Gauss Fuchs rigid rigid rigid Nicholas Katz Rigid local systems [6] Fuchs Katz Crawley- Boevey[1] [7] Katz rigid rigid Katz middle convolu

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1 rigidity Fuchs 1 Introduction y + p(x)y + q(x)y = 0, y 2 p(x), q(x) p(x) q(x) Fuchs 19 Fuchs 83

2 Gauss Fuchs rigid rigid rigid Nicholas Katz Rigid local systems [6] Fuchs Katz Crawley- Boevey[1] [7] Katz rigid rigid Katz middle convolution middle convolution Fuchs middle convolution Fuchs A: transitive rigidity rigid A: rigid 1? 1 1 middle convolution Riemann-Liouville 84

3 1: Fuchs Katz holonomic rigid rigidity rigid Katz 85

4 2 Fuchs 2.1 Fuchs dy dx = p j=1 A j x a j Y, a 1,..., a p C, A 1,..., A p M(n; C) (1) Fuchs Schlesinger Fuchsian system x = a j a 1,..., a p a 0 = a j 1 A j t = 1 x t = 0 t A j A 0 A 0 = p j=1 A j Jordan A 0, A 1,..., A p (1) a j Y j (x) = F (x)(x a j ) A j F (x) x = a j F (a j ) = I n (x a j ) A j = e A jlog(x a j ) a j x 0 a j 1 γ γ F (x a j ) A j e 2πiA j γ Y j (x) = Y j (x)e 2πiA j. 86

5 Riemann P 1 a 0,..., a p x 0 b b X = P 1 \{a 0,..., a p } b X π 1 (X, b) π 1 (X, b) γ Y(x) x = b Y(x) γ b M Y(x)M A: (1) A: Y = t (Y 1,..., Y n ) n n n n : Y Y Y ρ : π 1 (X, b) GL(n, C) γ M γ M γ ρ C:? γ 1 γ 2 γ 1 γ 2 87

6 C: M γ2 M γ1 π 1 (X, b) = γ 0, γ 1,..., γ p γ 0 γ 1 γ p = 1 b a 0 γ 0 a 1,..., a p γ 1,..., γ p 1 p + 1 γ 0 γ 1 γ 2 γ p = 1 ρ(γ j ) = M j M p M 1 M 0 = I n (2) p + 1 (M 0, M 1,..., M p ) ρ (2) ρ p γ, γ π 1 (X, b) a j 1 a k = γ γ in π 1 (X, b). µ 2 a k 1 γ = µγ µ 1 γ j ρ M j [M j ] a j γ j π 1 (X, b) γ j GL(n, C) [M j ] a j 2.2 rigidity γ j M j Y j e 2πiA j M j e 2πiA j M j (2) p + 1 (2) (M 0, M 1,..., M p ) rigid 88

7 2: γ γ 89

8 2. ρ = (M 0, M 1,..., M p ) rigid [ρ] [ρ] ρ = (N 0, N 1,..., N p ) ρ ρ j D M j = DN j D 1 rigid ρ ρ M j = D j N j D 1 j rigid D j j D M 0,..., M p N 0,..., N p ρ rigid rigidity 3. rigidity ı = (1 p)n 2 + p dimz(m j ). j=1 Z(M) = {B GL(n, C) BM = MB} Z(M j ) dimz(a) A A Jordan α A e j (A : α) A Jordan α Jordan j j j 4. α 1 α 1 A = α α β 1 β e 1 (A : α) = 2, e 2 (A : α) = 1, e 3 (A : α) = 1 4 (211) α Jordan A A β (11) A = ((211), (11)) 90

9 A e j A dimz(a) = e j (A : α) 2 α Jordan rigidity rigidity Katz 5. (Katz) ρ rigidity 2 ρ rigid rigidity 2 ρ = (M 0, M 1,..., M p ) M j p n ı 2 rigid Fuchs : M j M j M j e 2πiA j (2) M j ı = 2, j=1 : ρ 2 M αi n1 βi n2 dimz(m) = n n 2 2 dimz(m) = n ı ı n 2 rigidity 91

10 A: ı 2 ı ı = 2 rigid p n 6. (1) n = 2 (11), ((11)), (2) 3 (2) = 2 Jordan e j z j ı = (1 p) z j = 2 ı = 6 2p p = 2 n = 2 p = 2 rigid ((11), (11), (11)) (11) ((11)) (11) Jordan n = 3 (2) n = 3 (111) (21) 3 5 ı = (1 p) 9 + p j=0 z j rigid p j=0 z j = 9p 7 p = 2 z 0 + z 1 + z 2 = 11 ((211), (111), (111)) p = 3 z 0 + z 1 + z 2 + z 3 = 20 ((21), (21), (21), (21)) p 4 (3) n = 4 (31), (22), (211), (1111) p = 2 rigid ((31), (1111), (1111)), ((22), (211), (1111)), ((211), (211), (211)) p = 3 ((31), (31), (31), (1111)), ((31), (31), (22), (211)), ((31), (22), (22), (22)), p = 4 ((31), (31), (31), (31)) ((31), (31), (31), (1111)) p j=0 z j 92

11 ı = 2 Crawley- Boevey ı = 2 :? : : : : : 2.3 middle convolution middle convolution Introduction Fuchs Fuchs I λ a f(x) = 1 Γ (λ) x a 93 f(t)(x t) λ dt

12 a C f λ C f Riemann-Liouville From Gauss to Painlevé λ = n f n I λ a I µ a = I λ+µ a : n 1 0 Γ (λ) n (x t) n x Γ C: a? a D D a x a x C: x n a a x a x x 2 3 Γ n 0 a x 0 x n a (1) Z(x) = I λ a Y (x) Z(x) Z(x) middle convolution A j 94

13 3: Riemann-Liouville Riemann-Liouville Riemann-Liouville middle convolution λ µ middle convolution λ + µ 1 middle convolution λ = 0 middle convolution middle convolution middle convolution rigidity Introduction rigid middle convolution addition 1 1 middle convolution rigid 1 (x a j ) α j rigid rigid middle convolution A j middle convolution middle convolution Appell rigid 95

14 : rigid 4 rigid 5 rigid 6 n = 4 n = 5 7 Riemann-Liouville 1 Fuchs dy dx = ( p j=1 A j x a j ) Y A j M(n, C) t = 1 A x 0 = p j=1 A j t = 0 2 A M(n, C) x 0 1 x 0 γ γ x A = x A e 2πiA 3 dimz(a) = e j (A; α) α 4 ((11), (11), (11)) rigid a 1, a 2, b 1, b 2, c 1, c 2 C, a 1 a 2, b 1 b 2, c 1 c 2, a 1 a 2 b 1 b 2 c 1 c 2 = 1 ) ) ) (a 1 (b 1 (c 1 a2 b2 c2 A, B j 1, C 96

15 (A, B, C) [(A, B, C)] (A, B, C) (A, B, C ) def P GL(2, C), A = P 1 AP, B = P 1 BP, C = P 1 CP [(A, B, C)] 5 (1) ((21), (111), (111)) rigid (2) (1) 6 n = 5 rigid 7 (I λ 0 f)(x) = x 0 f(t)(x t) λ dt I λ 0 I µ 0 = I λ+µ 0 ( : ) 2 haraoka@kumamoto-u.ac.jp midlle convolution [5] 2.4 middle convolution middle convolution Riemann-Liouville Fuchs (1) λ C O O O G j = A 1 A j + λ A p M(pn; C) O O O λ λ = λi n v 1 p K =. C pn v j KerA j, L = KerG 0 = KerG j, v p j=1 G 0 = p j=1 G j K G j v 1 0 G j. = λ v j K 0 v p 97

16 K G j L G j C pn /K + L G j G j C pn /K + L B j pn dim(k + L) n mc λ : (A 1,..., A p ) (B 1,..., B p ) middle convolution Riemann-Liouville Riemann-Liouville G j B j 2.5 middle convolution (1) Y (x) W (x) = Y (x) x a 1. Y (x) x a p 1 W G = (x T ) dw dx A 1... A p.., T = = (G 1)W (3) a 1 I n... A 1... A p a p I n (3), Laplace Riemann-Liouville U(x) = W (t)(x t) λ dt 98

17 (3) Riemann-Liouville Riemann- Liouville (X T ) du dx = (G + λ)u Fuchs du dx = p j=1 G j x a j U Riemann-Liouville G j K + L = 0 K + L = 0 dv dx = p j=1 B j x a j V middle convolution 7. (i) mc 0 = id (ii) mc λ mc µ = mc λ+µ (iii) (A 1,..., A p ) : mc λ (A 1,..., A p ) : (iv) mc λ rigidity middle convolution Katz addition (A 1,..., A p ) (A 1 + α 1,..., A p + α p ), (α 1,..., α p ) C p Y (x) p (x a j ) α j Y (x) j=1 rigidity addition middle convolution 99

18 addition middle convolution middle convolution addition middle convolution addition middle convolution basic 8. (Katz) rigid 1 rigid additon middle convolution 1 1 dy dx = p j=1 α j x a j y y = p j=1 (x a j) α j Riemann- Liouvile addition rigid rigid Euler A: 1 rank A: p a 1,..., a p p A: 100

19 B: generic A: generic Riemann-Liouvile a j A: A: W (t) rank local system (x t) λ x a j 2.6 Fuchs : Katz middle convolution middle convolution λ λ addition K + L KerA j KerG 0 A j addition KerA j λ KerG 0 rigidity 2 1 rigidity rigidity 2 4 middle convolution Fuchs 101

20 (A 0, A 1..., A p ) O Kostov Deline-Simpson Kosov Craweley-Boevey quiver quiver quiver (Victor Kac) Crawley-Boevey O (Crawley-Boevey) Step 1. ı 2 ı = 0 ( ) 1 Step 2. (A 0, A 1,..., A p ) (e (0), e (1),..., e (p) ) e (i) = ((e (i,1) j ) j 1, (e (i,2) j ) j 2,...), e (i) = k j e (i,k) j = n, (i = 0, 1,..., p) (e (i,k) j ) j 1 A i Jordan i = 1,..., p e (i,k) 1 e (i,1) 1 d = max e (i,k) 1 = e (i,1) 1 k p i=0 e (i,1) 1 (p 1)n d 0 { (i,1) e 1 < d (e (0), e (1),..., e (p) ) d > 0 e (i,1) 1 < d Step 3 Step 3. e (i) = ((e (i,1) 1 d, e (i,1) 2, e (i,1) 3,...), (e (i,2) 102 j ) j 1, (e (i,3) j ) j 1,...)

21 e (i,1) 1 d = 0 n = e (i) = n d < n n = 1 rigid n > 1 Step 2 1 n n = 1 d 0 : Step2? 2 : 1? addition middle convolution ı : ı 2 Step2 Step1 Step1 Step1 : middle convolution G j A 1,..., A p O (j, j) A j + λ 0 0 G j B j λ KerA j A j λ (j, j) 0 0 Jordan 1 d V = C n B j mc λ (V ) λ 0 K + L K L dim mc λ (V ) = pn dim(k + L) = pn dimk diml p = pn dimkera j dimkerg 0 j=1 103

22 e (j,1) 1 0 dimkera j = e (j,1) 1, dimkerg 0 = e (0,1) 1 0 addition p dimkera j = e (j,1) 1 1 middle convolution λ dimkerg 0 = e (0,1) 1 middle convolution 1 d middle convolution addition d d 1 rigid d 0 Crawley-Boevey quiver 2 Jordan e (i,1) 1 j 2 : rigid ı = 2 Katz? : :? ı = 2 : Step1 ı = 2 ı = 2 104

23 ı = 2 n = 4 rigid 1 ı = 2 ı = 2 d e (i,1) 1 : Katz 1? d e (i,1) 1 : d 0 rigid d 0? 0 non-rigid rigid d > 0 1 rigid 3 D C n n du = Ωu, Ω = A k (x)dx k k=1 u = t (u 1,..., u N ), A k (x) = (a k ij(x)) 1 i,j N a k ij(x) D u u = A k (x)u (1 k n), x k u i x k = N a k ij(x)u j (1 k n, 1 i N) j=1 Pfaff N 105

24 a k ij C 2 xl xk u = xk xl u x l (A k u) = x k (A l u) A k x l + A k A l = A l x k + A l A k Pfaff A k a k ij(x) S CP n S = S j S S j γ, γ π 1 (CP n \S, b) S j 1 S k 1 S j 1 γ γ ρ : π 1 (CP n \S, b) GL(N, C) S j 1 γ [ρ(γ)] S j S j rigidity rigidity 2 rigid 106

25 p + 1 γ 0, γ 1,..., γ p γ 0 γ 1 γ p = 1 p + 1 π 1 (CP n \S, b) S S rigidity CP 2 Appell F 1, F 2, F 3, F 4 4 4: xy(x 1)(y 1)(x y) = 0 Gérard-Levelt : 1 107

26 π ( 5) b b γ 1,..., γ 5 γ 1 γ 2 γ 4 γ π 1 (CP 2 \ S, b) = γ 1,..., γ 5 γ 1 γ 2 = γ 2 γ 1, γ 4 γ 5 = γ 5 γ 4 γ 1 γ 3 γ 5 = γ 3 γ 5 γ 1 = γ 5 γ 1 γ 3 γ 2 γ 3 γ 4 = γ 3 γ 4 γ 2 = γ 4 γ 2 γ 3 Zariski-van Kampen 4 [8] Pfaff γ j M j M 1,..., M 5 rigid Appell F 1, F 2.F 3, F 4 rigid [4] middle convolution 1 x i middle convolution (A 1,..., A p ) (G 1,..., G p ) (B 1,..., B p ) Riemann-Liouville Pfaff 1 ( dx du = A 1 x + A dy 2 y 1 + A d(x y) 3 x y + A 4 dx x 1 + A ) dy u, A 1,..., A 5 M(n, C) y

27 5: π 1 (CP 2 \ S, b) 109

28 Pfaff 3 x middle convolution x middle convolution A 1, A 3, A 4 G 1, G 3, G 4 G 1 = A 1 + λ A 3 A 4 O O O O O O O O O, G 3 = A 1 A 3 + λ A 4, G 4 = O O O O O O O O O A 1 A 3 A 4 + λ y G 2 G 5 A 2 O O A 3 + A 5 A 3 O G 2 = O A 2 + A 4 A 4, G 5 = A 1 A 1 + A 5 O O A 3 A 2 + A 3 O O A 5 (A 1,..., A 5 ) (G 1,..., G 5 ) x G 1, G 3, G 4 K + L G 2 G 5 K + L K + L G 2, G 5 middle convoution : G 2, G 5? Riemann-Liouville (x t) λ y y : Appell F 1, F 2, F 3, F 4 A 1 addition middle convolution 6 4 x = 0, 1, 4 x 4 4 middle convolution A 2, A 5 110

29 6: rigid middle convolution [2] Appell F 4 π 1 rigid [3] [1] W. Crawley-Boevey: On matrices in prescribed conjugacy classes with no common invariant subspace and sum zero, Duke Math. J. 118 (2003), [2] Y. Haraoka: Middle convolution for completely integrable systems with logarithmic singularities along hyperplane arrangements, Adv. Stud. Pure Math. 62 (2012), [3] Y. Haraoka and Y. Ueno: Rigidity for Appell s hypergeometric series F 4, Funk. Ekvac. 51 (2008), [4] Y. Haraoka and T. Kikukawa: Rigidity of monodromies for Appell s hypergeometric functions, Opuscula Mathematica. 35 (2015), [5] :,, [6] N. M. Katz: Rigid Local Systems, Princeton Univ. Press, Princeton, NJ,

30 [7] T. Oshima: Fractional calculus of Weyl algebra and Fuchsian differential equations, MSJ Memoirs, 28. Mathematical Society of Japan, Tokyo, [8] :, 4,, 2001,

Fuchs Fuchs Laplace Katz [Kz] middle convolution addition Gauss Airy Fuchs addition middle convolution Fuchs 5 Fuchs Riemann, rigidity

2010 4 8 7 22 2 Fuchs Fuchs Laplace Katz [Kz] middle convolution addition Gauss Airy Fuchs addition middle convolution Fuchs 5 Fuchs Riemann, rigidity 0 Fuchs addition middle convolution Riemann Fuchs