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
Stokes structures
Global behaviour of vanishing cycles of functions
X → C in alg. geom. (Pham, Berry...)
Analogy with the theory of wild ramification in
Arithmetic (Deligne...).
I: dimension one
Frobenius manifolds and quantum cohomology
(Dubrovin...).
Claude Sabbah
Centre de Mathématiques Laurent Schwartz
tt∗ geometry (Cecotti & Vafa...).
École polytechnique, CNRS, Université Paris-Saclay
Geometric Langlands correspondence with wild
ramification (Frenkel & B. Gross...).
Palaiseau, France
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
Aim: RH corresp. for merom. ODE’s
Introduction toStokes structures – p. 2/22
3
Riemann-Hilbert corresp. (categorical) on a
punctured Riemann surf. X ∗ = X r S :




 Merom. flat bdles
 loc. cst.
⇐⇒ sheaves of
on (X, S)


 with reg. sing. at S
 finite rk on X ∗
Explicit computation of sols (integral formulas)

Lin. repres.



 π (X ∗ )
1
⇐⇒

↓



GLn (C)
realizing Stokes data with effective solutions
( theory of multisummation)
Constructing moduli spaces of diff. eqns and realizing
the RHB corresp. by a map between moduli spaces.
replacing the group GLn (C) with other reductive
algebraic groups.
Riemann-Hilbert-Birkhoff corresp. (categorical) on a
punctured Riemann surf. X ∗ = X r S :


(


Stokes-filt.

 Generalized
Merom. flat bdles
⇐⇒ loc. syst. ⇐⇒ monodromy data


on (X, S)


f
(Stokes data)
on X
Extending the categorical approach to the Tannakian
aspect ( Differential Galois theory).
Introduction toStokes structures – p. 3/22
Stokes phenomenon in dim. one
Introduction toStokes structures – p. 4/22
5
Linear cplx diff. eqn. df /dz = A(z) · f ,
A(z) matrix of size d × d, merom. pole at z = 0.
Gauge equiv.: P ∈ GLd (C({z})),
Sheaf
A ∼ B = P [A] := P −1 AP + P −1 P ′

 C
+
z
′
ϕd
Asympt. analysis in dim. one
∞
∞
A∆
e = ker z∂z : C e → C e
∆
∆
(A∆
e ∗ = O∆
e ∗)
ϕk ∈ z1 C[ z1 ]
0 ⊂ A 1 ⊂ A mod 0 .
Sheaves ASrd
1
S
S1
C = const.
non reson.
Basic exact sequence:
0
0 −→ ASrd
−→ AS 1 −→ ̟ −1 C[[z]] −→ 0
1
Theorem (Levelt-Turrittin). Given A, ∃ a formal gauge
transf. Pb ∈ GLd (C((z 1/p ))) s.t. B = Pb [A] is a normal
form.
Introduction toStokes structures – p. 5/22
Asympt. analysis in dim. one
e = [0, ε) × S 1 , coord. ρ, eiθ .
Real or. blow-up: ∆
(
e →∆
∆
(ρ, eiθ ) 7−→ z = ρeiθ
̟:
S1 → 0
Sheaf
∞
∞
A∆
e = ker z∂z : C e → C e
∆
∆
Introduction toStokes structures – p. 6/22
7
Asympt. analysis in dim. one
Theorem (Hukuhara-Turrittin).
Locally on S 1 , ∃ a lifting Pe ∈ GLd (AS 1 [1/z]) of Pb s.t.
Pe [A] = Pb [A] = B normal form.
e of sols of
Corollary. The sheaf on ∆
df /dz = A(z) · f
(A∆
e ∗ = O∆
e ∗)
0 ⊂ A 1 ⊂ A mod 0 .
Sheaves ASrd
1
S
S1
Example: ϕ =
s.t. u(z) ∈ C[z], q > 1, and
u(0) 6= 0 or u(z) ≡ 0. Then ∀α ∈ C and ∀eiθo ∈ S 1
( rd 0
Aθ o
⇐⇒ Re(u(0)e−ikθo ) < 0,
α ϕ
z e ∈
0 ⇐⇒ idem or u(z) ≡ 0.
Aθmod
o
u(z)/z q
Introduction toStokes structures – p. 6/22
6
e = [0, ε) × S 1 , coord. ρ, eiθ .
Real or. blow-up: ∆
(
e →∆
∆
(ρ, eiθ ) 7−→ z = ρeiθ
̟:
S1 → 0
∆ = complex disc, complex coord. z .
 ′
ϕ1

Norm. form: B =  . . .
4
Other approaches
having entries in A erd 0 , resp. in A emod 0 , is a real constr. sheaf,
∆
∆
constant on any open interval I of S 1 s.t.
∀k, Re(ϕk ) does not vanish.
Example. ϕ = z −q u(z), u(0) 6= 0,
1
On S 1 , Re ϕ = 0 ⇐⇒ θ = (arg u(0)+π/2) mod Z·π/q.
q
Introduction toStokes structures – p. 7/22
8
The Malgrange-Sibuya theorem
9
Stokes-filtered loc. syst. (non-ramif. case) 10
Fix a norm. form (irregular type), e.g. non-ramified:
B = diag(ϕ′1 , . . . , ϕ′d ) +
Aim: To specify the struct. of sol. space of a merom.
ODE without
making explicit the realization as functions,
fixing the normal form.
C
.
z
B -marked connections (∼: holom. gauge equiv.):
n
o.
Iso(B) = (A, Pb ) | B = Pb [A]
∼
The local system L on S 1 : Sols of df /dz = A(z)f
e = [0, ε) × S 1 and restricted to
on ∆∗ , extended to ∆
1
{0} × S . Hence L ⇐⇒ monodromy of sols.
Stokes sheaf St(B) on S 1 :
n
o
St(B)θ = Id +Q | Q ∈ End(Aθrd 0 ), (Id +Q)[B] = B
For every ϕ ∈ z −1 C[z −1 ], a pair of nested
subsheaves L<ϕ ⊂ L6ϕ of L .
L6ϕ,θ = {fθ | e−ϕ f (z) ∈ Aθmod 0 }
Theorem (Malgrange-Sibuya).
L<ϕ,θ = {fθ | e−ϕ f (z) ∈ Aθrd 0 }
Iso(B) ≃ H 1 (S 1 , St(B))
Hukuhara-Turrittin ⇒ L<ϕ = L6ϕ except if ϕ = ϕk
for some k = 1, . . . , d.
Introduction toStokes structures – p. 9/22
Introduction toStokes structures – p. 8/22
Stokes-filtered loc. syst. (non-ramif. case) 11
Stokes-filtered loc. syst. (non-ramif. case) 12
Let (L , L• ) be a non-ramif. Stokes-filt. loc. syst.
Aim: To give an intrinsic characterization of the
category of Stokes-filtered local systems.
Φ := {ϕ | rk grϕ L 6= 0} is finite and
P
ϕ∈Φ rk grϕ L = rk L .
Definition. A (non-ramif.) Stokes-filt. loc. syst. on S 1 :
A loc. syst. L on S 1 ,
∀ϕ ∈ z −1 C[z −1 ], an R-const. subsheaf L6ϕ ⊂ L
s.t.
ψ = ϕ, or
,
∀θ ∈ S 1 , L6ψ,θ ⊂ L6ϕ,θ ⇐⇒
Re(ψ − ϕ) < 0 near θ
P
setting ∀θ , L<ϕ,θ = ψ<θ ϕ L6ψ,θ
L<ϕ
∀ϕ ∈ Φ, ∀θ ,
Remark: can define (L , L• ) over Z, Q, . . .
L
ψ6θ ϕ
grψ Lθ .
Level structure
Levels of B (hence A) : {q1 < · · · < qr }
qi := pole ord. of some ψ − ϕ,
and grϕ L := L6ϕ /L<ϕ
one asks that ∀ϕ,
grϕ L is a local system on S 1 ,
P
∀θ , dim L6ϕ,θ = ψ6θ ϕ rk grψ L .
(∗)
L6ϕ,θ ≃
ϕ 6= ψ ∈ Φ.
#Levels(A) = 1.
theory of summability.
2q Stokes directions for each (ϕ, ψ).
#Levels(A) > 1.
theory of multisummability.
Principal and Secondary Stokes directions.
Introduction toStokes structures – p. 10/22
Introduction toStokes structures – p. 11/22
Stokes-filtered loc. syst. (non-ramif. case) 13
Theorem
∀ open I ⊂ S 1 which ∋ at most one Stokes dir.
∀ pair in Φ, then (∗) holds on I (e.g. |I| 6 π/qr +ε).
Deligne’s RH correspondence
Theorem (Deligne’s RH corresp.).
(
Any morphism λ : (L , L• ) → (L ′ , L•′ ) graded on I
w.r.t. some iso (∗) and (∗)′ , hence is strict, i.e.,
′ ∩ λ(L ).
∀ϕ, λ(L6ϕ ) = L6ϕ
Uniqueness of the splitting if #Level(A) = 1 and
moreover |I| = π/q + ε.
Duality.
The exact sequences
0 −→ L6ϕ −→ L −→ L >ϕ −→ 0
0 −→ L<−ϕ −→ L −→ L >−ϕ −→ 0
are switched by duality H omC (•, C).
⇒ grϕ (L ∨ ) ≃ (gr−ϕ L )∨ . Extk (• , C) = 0 if k > 1.
Merom. flat bdles
on (∆, 0)
norm. form


 Merom. flat bdles
of norm. form

 on (∆, 0)
≀
≀


 Stokes-filt.
loc. syst.

 on S 1


 graded

 Stokes-filt.
 loc. syst.



on S 1
gr
Introduction toStokes structures – p. 12/22
Stokes data (non-ramif. case, pure level)
Case #Level(A) = 1 (level = q )
∼
∼
L2µ+1 =
L
L
τ2µ+1 : L2µ+1 −→ grF L2µ+1 =
k
grF
k L2µ
L
k
grkF L2µ+1
Stokes multipliers
Opposedness property:
k
Stokes data (non-ramif. case, pure level)
τ2µ : L2µ −→ grF L2µ =
∼
L
15
Opposed filtrations ⇒ unique splittings
Isoms Sℓℓ+1 : Lℓ −→ Lℓ+1
(
F• L2µ
ր
Exhaustive filtrations
•
F L2µ+1 ց
L2µ =
Introduction toStokes structures – p. 13/22
Case #Level(A) = 1 (level = q )
Stokes data
(Lℓ )ℓ∈Z/2qZ : C-vect. spaces,
2µ
Fk L2µ ∩ S2µ−1
(F k L2µ−1 )
2µ+1
k
(Fk L2µ )
k F L2µ+1 ∩ S2µ
Introduction toStokes structures – p. 14/22
14
Σℓ+1
:= τℓ+1 ◦ Sℓℓ+1 ◦ τℓ−1 : grF Lℓ −→ grF Lℓ+1
ℓ
Σℓ+1
block lower/upper triangular,
ℓ
diag. blocks (Σℓ+1
)jj are isos.
ℓ
Introduction toStokes structures – p. 15/22
16
Stokes data (non-ramif. case, pure level)
17
(L , L• ) Stokes-filt. loc. syst. pure level q
Stokes data (non-ramif. case, pure level)
(L , L• ) Stokes-filt. loc. syst. pure level q (e.g. q = 2)
θ0
θ0
⇐⇒ Stokes data of pure level q .
⇐⇒ Stokes data of pure level q . Li = Γ(Ii , L )
Fix θ0 ∈ S 1 not a Stokes dir. ⇒ numbering of Φ s.t.
b1
ϕ1 <θ0 · · · <θ0 ϕr
Lθ 2
2q Stokes dirs (θℓ := θ0 + ℓπ/q)ℓ∈Z/2qZ on S 1 .

 ϕ1 <θ2µ · · · <θ2µ ϕr
=⇒

ϕr <θ2µ+1 · · · <θ2µ+1 ϕ1
=⇒ (L6ϕj ,θℓ )j :
(
θ2
a2
L2
L1
a1
Lθ 1
I1
θ1
b0
I0
I2
b2
filt. ր if ℓ = 2µ
filt. ց if ℓ = 2µ + 1
θ3
Lθ 3
a3
I3
L3
L0
a0
θ0
Lθ 0
b3
Introduction toStokes structures – p. 16/22
Stokes data (non-ramif. case, pure level)
Introduction toStokes structures – p. 17/22
19
(L , L• ) Stokes-filt. loc. syst. pure level q (e.g. q = 2)
Stokes data (non-ramif. case)
L1 F •
Lθ 2
F•
L2
θ2
Lθ 1
S12
S01
S23
S30
F•
Lθ 3
F•
I1
θ1
I0
I2
L0
F•
20
(L , L• ) Stokes-filt. loc. syst. max level qr
(e.g. q1 = 1, q2 = 2)
θ0
⇐⇒ Stokes data of pure level q . Li = Γ(Ii , L )
F•
18
θ3
F•
Lθ 0
I3
θ0
→ more difficult to describe the char. properties of
Stokes data (Lℓ , Sℓℓ+1 )
F • L3
Introduction toStokes structures – p. 17/22
Introduction toStokes structures – p. 18/22
21
Examples
22
Dubrovin’s conjecture
How to compute Stokes data?
X : smooth proj. Fano var. which admits a full
exceptional collection:
Use of the Fourier transf. (Marco’s talk):
more complicated ⇐= simpler.
(E1 , . . . , Em ) ∈ Db (Coh(X)) generate as a
triang. category,
Explicit procedures (..., Mochizuki), but difficult to
obtain closed formulas for Stokes data.
e.g. Airy diff. eq. (ramified)
!
!
0 −1
0 0
z = 1/y
0 1
1 0
(∂y2 −y)u = 0
A(z) = −
+
z4
z
for i 6= j , Extk (Ei , Ej ) = 0 except i < j and
(
k = k(i, j),
0, Hom = C,
k
Ext (Ei , Ei ) = 0 except k =
dim X
SX = (Sij ) ∈ Mm (Z),
X
Sij = χ(Ei , Ej ) :=
(−1)k dim Extk (Ei , Ej ).
Unramified case, q = 1: Fourier transform of a
reg. sing. diff. eq. on P1 .
Gaussian type: irreg. sing. at ∞, unramif., q = 2.
FT of E ϕ : irreg. sing. at ∞, possibly ramif.
k
Introduction toStokes structures – p. 19/22
23
Dubrovin’s conjecture
f : U → C a tame reg. fnct. on a smooth affine var.
Ωmax (U )[z, z −1 ]
(zd + df )Ωmax −1 (U )[z, z −1 ]
,
z 2 ∂z −f
24
Dubrovin’s conjecture
Dubrovin’s conjecture.
If f is the Landau-Ginzburg potential mirror to X good
Fano, then ∃ choices a bases of vanishing cycles of f s.t.
finite # of crit. pts.
GM syst.:
GM :=
Introduction toStokes structures – p. 20/22
GM at z = 0: one level, q = 1, Stokes matr. Sf± , can
be chosen with entries in Z.
Diag. blocks ↔ monodr. of vanishing cycles of f .
Introduction toStokes structures – p. 21/22
Sf+ = SX
X = Pn (Dubrovin, Guzzetti)
...
Introduction toStokes structures – p. 22/22