0 +B1z, where the tracefree 2×2 matrices B−1, B0, B1 depend on t only. This leads to the equation( − z−1+z 2 0 −q′ z−1+z 2 ) = −B1z −1 +B1z − [ B−1z −1 +B0 +B1z, ( − tz−1+tz 2 − d 0 z − q z−1+z 2 + d )] . A computation yields the equation q′ = −2q2 − 4d−1 t q − 2. The solutions of this equation have the form 1 2 y′ y , where y is a non zero solution of the Bessel equation y′′ + 4d−1 t y′ + 4y = 0. One obtains in a similar way for an isomonodromic family of reducible modules of type (ε1, ε2) the equation q′ = −2ε2q 2 − 4d− 1 t q − 2ε1. 18 M. van der Put and J. Top The solutions are q = ε2 2 y′ y where y is a solution of the Bessel equation y′′+ 4d+1 t y′+ 4ε1ε2y = 0. These equations are consistent with the formula of Theorem 3.3 q′′ = (q′)2 q − q′ t − 4(θ0 − 1) t + 4θ∞q 2 t + 4q3 − 4 q for isomonodromic families in M(θ0, θ∞). According to [5] we found in this way all Riccati solutions for PIII(D6), up to the action of the Bäcklund transformations. Remark 3.8. The assumption that a function q satisfies two distinct PIII(D6) equations leads to q4 = 1. Thus we found the algebraic solutions q = ±1 for θ∞ = θ0 − 1 and q = ±i for −θ∞ = θ0−1. According to [5] these are all the algebraic solutions of PIII(D6), up to the action of the Bäcklund transformations. 4 Bäcklund transformations for PIII(D6) 4.1 Automorphisms of S We start with a table of generators for the group Aut(S) of ‘natural’ automorphism of S, in terms of their action on the parameters α, β and on t, z. α β t z σ1 α−1 β−1 −t z σ2 −α −β t z σ3 α β−1 it iz σ4 β α t z−1 These generators are defined as follows. 1. σ1 : (M, t) 7→ (M,−t). This induces bijections S(α, β) → S(α−1, β−1). Indeed, the basis vectors e1, e2 of V (0) are interchanged and the same holds for the basis f1, f2 of V (∞). 2. Define the differential module N = C(z)b by δb = 1 2b. Then σ2 : (M, t) 7→ (M ⊗ N, t). Since Λ2(M ⊗N) = N⊗2 is the trivial module, (M ⊗N, t) belongs to S. Let E1, E2 be a basis of C((z)) ⊗M such that δE1 = − tz−1+θ0 2 E1 and δE2 = tz−1+θ0 2 E2. Then the formal module C((z))⊗ (M ⊗N) has basis E1⊗ b, E2⊗ b and δ(E1⊗ b) = (− tz−1+θ0 2 + 1 2)(E1⊗ b) and similarly δ(E2⊗b) = ( tz −1+θ0 2 + 1 2)(E2⊗b). Thus eπi(θ0+1) = −α is the eigenvalue of the formal monodromy at z = 0. The same argument shows that −β is the eigenvalue of the formal monodromy at z =∞. 3. Let φ be a C-linear automorphism of the field C(z), such that z d dz ◦ φ = µ · φ ◦ z d dz for some µ ∈ C∗. There are two possibilities: (a) φ(z) = cz with c ∈ C∗, µ = 1, (b) φ(z) = cz−1 with c ∈ C∗, µ = −1. For (M, t) ∈ S one considers (C(z) ⊗φ M, . . . ). As additive group C(z) ⊗φ M is identified with M . Its structure as vector space is given by the new scalar multiplication f ∗m := φ(f)m. In case (a), the differential structure is given by the original δ and in case (b) the differential structure is given by −δ. (a) φ(z) = cz. The formal local module C((z))⊗M with basis E1, E2 and δE1 = − tz−1+θ0 2 E1, δE2 = tz−1+θ0 2 E2 is transformed into C((z)) ⊗φ M . Now δE1 = − tcz−1+θ0 2 ∗ E1 and δE2 = tcz−1+θ0 2 ∗ E2. The basis F1, F2 of C((z−1)) ⊗M with δF1 = − tz+θ∞ 2 F1 and δF2 = tz+θ∞ 2 F2, yields for the new structure the formulas δF1 = − tc−1z+θ∞ 2 ∗ F1 and δF2 = tc−1z+θ∞ 2 ∗ F2. The condition tc = ±tc−1 implies that c4 = 1. We define σ3 by φ(z) = iz. This yields the new module (C(z)⊗φM, it) with new α equal to eπiθ0 and new β equal to e−πiθ∞ . Geometric Aspects of the Painlevé Equations PIII(D6) and PIII(D7) 19 (b) We define σ4 by φ(z) = z−1. The new module (C(z) ⊗φ M, t) has new α = eπiθ∞ and new β = eπiθ0 . Comments. Using the first three columns of the table we will consider Aut(S) as an automor- phism group of the algebraic variety C∗ × C∗ × C∗. A straightforward computation shows that Aut(S) is the product 〈σ1〉 × 〈σ2〉 × 〈σ3, σ4〉, where 〈σ1〉 and 〈σ2〉 have order two and 〈σ3, σ4〉 is the dihedral group D4 of order eight. The action of Aut(S) on the monodromy data. 1. σ1. The map t 7→ −t has as consequence that the basis vectors e1, e2 of V (0) are permuted and the same holds for V (∞). The singular directions and the Stokes maps do not change. The new topological monodromies are ( 1+a1a2 α a1 α αa2 α ) at z = 0 and (1+b1b2 β b1 β βb2 β ) at z = ∞. The matrix of the link L is now ( `4 `3 `2 `1 ) . The matrix relation remains the same. This amounts to the change α 7→ α−1, β 7→ β−1, a1 ↔ a2, b1 ↔ b2, `1 ↔ `4, `2 ↔ `3. This induces an automorphism of R given by the formula (α, β, x1, x2, x3) 7→ (α−1, β−1, x1α −1β−1, x2α −1β−1, x3). 2. σ2. The map (M, t) 7→ (M ⊗ N, t) has as consequence that the formal monodromies at z = 0 and z = ∞ are multiplied by ( −1 0 0 −1 ) (and thus α 7→ −α, β 7→ −β). The Stokes matrices do not change. The same holds for the link L. The induced automorphism of R is given by the formula (α, β, x1, x2, x3) 7→ (−α,−β, x1,−x2,−x3). 3. σ3. The effect of this transformation is: the singular directions change over π 2 ; the basis vectors of V (0) are permuted; the basis of V (∞) is unchanged; the Stokes map and the link remain the same, however the corresponding matrices change. The induced automorphism of R is given by (α, β, x1, x2, x3) 7→ (α−1, β, α−1x1, α −1x2, x3). 4. σ4. The spaces V (0) and V (∞) are interchanged; the other data are unchanged. The induced automorphism of R is given by (α, β, x1, x2, x3) 7→ (β, α, x1, x2, x3). 4.2 Bäcklund transformations Define the map exp : C3 → (C∗)3 by (θ0, θ∞, t̃) 7→ (α, β, t) := (eπiθ0 , eπiθ∞ , et̃). We will define the group B(S) of the Bäcklund transformations as the affine automorphisms of C3 which respect the equivalence relation defined by exp and which map to elements of Aut(S). By definition there is an exact sequence of groups 0→ B(S)0 → B(S)→ Aut(S)→ 1, where B(S)0 is the group of affine transformations of C3, generated by: B1 : (θ0, θ∞, t̃) 7→ (2 + θ0, θ∞, t̃), B2 : (θ0, θ∞, t̃) 7→ (θ0, 2 + θ∞, t̃) and B3 : (θ0, θ∞, t̃) 7→ (θ0, θ∞, 2πi+ t̃). The aim is to give each Bäcklund transformation the interpretation of a morphism between the various moduli spaces M̃(∗, ∗), preserving the foliations by isomonodromic families, and to compute the effect on solutions of PIII(D6). First we investigate the group B(S). The affine map B3 is not considered in the literature. Its action on M̃(θ0, θ∞) is obvious since M̃(θ0, θ∞) =M(θ0, θ∞)×T T̃ and T̃ = C→ T = C∗ is the map t̃ 7→ et̃. The effect of B3 on solutions of PIII(D6) is far from obvious. A solution is a function q(t̃) = “q(et̃)”. Since the equation depends only on t, the function q(2πi + t̃) satisfies the same equation. It seems that no formula for q(2πi+ t̃) in terms of q(t̃) and its derivative is present in the literature. Generators for B(S) are given by their action on C3 and the variables t, z. θ0 θ∞ t̃ t z s1 2− θ0 −θ∞ πi+ t̃ −t z s2 1 + θ0 1 + θ∞ t̃ t z s3 θ0 −θ∞ πi 2 + t̃ it iz s4 θ∞ θ0 t̃ t z−1 20 M. van der Put and J. Top These elements generate B(S) because: s∗ is mapped to σ∗ for ∗ = 1, 2, 3, 4; B3 = s21 = s43; B1 = s3s −1 1 s4s3s4 and B2 = B−11 s22. The group 〈B3〉, generated by B3, is isomorphic to Z and lies in the center of B(S). Put B(S) = B(S)/〈B3〉 and let s∗ denote the image of s∗ in this quotient. Then s23 has order two and lies in the center of B(S). We compare this with Okamoto’s paper [7]. The group of the Bäcklund transformations B of equation PIII′(D6) d2Q dx2 = 1 Q ( dQ dx )2 − 1 x dQ dx + Q2(γQ+ α) 4x2 + β 4x + δ 4Q is computed to be the affine Weyl group of type B2. The substitution x = t2, Q = tq transforms the equation into d2q dt2 = 1 q ( dq dt )2 − 1 t dq dt + αq2 + β t + γq3 + δ q , and thus in our notation α = 4θ∞, β = −4(θ0 − 1), γ = 4, δ = −4. In a sense, PIII(D6) is a degree two covering of PIII′(D6). It can be seen that there is a surjective homomorphism B(S)→ B with kernel 〈s23〉. Our approach using moduli spaces explains the Bäcklund transformations presented in [4]. In contrast to this, the new transformations of [13] do not seem to have a simple modular interpretation. 4.3 Bäcklund transformations of the moduli spaces Let s ∈ B(S) have image σ ∈ Aut(S). Choose α, β and write α′ = σ(α), β′ = σ(β). Choose θ0, θ∞, θ′0, θ ′ ∞ such that eπiθ0 = α, . . . , eπiθ ′ ∞ = β′ and s(θ0) = θ′0, s(θ∞) = θ′∞. Now σ induces a bijection S(α, β) σ→ S(α′, β′) and consider M(θ0, θ∞)→ S(α, β) σ→ S(α′, β′)←M(θ′0, θ ′ ∞). The first and the last arrow are injective. Their images contain the locus of the irreducible modu- les and the loci of the (ε1, ε2)-reducible modules depending on the θ0, . . . , θ ′ ∞ (see Section 3.4.3). Thus we obtain a, maybe partially defined, map, again denoted by s :M(θ0, θ∞)→M(θ′0, θ ′ ∞). It can be seen from the definitions of the elements of Aut(S) that the Bäcklund transformation s is a birational algebraic map. Using the expression for s(t̃) one obtains a birational morphism, again denoted by s : M̃(θ0, θ∞) → M̃(θ′0, θ ′ ∞) which respects the foliations (i.e., the isomo- nodromic families). In particular, s maps ‘generic’ solutions for the parameters θ0, θ∞ and the variable t̃ to ‘generic’ solutions for the parameters θ′0, θ ′ ∞ and the variable s(t̃). The term ‘generic’ means here the solutions corresponding to irreducible modules and the ones for (ε1, ε2)-reducible modules (i.e., Riccati solutions) which are present in both M(θ0, θ∞) and M(θ′0, θ ′ ∞). 4.4 Formulas for the Bäcklund transformations 4.4.1 s1 : M̃(θ0, θ∞)→ M̃(2− θ0,−θ∞) Using the first charts of the two spaces and their variables, s1 has the form (a−1, b−2, . . . , b1, c0, t̃) 7→ (a−1, b−2, . . . , b1, c0, t̃+ iπ). The formula for s1 on the second charts is similar. The induced map for the PIII(D6) equations is t 7→ −t, d dt 7→ − d dt , q(t̃) 7→ q(t̃+ iπ). Geometric Aspects of the Painlevé Equations PIII(D6) and PIII(D7) 21 4.4.2 s2 :M(θ0, θ∞)→M(1 + θ0, 1 + θ∞) A point on the first chart of the first space is represented by the differential operator z d dz +( az−1b z − q − az−1 ) , where a := a−1, q := −c0, b = b1z + b0 + b−1z −1 + b−2z −2 and the b1, . . . , b−2 are polynomials in t, q, q−1, using the notation of Section 3.3.1. This is transformed by s2 into the operator z d dz + A with A = ( az−1 + 1 2b z − q − az−1 + 1 2 ) . We want to compute an operator z d dz + Ã with Ã = ( ãz−1 b̃ z − q̃ −ãz−1 ) , b̃ = b̃1z+ b̃0 + b̃−1z −1 + b̃−2z −2 and b̃1, . . . , b̃−2 polynomials in t, q̃, q̃−1, representing a point on the first chart of M(12 + θ0 2 , 1 2 + θ∞ 2 ), which is equivalent to z d dz + A. Thus we have to solve an equation of the type {z d dz + A}T = T{z d dz + Ã} with T ∈ GL(2,C(z)). A local computation shows that T has the form T0 + T−1z −1 + T−2z −2 6= 0 with ‘constant’ matrices T0, T−1, T−2. A Maple computation yields the solution q̃ = − tq2 − qθ0 − t+ 2a q(tq2 + qθ∞ − t+ 2a) , ã = long 2q2(tq2 + qθ∞ − t+ 2a)2 , long = 8a3 − 4aq2t2 + 8a2q2t− qt2 + 2aq4t2 − 8a2t+ 2at2 − 4a2q + 4aqt− q5t+ qt2θ0 − q5t2θ0 + q2tθ20 − 4a2qθ0 + 2aq2θ0 + q4tθ0 − q2tθ0 − q5t2θ∞ − 4q4tθ2∞ + 4a2qθ∞ + qt2θ∞ − 2aq2θ∞ − q4tθ∞ + q2t2θ∞ + q3θ0θ∞ − 4aq3tθ0 − 4aqtθ∞ + q2tθ0θ∞ − q4tθ0θ∞ − 2aq2θ0θ∞ − 4aq3t+ 2q3t. The induced map for solutions of PIII(D6) is obtained from the formula for q̃ and the equality q′ = 4a−q t . Comments on the formulas. The term q in the denominator of the formulas is due to our choice of working on the first charts of the spacesM(θ0, θ∞) andM(θ0 + 1, θ∞+ 1). This term does not produce singularities for s2. The denominator (tq2 + qθ∞ − t + 2a) is due to a reducible locus of type (ε1, ε2) = (−1, 1). More precisely, tq2 + qθ∞ − t+ 2a = 0 describes the reducible locus of M(θ0, θ∞) if and only if θ0 + θ∞ ∈ 2Z and θ0 + θ∞ ≤ 0. A priori, s2 is not defined on this locus if moreover θ0 + θ∞ = 0. One computes that for θ∞ + θ0 = 0 the formulas reduce to the rational map q̃ = −q−1, ã = −q+2a 2q2 and thus s2 is well defined on this locus of type (ε1, ε2) = (−1, 1). In fact, s2 maps this locus to the reducible locus of type (ε1, ε2) = (1,−1), which is present in M(θ0 + 1, θ∞ + 1). 4.4.3 s3 : M̃(θ0, θ∞)→ M̃(θ0,−θ∞) On the first charts the map is given by: (a−1, b−2, . . . , b1, c0, t̃) 7→ (−ia−1,−ib−2, b−1, ib0,−b1, −ic0, t̃+ iπ2 ) and similarly on the second charts. For the PIII(D6) equations, the map is t 7→ it, d dt 7→ −i d dt , q(t̃) 7→ −iq(t̃+ iπ2 ). 4.4.4 s4 :M(θ0, θ∞)→M(θ∞, θ0) Consider a point on the chart c1 6= 0 ofM(θ0, θ∞) (see Section 3.3.1) represented by z d dz +A(z), where the entries of A(z) are polynomial in z, z−1, t, a := a−1, q := −c0, q−1. Now s4 transforms this operator into z d dz −A(z−1). We suppose that a transformation T := T−1z −1 +T0 +T1z 6= 0 brings this operator into a point of the chart c1 6= 0 of M(θ∞, θ0) represented by an operator z d dz + Ã(z), where the entries of Ã(z) are polynomials in z, z−1, t, ã = ã−1, q̃ = −c̃0, q̃−1. A Maple computation shows that there is a unique solution in terms of q̃ and ã of the equation 22 M. van der Put and J. Top {z d dz −A(z−1)}T = T{z d dz + Ã(z)}, namely q̃ = q(−q2t− θ∞q − t+ 2a) (−q2t− θ0q − t+ 2a) , ã = long 2(−q2t− θ0q − t+ 2a)2 , long = 4q2at2 + qt2θ0 + q5t2θ∞ + 4q3atθ0 − q5t2θ0 − q4tθ∞θ0 − q2tθ0θ∞ + 4qtθ∞a + q2tθ20 + q4tθ2∞ − t2qθ∞ − 4qθ0a 2 − 4a2θ∞q + 2aθ0q 2θ∞ − 8a2q2t+ 2aq4t2 + 2at2 − 8a2t+ 8a3. The induced map for solutions of PIII(D6) is obtained from the formula for q̃ and the equality q′ = 4a−q t . Comments on the formulas. The denominator in these formulas is due to a possible reducible locus of type (ε1, ε2) = (−1,−1). This locus is present inM(θ0, θ∞) if and only if θ0 − θ∞ ∈ 2Z and θ0 − θ∞ ≤ 0. This locus is not present in M(θ∞, θ0) if moreover θ0 − θ∞ < 0. However, in the critical case θ0 = θ∞, s4 turns out to be the identity. If θ0 − θ∞ ∈ 2Z and θ0 − θ∞ ≥ 2, then the reducible modules of type (1, 1) are present in M(θ0, θ∞) and not in M(θ∞, θ0). The corresponding term 2a + tq2 + (2d − 1)q + t in the denominator of the map s4 does not occur, because s4 maps this reducible locus to the reducible locus of type (−1,−1) of M(θ∞, θ0). Using the formulas for s1, . . . , s4 one can deduce formulas for B1 and B2. We will however derive these by the direct method used for s2 and s4. 4.4.5 The transformation B1 : θ0 7→ 2 + θ0, θ∞ 7→ θ∞, t̃ 7→ t̃ We compute the birational map B1 on the open part of the chart c1 6= 0 of M(θ0, θ∞) where q = −c0 6= 0 (see Section 3.3.1). A point is represented by an operator z d dz +A. The entries of A are polynomials in q = −c0, q−1 and a = a−1. Further we suppose that the image under B1 lies in the open part of M(2 + θ0, θ∞) given by c1 6= 0 and has the form z d dz + Ã, where the entries of Ã are polynomials in q̃, q̃−1 and ã = ã−1. Since the two differential operators represent the same differential module, there exists a T ∈ GL(2,C(z)) such that z d dz + Ã = T−1(z d dz +A)T . Local calculations at z = 0 and z =∞ predict that T has the form T = T−2z −2+T−1z −1+T0, T−2 = ( 0 ∗ 0 0 ) and detT ∈ C∗. Further it is assumed that z d dz + A is not (ε1, ε2)-reducible for the critical cases (ε1, ε2) = (−1,−1), θ∞ = θ0 and (ε1, ε2) = (−1, 1), θ∞ = −θ0, where this reducible locus is present in M(θ0, θ∞) and not in M(2 + θ0, θ∞). Maple produces the formulas q̃ = q(−4a2 + 4at− t2 + θ20q 2 + 2tθ∞q 3 + t2q4) (2a− t− θ0q + tq2)(2a− t− θ0q − tq2) , ã = long (2a− t− θ0q + tq2)2(2a− t− θ0q − tq2)2 , where “long” means too long for copying. Substitution of a = tq′+q 4 in the formula for q̃ yields the formula for B1 with respect to solutions. In the first critical case (ε1, ε2) = (−1,−1), θ∞ = θ0 one obtains q̃ = −q(2a− t+ θ0q + tq2) 2a− t− θ0q + tq2 , ã = long (2a− t− θ0q + tq2)2 . The reducible locus is given by a = t 2 + θ0 2 q+ t 2q 2. Thus the above map extends to the reducible locus and produces there the formulas q̃ = −q(2tq + 2θ0 + 1) 2tq + 1 , ã = long (2tq + 1)2 . Geometric Aspects of the Painlevé Equations PIII(D6) and PIII(D7) 23 For the second critical case (ε1, ε2) = (−1, 1), θ∞ = −θ0 one finds q̃ = q(−2a+ t− θ0q + tq2) 2a− t− θ0q − tq2 , ã = long (2a− t− θ0q + tq2)2 . The reducible locus is given by a = t 2 + θ0 2 q − t 2q 2. On this locus one has q̃ = −q + θ0 t . Comment. As in Sections 4.4.2 and 4.4.4, the map B1 is well defined on the reducible loci because B1 changes the type (ε1, ε2) of the reducible loci. 4.4.6 The transformation B2 : θ0 7→ θ0, θ∞ 7→ 2 + θ∞, t̃ 7→ t̃ Let an object of M(θ0, θ∞) be represented by a standard operator z d dz + A. We expect that the transformation B2 yields an object of M(θ0, 2 + θ∞), represented by a standard operator z d dz + Ã. Then z d dz + Ã = T−1(z d dz +A)T for a certain T ∈ GL(2,C(z)). Local calculations at z = 0 and z = ∞ show that T has the form T−1z −1 + T0 + T1z with T−1 = ( 0 ∗ 0 0 ) , detT1 = 0, detT ∈ C∗. The matrix A depends on q and a := a−1 and the matrix Ã depends on q̃ and ã := ã−1. We have to solve the equation T (z d dz + Ã) = (z d dz +A)T . Maple produced the formula q̃ = − (2a+ t+ θ∞q + tq2)(2a− t+ θ∞q + tq2)q 4aq2t+ 2qt− t2 − 2q3t− q2θ2∞ + 4a2 − 4aq − 2qtθ0 − 2q2θ∞ + t2q4 . The denominator of ã is the square of the denominator of q̃ and the numerator of ã is too large to copy here. The substitution of a = tq′+q 4 in the formula for q̃ yields the B2 map for the solutions of PIII(D6). The two cases where B2 is, a priori, not defined on the reducible locus are: 1. (ε1, ε2) = (1, 1) and θ0 2 = θ∞ 2 + 1. The reducible locus is given by 2a+ t+ θ∞q + tq2 = 0. After substitution of θ0 2 = θ∞ 2 + 1, the denominator of q̃ factors as (2a + t + θ∞q + tq2)(2a − t− θ∞q− 2q+ tq2). Further q̃ = − (2a−t+θ∞q+tq2)q 2a−t−θ∞q−2q+tq2 On the reducible locus one has q̃ = tq t+θ∞q+q and B2 is well defined. 2. (ε1, ε2) = (−1, 1) and θ0 2 = − θ∞ 2 . The reducible locus is given by 2a − t + θ∞q + tq2 = 0. After substitution of θ0 2 = − θ∞ 2 , the denominator of q̃ factors as (2a − t + θ∞q + tq2)(2a + t − θ∞q − 2q + tq2). Further q̃ = − (2a+t+θ∞q+tq2)q 2a+t−θ∞q−2q+tq2 . On the reducible locus one has q̃ = tq t−θ∞q−q and B2 us well defined. Comment. As in Sections 4.4.2, 4.4.4 and 4.4.5, the map B2 is well defined because B2 changes the types (ε1, ε2) of the reducible loci. Acknowledgments The authors thank Yousuke Ohyama for his helpful answers to our questions and his remarks concerning the tau-divisor (see Section 3.3.4). References [1] Inaba M., Saito M.-H., Moduli of unramified irregular singular parabolic connections on a smooth projective curve, Kyoto J. Math. 53 (2013), 433–482, arXiv:1203.0084. [2] Kaneko K., Ohyama Y., Meromorphic Painlevé transcendents at a fixed singularity, Math. Nachr. 286 (2013), 861–875. [3] Kaup L., Kaup B., Holomorphic functions of several variables. An introduction to the fundamental theory, de Gruyter Studies in Mathematics, Vol. 3, Walter de Gruyter & Co., Berlin, 1983. http://dx.doi.org/10.1215/21562261-2081261 http://arxiv.org/abs/1203.0084 http://dx.doi.org/10.1002/mana.200810241 http://dx.doi.org/10.1515/9783110838350 24 M. van der Put and J. Top [4] Milne A.E., Clarkson P.A., Bassom A.P., Bäcklund transformations and solution hierarchies for the third Painlevé equation, Stud. Appl. Math. 98 (1997), 139–194. [5] Ohyama Y., Rational transformations of confluent hypergeometric equations and algebraic solutions of the Painlevé equations: P1 to P5, in Algebraic, Analytic and Geometric Aspects of Complex Differential Equations and their Deformations. Painlevé Hierarchies, RIMS Kôkyûroku Bessatsu, Vol. B2, Res. Inst. Math. Sci. (RIMS), Kyoto, 2007, 137–150. [6] Ohyama Y., Kawamuko H., Sakai H., Okamoto K., Studies on the Painlevé equations. V. Third Painlevé equations of special type PIII(D7) and PIII(D8), J. Math. Sci. Univ. Tokyo 13 (2006), 145–204. [7] Okamoto K., Studies on the Painlevé equations. IV. Third Painlevé equation PIII, Funkcial. Ekvac. 30 (1987), 305–332. [8] van der Put M., Families of linear differential equations related to the second Painlevé equation, in Algebraic Methods in Dynamical Systems, Banach Center Publ., Vol. 94, Polish Acad. Sci. Inst. Math., Warsaw, 2011, 247–262. [9] van der Put M., Families of linear differential equations and the Painlevé equations, in Geometric and Differential Galois Theories, Sémin. Congr., Vol. 27, Soc. Math. France, Paris, 2012, 203–220. [10] van der Put M., Saito M.-H., Moduli spaces for linear differential equations and the Painlevé equations, Ann. Inst. Fourier (Grenoble) 59 (2009), 2611–2667, arXiv:0902.1702. [11] van der Put M., Singer M.F., Galois theory of linear differential equations, Grundlehren der Mathematischen Wissenschaften, Vol. 328, Springer-Verlag, Berlin, 2003. [12] van der Put M., Top J., A Riemann–Hilbert approach to Painlevé IV, J. Nonlinear Math. Phys. 20 (2013), suppl. 1, 165–177, arXiv:1207.4335. [13] Witte N.S., New transformations for Painlevé’s third transcendent, Proc. Amer. Math. Soc. 132 (2004), 1649–1658, math.CA/0210019. http://dx.doi.org/10.1111/1467-9590.00044 http://dx.doi.org/10.4064/bc94-0-18 http://dx.doi.org/10.5802/aif.2502 http://arxiv.org/abs/0902.1702 http://dx.doi.org/10.1080/14029251.2013.862442 http://arxiv.org/abs/1207.4335 http://dx.doi.org/10.1090/S0002-9939-04-07087-X http://arxiv.org/abs/math.CA/0210019 1 Introduction 2 The family (1,-,1/2) and PIII(D7) 2.1 The construction of the monodromy space RP 2.2 The construction of the moduli space M() 2.3 Isomonodromy and the Okamoto-Painlevé space 2.4 Automorphisms of S and Bäcklund transforms 2.5 Remarks 3 The family (1,-,1) 3.1 Definition of the family 3.2 The monodromy space 3.3 The general case =1 3.3.1 The moduli space M(0,) 3.3.2 The Okamoto–Painlevé space (0,) for 022Z 3.3.3 Verification of the formula in Theorem 2.3 3.3.4 After a remark by Yousuke Ohyama 3.4 The cases =1 3.4.1 Geometric quotients of the monodromy data 3.4.2 Reducible modules in S 3.4.3 The reducible connections in M(0,) 3.4.4 M(0, ) for =1 3.4.5 Isomonodromy for reducible connections 4 Bäcklund transformations for PIII(D6) 4.1 Automorphisms of S 4.2 Bäcklund transformations 4.3 Bäcklund transformations of the moduli spaces 4.4 Formulas for the Bäcklund transformations 4.4.1 s1:(0,)(2-0,-) 4.4.2 s2: M(0,)M( 1+0, 1+) 4.4.3 s3: (0,)(0,-) 4.4.4 s4:M(0,)M(,0) 4.4.5 The transformation B1: 02+0, , 4.4.6 The transformation B2:00, 2+, References

Geometric Aspects of the Painlevé Equations PIII(D₆) and PIII(D₇)

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