Moduli of stable parabolic connections, Riemann-Hilbert correspondences and Geometry of Painlevé equations

Transcription

1 RHPIA-2005 Moduli of stable parabolic connections, Riemann-Hilbert correspondences and Geometry of Painlevé equations Masa-Hiko SAITO (Kobe University) Based on Joint Works with Michi-aki Inaba and Katsunori Iwasaki (Kyushu Univ.) SISSA, Trieste, 20-September :00 10:45 Conference on Riemann-Hilbert Problems, Integrability and Asymptotics 1

2 The purposes To understand the Painlevé property of equations of Painlevé type (Painlevé equations or Garnier equations, etc) from the viewpoint of the isomonodromic or isostokes deformations of linear diff. equations. To understand many nice properties of equations of Painlevé type like 1. Spaces of initial conditions due to Okamoto (Sakai s work or Saito Takebe Terajima s work on Okamoto Painlevé pair(s, Y)). 2. Symmetries of equations = Bäklund transformations 3. Non-autonomous Hamiltonian structures 4. Special solutions Riccati type solutions, Rational solutions, etc 5. τ-functions

3 To understand the WKB analysis through the compactification of the moduli space of the stable parbolic connections. (Full story will be in future).

4 Today s talk Restricted to the case of regular singularities Papers M. Inaba, K. Iwasaki and M.-H. Saito, Bäcklund transformations of the sixth Painlevé equation in terms of Riemann-Hilbert Correspondence, Internat. Math. Res. Notices 2004:1 (2004), M. Inaba, K. Iwasaki and M.-H. Saito, Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equationoftypevi.parti, math.ag/ (2003); Part II, inpreparation. M. Inaba, K. Iwasaki and M.-H. Saito, Dynamics of the Sixth Painlevé Equation, to appear in Angers proceedings, math.ag/

5 Translations of the terminology Analysis Geometry C: a compact R. surface of genus g C: a nonsing. proj. curve of genus g t =(t 1,,t n ); n-distinct pts on C t =(t 1,,t n ); n-distinct pts on C dx dz = P n A i (z) i=1 z t i x : E E Ω 1 C (D(t)) Linear D.E. on C with A connection on vect. bdl E of rank r at most regular sing. at t. on C with at most 1 st order poles at t. λ (i) j :Eigenvalues of A i(t i ) λ (i) j : Eigenvalues of res t i ( ) End(E ti ) Time varaiables T = M g,n = {(C, t)} (s 1,...,s 3g 3,t 1,...,t n ) Moduli of n-pointed curves of genus g Space of initial conditions Moduli space of stable parabolic S (C,t,λ) connections M α (C, t) λ Phase space Family of moduli spaces S T Λ r n M T Λ r n Riemann-Hilbert correspondence RH λ : M α λ R a Isomonodromic deformations of L.D.E. Pullback of local constant section Schlessinger equation Zero curvature equations on M

6 Translations of Properties Analysis Geometry Painlevé property Properness + Surjectivity of RH λ : M α λ R a Symmetry ( Bäklund transformation) Elementary transformations of s.p. conn. Special Birational map (Flop) Simple reflections in Bäcklund transf. s : M M appeared in the resol. of simult. sing. of R a Hamitonian Structures Symplectic str. on M α (C, t) λ on Ra smooth and RH λ is a symmplectic map Special solutions like Riccati solution Singylarities of R a Poincaré returnmapor Natural actions of π 1 (M g,n, ) non-linear monodromy on isomonodromic flows, R (C0,t 0 ),a and of equations of Painlevé type on M α ((C 0, t 0 )) λ τ-functions Sections of the determinant line bundle on M which are flat on isomonod. flows

7 Stable Parabolic connections Setting Fix the following data (1) (C, t, (L, L ), (λ (i) j )) which consists of C : a complex smooth projective curve of genus g, t =(t 1,,t n ):asetofn-ditinct points on C. (PutD(t) =t t n ). (L, L ): a line bundle on C with a logarithmic connection L : L L Ω 1 C (D t). λ =(λ (i) j ) 1 i n,0 j r 1 C nr such that P r 1 j=0 λ(i) j = res ti ( L ).

8 Moduli space of stable parabolic connections We can consider the moduli space of stable parabolic connections on C with logarithmic singularities at D(t): (2) M α (C, t,l) λ = {(E, E, {l (i) j } 1 i n,0 j r 1, Ψ)}/ ' E : a vector bundle of rank r on C : E E Ω C (D(t)) :a logarithmic connection Ψ : r E ' L : a horizontal isomorphism (Fixing the determinant) E ti = l (i) 0 l (i) 1 l (i) r 1 l r =0:afiltration of the fiber at ³ t i such that dim l (i) j /l(i) j+1 =1and ³ res ti ( ) λ (i) Id j (l (i) j ) l(i) j+1

9 Moduli space of SL r -rep. of the fundamental group Take the categorical quotient of affine variety (3) Rep(C, t,r)={ρ : π 1 (C \ D(t)) SL r (C)}//Ad(SL r (C)) (ρ 1, ρ 2 Hom(π 1 (C \ D t ),SL r (C)) are Jordan equivalent iff sem(ρ 1 ) ' sem(ρ 2 )). Fix: ³ a = a (i) j A r,n = C n(r 1) 1 i n,1 j r 1 Then we define another moduli space of SL r -representations with fixed characteristic polynomial of monodromies around t i : n o Rep(C, t,r) a = [ρ] Rep(C, t,r), det(si r ρ(γ i )) = χ a (i)(s) where χ a (i)(s) =s r + a (i) r 1 sr a (i) 1 s +( 1)r.

10 Riemann-Hilbert correspondence Assume that r 2, n 1 and nr 2r 2 > 0 when g =0, n 2. (Moreover the weight α is generic). Then the Riemann-Hilbert correspondence (4) RH (C,t,λ) : M α (C, t,l) λ Rep(C, t,r) a can be defined by (E, E, {l (i) j }, Ψ) 7 ker( an ) C\D t where χ a (i)(s) = r 1 Y j=0 (s exp( 2π 1λ (i) j )) Note that dim M α (C, t,l) λ =(r 1)(2(r +1)(g 1) + rn)

11 Fundamental Results Theorem 1. (Inaba-Iwasaki-Saito (r = 2,g = 0,n 4), Inaba (general case)) Under the notation as above, we have the following. 1. The moduli space M α (C, t,l) λ is a nonsingular algebraic manifold with a natural symplectic structure. 2. The modulis space M α (C, t,l) λ has a natural compactification M α (C, t,l) λ which is the moduli space of the φ-stable parabolic connections.

12 Theorem 2. (Inaba-Iwasaki-Saito (r = 2,g = 0,n 4), Inaba (general case)): Under the conditions above, the Riemann-Hilbert correspondense (5) RH C,t,λ : M α (C, t,l) λ Rep(C, t,r) a is a proper surjective bimeromorphic map. Hence the Riemann-Hilbert correspondence gives an (analytic) resolution of singularities. Moreover RH C,t,λ preserves the symplectic structures on Rep(C, t,r) a M α (C, t,l) λ. Remark 1. Rep(C, t,r) a is an affine scheme which may have singularities for special a. Inthecaseofg = 0, we can show that dω =0. Moreover, we expect that dω =0ingeneral.

13 Varying time (C, t) and parameter λ, a Consider the open set of the moduli space of n-pointed curves of genus g M o g,n = {(C, t) =(C, t 1,,t n ),t i 6= t j,i6= j} and the universal curve π : C M o g,n. Fixing a relative line bundle L for π with logarithmic connection L wecanobtainthefamilyof moduli spaces over M o g,n Λ(L) (6) such that M α g,n (L) π n M o g,n Λ(L) π 1 n ((C, t,l,λ)) = M α (C, t,l) λ

14 We can also construct the fiber space Rep r,n g (7) such that φ r,n g M o g,n A r,n (φ r,n g ) 1 ((C, t, a)) = Rep(C, t,sl r ) a.

15 Riemann-Hilbert corr. in family We can obtain the following commutative diagram: RH n Rep r,n (8) M α (L) π y n g y φ r,n g Mg,n o (1 µ r,n ) Λ(L) Mg,n o A r,n where µ r,n canbegivenbytherelations χ a (s) = r 1 Y j=0 (s exp( 2π 1λ (i) j )) that is, a (i) k are (±1) kth fundamental symmetric functions of exp( 2π 1λ (i) j ).

16 Geometric Isomonodromic Deform. of L.D.E. The case of generic exponents λ Fix a generic λ Λ(L) and set a = µ r,n (λ) so that RH C,t,λ : M α ' (C, t,l) λ Rep(C, t,r)a is an analytic isomorphism for any (C, t) Mg,n. o Algebraic structure of Rep(C, t,r) a does not change under variation of (C, t), thatis, Rep(C, t,r) a ' Rep(C 0, t 0,r) a. Algebraic structure of Mα (C, t,l) λ change under variation of (C, t).

17 Taking the universal covering map ]M g,n o Mo g,n, and pulling back we obtain the diagram: ^ M α g,n(l) λ ( π n ) λ y RH n,λ ' µ ^ Rep r,n g y φr,n g,a ' Rep(C 0, t 0,r) a ]M g,n o a ]M g,n o {λ} (1 µ r,n) ]M g,n o a. r,n Since φ g,a is isomorphic to product family, it has a unique constant section s x passing through a point x Rep(C 0, t 0,r) a {t 0 }. Pulling back the section {s x } x Rep(C0,t 0,r) a {t 0 } via RH λ,weobtain the set of analytic sections of ( π n ) λ : { s x } x Rep(C0,t 0,r) a {t 0 }. ^ M α g,n(l) λ ]M o g,n {λ}

18 The family of sections { s x } x gives the splitting homomorphism ṽ λ :( π n ) λ (T ]M g,n o T {λ}) M^ α g,n(l) λ for the natural homomorphism T M ^ ( π α n ) λ g,n(l) (T λ ]M g,n o Then the subbundle (9) IF g,n,λ =ṽ λ (( π n ) λ (T ]M g,n o {λ})) T. M^ α g,n (L) λ {λ}). Take any local generators of the tangent sheaf of T M ] g,n o h,..., i. q 1 q N where N =3g 3+n =dim]m g,n. o Thensettingv i (λ) :=v λ ( we obtain the integrable differential system on M^ α g,n(l) λ IF g,n,λ 'hv 1 (λ),...,v N (λ)i. (locally). q i ),

19 Case of special exponents λ When the set of exponents λ is special, the R.H. corr. µ RH n,λ : M^ α g,n(l) λ Rep ^ r,n g a µ contracts some subvatieties to the singular locus on Rep ^ r,n g However, by Hartogs theorem, we can extend the isomonodromic foliation IF g,n,λ to the total space M^ α g,n(l) λ. a

20 Painlevé Property of Isomonodromic Flows Theorem 3. The isomonodromic flows IF λ satisfies the Painlevé property for all exponents λ. Hamiltonian strucure of Isomonodromic Flows Theorem 4. The isomonodromic flows IF λ canbewrittenina Hamiltonian system locally.

21 In the case of generic λ, the differential system on IF g,n,r := hv 1 (λ),...,v N (λ)i. ^ M α g,n(l) λ has cleary solution manifolds or integrable manifolds = the images of ]M o g,n by { s x } x.byconstruction, These integrable submanifolds are isomonodromic flow of connections. Even in the case of special λ, the properness of RH λ,n implies the theorem. IF (0,4,2) is equivalent to a Painlevé VIequation. IF (0,n,2) with n 5 are Garnier systems.

22 Painlevé VI case We will see what is happening in the case of Painlevé VI equations.

23 Isomonodromic flows = Painlevé or Garnier flows RH λ ' R(P n,t ) a R(P n,t0 ) a M α n (t 0, λ,l) M α n (t, λ,l) t 0 T n {λ} t = t 0 Tn {a} t Figure 1. Riemann-Hilbert correspondence and isomonodromic flows for generic λ

24 Riccati flows are tangent to family of ( 2)-curves M α 4 (t 0, λ,l) RH λ contraction M α 4 (t, λ,l) R(P 4,t ) a R(P 4,t0 ) a t 0 T 4 {λ} t = t 0 T4 {a} t ( 2)-rational curve Case of Painlevé VI Figure 2. Riemann-Hilbert correspondence and isomonodromic flows for special λ

25 Hamiltonian systems of Painlevé P VI P VI is equivalent to a Hamiltonian system H VI. dx (H VI ): dt = H VI y, dy dt = H VI x, Hamiltionian in suitable coordinates can be given H VI = H VI (x, y, t, λ 1, λ 2, λ 3, λ 4 ) C(t)[x, y, λ i ] 1 H VI (x, y, t) = x(x 1)(x t)y 2 {2λ 1 (x 1)(x t) t(t 1) +2λ 2 x(x t)+(2λ 3 1)x(x 1)} y + λ(x t)] λ := (λ1 + λ 2 + λ 3 1/2) 2 λ 2 ª 4.

26 Bäcklund transformations for Painlevé VI. P VI (λ) have non-trivial birational automorphisms which are called Bäcklund transformations. ThegroupofallBäcklund transformations form the affine Weyl group W of type D (1) 4. Proposition 1. The group of Bäcklund transformations which can be obtained from elementary transformations of stable parabolic connections is a proper subgroup of W (D (1) 4 ) whose index is finite. The invloution s 0 of W (D (1) 4 ) is not in the group.

27 Theorem For a suitable choice of a weight α 0, the morphism π 4 : M4 α0( 1) T 4 Λ 4 is projective and smooth. Moreover for any (t, λ) T 4 Λ 4 the fiber π 1 4 α0 (t, λ) :=M4 (t, λ, 1) is irreducible, hence a smooth projective surface. 4 ( 1) \ M 4 α ( 1) be the complement of M4 α ( 1) in M4 α0( 1). 2. Let D = M α0 (Note that α = α 0 /2). Then D is a flat reduced divisor over T 4 Λ For each (t, λ), set S t,λ := π 1 4 α0 (t, λ) :=M4 (t, λ, 1). Then S t,λ is a smooth projective surface which can be obtained by blowingups at 8 points of the Hirzeburch surface F 2 =Proj(O P 1( 2) O P 1) of degree 2. The surface has a unique effective anti-canonical divisor K St,λ = Y t,λ whose support is D t,λ.thenthepair (10) (S t,λ, Y t,λ ) is an Okamoto-Painlevé pair of type D (1) 4. That is, the anti-canonical divisor Y t,λ consists of 5-nodal rational curves whose configuration is same as Kodaira Néron degenerate elliptic curves of type D (1) 4 (=Kodaira type I0). Moreover we have (M4 α( 1)) t,λ =(M4 α0( 1)) t,λ \Y t,λ.

28 Okamoto Painlevé pairoftypep VI Y 1 Y 2 Y 3 Y 4 -section Y 0 F 2 π P 1 t 1 =0 t 2 =1 t 3 = t t 4 = Figure 3. Okamoto-Painlevé pairoftyped (1) 4

29 Proposition 2. The invariant ring (R 3 ) Ad(SL 2(C)) is generated by seven elements x 1,x 2,x 3,a 1,a 2,a 3,a 4 and there exist a relation (11) f(x, a) =x 1 x 2 x 3 + x x x 2 3 θ 1 (a)x 1 θ 2 (a)x 2 θ 3 (a)x 3 + θ 4 (a), where we set θ i (a) =a i a 4 + a j a k, (i, j, k) =a cyclic permutation of (1, 2, 3), θ 4 (a) =a 1 a 2 a 3 a 4 + a a a a Therefore we have an isomorphism (R 3 ) Ad(SL 2(C)) ' C[x1,x 2,x 3,a 1,a 2,a 3,a 4 ]/(f(z, a)). Hence Rep(P 1, (t 1,t 2,t 3,t 4 ), 2) a =Spec(R 3 ) Ad(SL 2(C)) is isomorphic to an affine cubic. (12) a i =2cos2πλ i for 1 i 4.

30 A 1 -singularity R(P 4,t ) a i =2 =0 A 4 ' C 4 a 1 =2 The family of affine cubic surfaces

31 (13) Mn α (L) π y n RH n R n y φ n T 0 n Λ n (1 µ n ) T 0 n A n. Here, we have 1 µ n (1 µ n )(t, λ) =(t, a) (14) a i =2cos2πλ i for 1 i n.