Ubiquity of the Stokes phenomenon 2 Various places where the Stokes - PDF document

1 Ubiquity of the Stokes phenomenon 2 Various places where the Stokes phenomenon occurs. Asympt. behaviour of sols of the Airy Eqn (Stokes...). Introduction to Global behaviour of vanishing cycles of functions Stokes structures X → C in alg. geom. (Pham, Berry...) I: dimension one Analogy with the theory of wild ramification in Arithmetic (Deligne...). Claude Sabbah Frobenius manifolds and quantum cohomology (Dubrovin...). Centre de Math´ ematiques Laurent Schwartz tt ∗ geometry (Cecotti & Vafa...). ´ Ecole polytechnique, CNRS, Universit´ e Paris-Saclay Geometric Langlands correspondence with wild Palaiseau, France ramification (Frenkel & B. Gross...). Programme SISYPH ANR-13-IS01-0001-01/02 Wild character varieties (Boalch...). Similarities with the theory of stability conditions on some Abelian categories (Bridgeland, Kontsevich...). Introduction toStokes structures – p. 1/22 Introduction toStokes structures – p. 2/22 Aim: RH corresp. for merom. ODE’s 3 Other approaches 4 Riemann-Hilbert corresp. (categorical) on a punctured Riemann surf. X ∗ = X � S : Explicit computation of sols (integral formulas)  realizing Stokes data with effective solutions   Lin. repres.   ( � theory of multisummation)  Merom. flat bdles  loc. cst.     π 1 ( X ∗ ) on ( X, S ) ⇐ ⇒ sheaves of ⇐ ⇒ Constructing moduli spaces of diff. eqns and realizing    ↓     the RHB corresp. by a map between moduli spaces. finite rk on X ∗ with reg. sing. at S  GL n ( C ) replacing the group GL n ( C ) with other reductive algebraic groups. Riemann-Hilbert-Birkhoff corresp. (categorical) on a Extending the categorical approach to the Tannakian punctured Riemann surf. X ∗ = X � S : aspect ( � Differential Galois theory).   �  Stokes-filt.  Generalized   Merom. flat bdles ⇐ ⇒ loc. syst. ⇐ ⇒ monodromy data on ( X, S )     on � X (Stokes data) Introduction toStokes structures – p. 3/22 Introduction toStokes structures – p. 4/22 Stokes phenomenon in dim. one 5 Asympt. analysis in dim. one 6 ∆ = complex disc, complex coord. z . Real or. blow-up: � ∆ = [0 , ε ) × S 1 , coord. ρ, e iθ . � � Linear cplx diff. eqn. d f/ d z = A ( z ) · f , ∆ → ∆ ( ρ, e iθ ) �− → z = ρe iθ ̟ : S 1 → 0 A ( z ) matrix of size d × d , merom. pole at z = 0 . Gauge equiv.: P ∈ GL d ( C ( { z } )) , ∆ = ker z∂ z : C ∞ ∆ → C ∞ Sheaf A e e e ∆ A ∼ B = P [ A ] := P − 1 AP + P − 1 P ′ ( A e ∆ ∗ = O e ∆ ∗ )   ϕ k ∈ 1 z C [ 1 ϕ ′ z ] Sheaves A rd 0 ⊂ A S 1 ⊂ A mod 0 . 1 ... S 1 S 1  + C   Norm. form: B =  C = const. Basic exact sequence: z ϕ ′ non reson. d → A rd 0 → ̟ − 1 C [ 0 − − → A S 1 − [ z ] ] − → 0 S 1 Theorem (Levelt-Turrittin). Given A , ∃ a formal gauge transf. � )) s.t. B = � ( z 1 /p ) P ∈ GL d ( C ( P [ A ] is a normal form. Introduction toStokes structures – p. 5/22 Introduction toStokes structures – p. 6/22 Asympt. analysis in dim. one 7 Asympt. analysis in dim. one 8 Real or. blow-up: � Theorem (Hukuhara-Turrittin). ∆ = [0 , ε ) × S 1 , coord. ρ, e iθ . � � Locally on S 1 , ∃ a lifting � P ∈ GL d ( A S 1 [1 /z ]) of � P s.t. ∆ → ∆ ( ρ, e iθ ) �− → z = ρe iθ ̟ : P [ A ] = � � P [ A ] = B normal form. S 1 → 0 Corollary . The sheaf on � ∆ of sols of ∆ = ker z∂ z : C ∞ ∆ → C ∞ Sheaf A e e e ∆ d f/ d z = A ( z ) · f ( A e ∆ ∗ = O e ∆ ∗ ) Sheaves A rd 0 ⊂ A S 1 ⊂ A mod 0 . having entries in A rd 0 , resp. in A mod 0 , is a real constr. sheaf, S 1 S 1 e e ∆ ∆ Example: ϕ = u ( z ) /z q s.t. u ( z ) ∈ C [ z ] , q � 1 , and constant on any open interval I of S 1 s.t. u (0) � = 0 or u ( z ) ≡ 0 . Then ∀ α ∈ C and ∀ e iθ o ∈ S 1 ∀ k , Re( ϕ k ) does not vanish. � A rd 0 ⇒ Re( u (0) e − ikθ o ) < 0 , Example. ϕ = z − q u ( z ) , u (0) � = 0 , ⇐ z α e ϕ ∈ θ o ⇒ θ = 1 A mod 0 ⇐ ⇒ idem or u ( z ) ≡ 0 . On S 1 , Re ϕ = 0 ⇐ q (arg u (0)+ π/ 2) mod Z · π/q. θ o Introduction toStokes structures – p. 6/22 Introduction toStokes structures – p. 7/22

The Malgrange-Sibuya theorem 9 Stokes-filtered loc. syst. (non-ramif. case) 10 Fix a norm. form ( irregular type ), e.g. non-ramified: Aim : To specify the struct. of sol. space of a merom. ODE without d ) + C B = diag( ϕ ′ 1 , . . . , ϕ ′ making explicit the realization as functions, z . fixing the normal form. B -marked connections ( ∼ : holom. gauge equiv.): The local system L on S 1 : Sols of d f/ d z = A ( z ) f � �� ( A, � P ) | B = � Iso( B ) = P [ A ] ∼ ∆ = [0 , ε ) × S 1 and restricted to on ∆ ∗ , extended to � { 0 } × S 1 . Hence L ⇐ ⇒ monodromy of sols. Stokes sheaf St( B ) on S 1 : For every ϕ ∈ z − 1 C [ z − 1 ] , a pair of nested � � subsheaves L <ϕ ⊂ L � ϕ of L . Id + Q | Q ∈ End( A rd 0 St( B ) θ = ) , (Id + Q )[ B ] = B θ L � ϕ,θ = { f θ | e − ϕ f ( z ) ∈ A mod 0 } θ Theorem (Malgrange-Sibuya). L <ϕ,θ = { f θ | e − ϕ f ( z ) ∈ A rd 0 } θ Iso( B ) ≃ H 1 ( S 1 , St( B )) Hukuhara-Turrittin ⇒ L <ϕ = L � ϕ except if ϕ = ϕ k for some k = 1 , . . . , d . Introduction toStokes structures – p. 8/22 Introduction toStokes structures – p. 9/22 Stokes-filtered loc. syst. (non-ramif. case) 11 Stokes-filtered loc. syst. (non-ramif. case) 12 Aim : To give an intrinsic characterization of the Let ( L , L • ) be a non-ramif. Stokes-filt. loc. syst. category of Stokes-filtered local systems. Φ := { ϕ | rk gr ϕ L � = 0 } is finite and � Definition . A (non-ramif.) Stokes-filt. loc. syst. on S 1 : ϕ ∈ Φ rk gr ϕ L = rk L . A loc. syst. L on S 1 , ≃ � ( ∗ ) ∀ ϕ ∈ z − 1 C [ z − 1 ] , an R -const. subsheaf L � ϕ ⊂ L ∀ ϕ ∈ Φ , ∀ θ , ψ � θ ϕ gr ψ L θ . L � ϕ,θ s.t. Level structure ⇒ ψ = ϕ, or ∀ θ ∈ S 1 , L � ψ,θ ⊂ L � ϕ,θ ⇐ Re( ψ − ϕ ) < 0 near θ , Levels of B (hence A ) : { q 1 < · · · < q r } setting ∀ θ , L <ϕ,θ = � q i := pole ord. of some ψ − ϕ, ϕ � = ψ ∈ Φ . ψ< θ ϕ L � ψ,θ L <ϕ and gr ϕ L := L � ϕ / L <ϕ � # Levels ( A ) = 1 . � theory of summability. 2 q Stokes directions for each ( ϕ, ψ ) . one asks that ∀ ϕ , # Levels ( A ) > 1 . � theory of multisummability. gr ϕ L is a local system on S 1 , ∀ θ , dim L � ϕ,θ = � Principal and Secondary Stokes directions. ψ � θ ϕ rk gr ψ L . Remark: can define ( L , L • ) over Z , Q , . . . Introduction toStokes structures – p. 10/22 Introduction toStokes structures – p. 11/22 Stokes-filtered loc. syst. (non-ramif. case) 13 Deligne’s RH correspondence 14 Theorem ∀ open I ⊂ S 1 which ∋ at most one Stokes dir. Theorem (Deligne’s RH corresp.). ∀ pair in Φ , then ( ∗ ) holds on I (e.g. | I | � π/q r + ε ).  � Any morphism λ : ( L , L • ) → ( L ′ , L ′  Merom. flat bdles • ) graded on I  Merom. flat bdles norm. form w.r.t. some iso ( ∗ ) and ( ∗ ) ′ , hence is strict , i.e. , of norm. form  on (∆ , 0)  λ ( L � ϕ ) = L ′ ∀ ϕ , � ϕ ∩ λ ( L ) . on (∆ , 0) Uniqueness of the splitting if # Level ( A ) = 1 and moreover | I | = π/q + ε . ≀ ≀ Duality .  The exact sequences   graded → L >ϕ −    0 − → L � ϕ − → L − → 0 Stokes-filt.   gr Stokes-filt. → L � − ϕ − loc. syst. 0 − → L < − ϕ − → L − → 0   loc. syst.   on S 1   are switched by duality H om C ( • , C ) . on S 1 E xt k ( • , C )=0 if k � 1 . ⇒ gr ϕ ( L ∨ ) ≃ (gr − ϕ L ) ∨ . Introduction toStokes structures – p. 12/22 Introduction toStokes structures – p. 13/22 Stokes data (non-ramif. case, pure level) 15 Stokes data (non-ramif. case, pure level) 16 Case # Level ( A ) = 1 (level = q ) Case # Level ( A ) = 1 (level = q ) Stokes data Opposed filtrations ⇒ unique splittings ( L ℓ ) ℓ ∈ Z / 2 q Z : C -vect. spaces, → gr F L 2 µ = � ∼ k gr F τ 2 µ : L 2 µ − k L 2 µ ∼ Isoms S ℓ +1 : L ℓ − → L ℓ +1 ℓ → gr F L 2 µ +1 = � � ∼ k gr k τ 2 µ +1 : L 2 µ +1 − F L 2 µ +1 F • L 2 µ ր Exhaustive filtrations F • L 2 µ +1 ց � Stokes multipliers Opposedness property : Σ ℓ +1 := τ ℓ +1 ◦ S ℓ +1 ◦ τ − 1 : gr F L ℓ − → gr F L ℓ +1 L 2 µ = � ℓ ℓ ℓ k F k L 2 µ ∩ S 2 µ 2 µ − 1 ( F k L 2 µ − 1 ) Σ ℓ +1 block lower/upper triangular, L 2 µ +1 = � k F k L 2 µ +1 ∩ S 2 µ +1 ℓ ( F k L 2 µ ) 2 µ diag. blocks (Σ ℓ +1 ) jj are isos. ℓ Introduction toStokes structures – p. 14/22 Introduction toStokes structures – p. 15/22