Wild ramiﬁcation in complex algebraic geometry

Wild ramification incomplex algebraic geometry

Claude Sabbah

Centre de Mathematiques Laurent Schwartz

UMR 7640 du CNRS

Ecole polytechnique, Palaiseau, France

Programme SEDIGA ANR-08-BLAN-0317-01

Wild ramification in complex algebraic geometry – p. 1/17

“Tame” complex algebraic geometry



Underlying space: complex algebraic variety (or cplxanalytic space)




Monodromy←→ local systems, coefficients ink = Q,R,C.





Monodr. + (tame ) sing. −→ k-perverse sheaf.






Riemann-Hilbert correspondence:






Riemann-Hilbert correspondence:Regular holon. D-modules←→ C-perverse sheaves







Hodge Theory in the singular setting:







Hodge Theory in the singular setting:Deligne’s Mixed Hodge complexes







Hodge Theory in the singular setting:Deligne’s Mixed Hodge complexesSaito’s Hodge D-modules







Hodge Theory in the singular setting:Deligne’s Mixed Hodge complexesSaito’s Hodge D-modules

Conversely, Hodge Theory requires tame sing.(Griffiths-Schmid) and C-Alg. Geom. produces tamesing. (Gauss-Manin systems)


“Wild” complex algebraic geometry


“Wild” complex algebraic geometryProblem: Extend these properties to diff. eqns. withirregular sing. (i.e., general holon. D-modules).



What for?



What for?

Exp. integrals: f : X −→ A1,∫

Γefω,



What for?


Γefω,

ω: alg. diff. form on X of degree d,Γ: locally closed cycle of dim. d



What for?


Γefω,


−→ cohom. H∗(X, f) of the complex (Ω•

X , d + df∧).



What for?


Γefω,



X , d + df∧).

Deligne:∫

Re−x

2

dx = π1/2↔

H1(A1,−x2) should have Hodge filtr. 1/2.



What for?


Γefω,



X , d + df∧).

Deligne:∫

Re−x

2

dx = π1/2↔

H1(A1,−x2) should have Hodge filtr. 1/2.Katzarkov-Kontsevich-Pantev: Periodic cycliccohom. of “smooth compact Z/2-graded dg.algebra” should underly an irreg. Hodge structure.



What for?


Γefω,



X , d + df∧).

Deligne:∫

Re−x

2

dx = π1/2↔

H1(A1,−x2) should have Hodge filtr. 1/2.Katzarkov-Kontsevich-Pantev: Periodic cycliccohom. of “smooth compact Z/2-graded dg.algebra” should underly an irreg. Hodge structure.−→ nc. Q-Hodge structure .



What for?


Γefω,



X , d + df∧).

Deligne:∫

Re−x

2

dx = π1/2↔

H1(A1,−x2) should have Hodge filtr. 1/2.Katzarkov-Kontsevich-Pantev: Periodic cycliccohom. of “smooth compact Z/2-graded dg.algebra” should underly an irreg. Hodge structure.−→ nc. Q-Hodge structure .

Better analogy with constr. Qℓ-sheaves on XFq.


A panorama of “wild” results



THEOREM (C.S.):



THEOREM (C.S.): A var. of polarized Q-Hodge struct.on A1 r p1, . . . , pr;



THEOREM (C.S.): A var. of polarized Q-Hodge struct.on A1 r p1, . . . , pr; Its Fourier transf. is a var. ofpolarized nc. Q-Hodge structure on A1 r 0.



THEOREM (C.S.): A var. of polarized Q-Hodge struct.on A1 r p1, . . . , pr; Its Fourier transf. is a var. ofpolarized nc. Q-Hodge structure on A1 r 0.(compare with Katz-Laumon 1985).




THEOREM (T. Mochizuki):




THEOREM (T. Mochizuki): X smooth projective /C,M a simple holon. DX -mod. =⇒ Hard Lefschetzholds for H∗(X,DR M ).





THEOREM (C. Hertling, H. Iritani, Reichelt-Sevenheck,C.S.):





THEOREM (C. Hertling, H. Iritani, Reichelt-Sevenheck,C.S.): Quantum cohom. of Fano toric varietiesunderlies a var. of polarized nc. Q-Hodge structureon a Zariski dense open set of the Kähler modulispace.


Irregular singularities on curves



X: smooth curve /C, M : holonomic DX -moduleD ⊂ X: singular set of M .




If M has regular sing. , Riemann-Hilbert corr.:




If M has regular sing. , Riemann-Hilbert corr.:DR : M 7−→ DR M is an equiv:Modhol-reg(DX)

∼−→ PervC(X).





∼−→ PervC(X).

If M is a vect. bdle with connection (D = ∅),possible irreg. sing. at∞:





∼−→ PervC(X).

If M is a vect. bdle with connection (D = ∅),possible irreg. sing. at∞: −→ Stokes phenom.





∼−→ PervC(X).


Formal decomposition (Levelt-Turrittin) + Stokesmatrices.





∼−→ PervC(X).


Formal decomposition (Levelt-Turrittin) + Stokesmatrices. Well-suited to Diff. Galois theory.





∼−→ PervC(X).


Formal decomposition (Levelt-Turrittin) + Stokesmatrices. Well-suited to Diff. Galois theory.Stokes-filtered local systems (Deligne, Malgrange)+ R-H corresp. Well-suited to holon. DX -modulesand higher dim.





∼−→ PervC(X).


Formal decomposition (Levelt-Turrittin) + Stokesmatrices. Well-suited to Diff. Galois theory.Stokes-filtered local systems (Deligne, Malgrange)+ R-H corresp. Well-suited to holon. DX -modulesand higher dim.

R-H : Modhol(DX)∼−→ Stokes-PervC(X)


Stokes-filtered local systems (dim. one)


Stokes-filtered local systems (dim. one)M = C(x)-vect. space with conn. ∇.


Stokes-filtered local systems (dim. one)M = C(x)-vect. space with conn. ∇.M = C((x))⊗M with conn. ∇


Stokes-filtered local systems (dim. one)M = C(x)-vect. space with conn. ∇.M = C((x))⊗M with conn. ∇Levelt-Turrittin (without ramif.): (M , ∇) ≃

⊕η(E

η ⊗ Rη)



⊕η(E

η ⊗ Rη)

η∈Φ⊂x−1C[x−1],



⊕η(E

η ⊗ Rη)

η∈Φ⊂x−1C[x−1], E η = (C((x)), d + dη),



⊕η(E

η ⊗ Rη)

η∈Φ⊂x−1C[x−1], E η = (C((x)), d + dη), Rη reg. sing.



⊕η(E

η ⊗ Rη)


monodromy

formalmonodromy

Stokesphenomenon



⊕η(E

η ⊗ Rη)


monodromy

formalmonodromy

Stokesphenomenon



Deligne (1978): Stokes filtration




Polar coord. x = |x|eiθ, S1 = |x| = 0





Order on x−1C[x−1] depending on eiθ ∈ S1:





Order on x−1C[x−1] depending on eiθ ∈ S1:η 6

θoψ iff Re(η − ψ)(x) 6 0 for arg x ∼ θo and

0 < |x| ≪ 1.







0 < |x| ≪ 1.

η not comparable to ψ







0 < |x| ≪ 1.


η not comparable to ϕ







0 < |x| ≪ 1.


η not comparable to ϕ

η not comparable to ω







0 < |x| ≪ 1.

L local syst. on S1 (e.g., L = (M )∇)







0 < |x| ≪ 1.


For each η ∈ x−1C[x−1], R-constr. subsheafL6η ⊂ L , s.t.







0 < |x| ≪ 1.


For each η ∈ x−1C[x−1], R-constr. subsheafL6η ⊂ L , s.t.η 6

θoψ =⇒ L6η,θo

⊂ L6ψ,θo(filtr. cond. )







0 < |x| ≪ 1.



θoψ =⇒ L6η,θo


Set L<η =∑ψ<η L6ψ,







0 < |x| ≪ 1.



θoψ =⇒ L6η,θo


Set L<η =∑ψ<η L6ψ, grη L = L6η/L<ψ.

Then







0 < |x| ≪ 1.



θoψ =⇒ L6η,θo



Then loc. on S1, L ≃⊕η grη L .







0 < |x| ≪ 1.



θoψ =⇒ L6η,θo



Then loc. on S1, L ≃⊕η grη L . (=⇒ each

grη L is a loc. syst. on S1)







0 < |x| ≪ 1.



θoψ =⇒ L6η,θo



Then loc. on S1, L ≃⊕η grη L . (=⇒ each

grη L is a loc. syst. on S1) (loc. grad. cond. )


Riemann-Hilbert corr. (local case)



(M ,∇): C(x)-vect. space with conn.




(L ,L•): Stokes-filtered C-loc. syst.





For each η ∈ x−1C[x−1], R-constr. subsheafL6η ⊂ L






filtr. cond. ,






filtr. cond. ,loc. grad. cond.







THEOREM (Deligne): ∃ a R-H corr. (equivalence)







THEOREM (Deligne): ∃ a R-H corr. (equivalence)

(M ,∇) ∼

C((x))⊗C(x)

(L ,L•)

gr

(M , ∇)∼ (gr L , gr L

•)


Stokes-perverse sheaves (dim. one)



(X,D) smooth proj. curve /C with div. D.




: X(D) −→ X = real oriented blow-up of X at D





I = −1(OX(∗D)/OX

)sheaf on X,

zero on X∗ :=XrD





I = −1(OX(∗D)/OX

)sheaf on X,

zero on X∗ :=XrD(should take “ramified polar parts” instead)





I = −1(OX(∗D)/OX

)sheaf on X,

zero on X∗ :=XrD

I: sheaf of ordered groups





I = −1(OX(∗D)/OX

)sheaf on X,

zero on X∗ :=XrD


Iét: étale space of I (not Hausdorff over D)





I = −1(OX(∗D)/OX

)sheaf on X,

zero on X∗ :=XrD


Iét: étale space of I (not Hausdorff over D):

point of Iét = germ η





I = −1(OX(∗D)/OX

)sheaf on X,

zero on X∗ :=XrD




k a field (coeffs)





I = −1(OX(∗D)/OX

)sheaf on X,

zero on X∗ :=XrD




k a field (coeffs)DEFINITION:





I = −1(OX(∗D)/OX

)sheaf on X,

zero on X∗ :=XrD




k a field (coeffs)DEFINITION:k-Loc. syst. on X∗ with Stokes filtr. at D ⇐⇒sheaf F6 on I

ét comp. with order s.t. F6|X∗ = k-loc syst.on X∗ and F6|−1(D) =: L6 = k-Stokes filtr.


Riemann-Hilbert corr. (global case)DEFINITION:k-Loc. syst. on X∗ with Stokes filtr. at D ⇐⇒sheaf F6 on I





PROPOSITION: Abelian category, morphisms strictlyfiltered.





THEOREM (Deligne): ∃ R-H equivalence“alg. bdles with connection on X∗”←→“C-Loc. syst. on X∗ with Stokes filtr. at D”.






Adding data at D −→ k-Stokes perverse sheaves on(X,D).







THEOREM (Deligne, Malgrange): ∃ R-H equivalence“hol. DX -modules on (X,D)”←→ “C-Stokes perversesheaves on (X,D)”.







THEOREM (Deligne, Malgrange): ∃ R-H equivalence“hol. DX -modules on (X,D)”←→ “C-Stokes perversesheaves on (X,D)”. Compatible with duality.


Levelt-Turrittin in dim. > 2


Levelt-Turrittin in dim. > 2QUESTION (2 variables):


Levelt-Turrittin in dim. > 2QUESTION (2 variables):(M , ∇): free C[[x1, x2]][x

−11 , x−1

2 ]-module with flat connect.



−11 , x−1

2 ]-module with flat connect.∃? (maybe after ramif. x1 = yr11 , x2 = yr22 ) a decomp.

(M , ∇) ≃⊕η(E

η ⊗ Rη)



−11 , x−1


(M , ∇) ≃⊕η(E

η ⊗ Rη)

η∈Φ⊂C[[x1, x2]][x−11 , x−1

2 ]/C[[x1, x2]],



−11 , x−1


(M , ∇) ≃⊕η(E

η ⊗ Rη)

η∈Φ⊂C[[x1, x2]][x−11 , x−1

2 ]/C[[x1, x2]],E η = (C[[x1, x2]][x

−11 , x−1

2 ], d + dη),



−11 , x−1


(M , ∇) ≃⊕η(E

η ⊗ Rη)

η∈Φ⊂C[[x1, x2]][x−11 , x−1

2 ]/C[[x1, x2]],E η = (C[[x1, x2]][x

−11 , x−1

2 ], d + dη),Rη reg. sing. along D = x1x2 = 0.



−11 , x−1


(M , ∇) ≃⊕η(E

η ⊗ Rη)

η∈Φ⊂C[[x1, x2]][x−11 , x−1

2 ]/C[[x1, x2]],E η = (C[[x1, x2]][x

−11 , x−1


ANSWER: no , ∃ counter-ex.



−11 , x−1


(M , ∇) ≃⊕η(E

η ⊗ Rη)

η∈Φ⊂C[[x1, x2]][x−11 , x−1

2 ]/C[[x1, x2]],E η = (C[[x1, x2]][x

−11 , x−1



TENTATIVE STATEMENT (C.S., 1993): Local formalexistence after a sequence of blowing-up ?



−11 , x−1


(M , ∇) ≃⊕η(E

η ⊗ Rη)

η∈Φ⊂C[[x1, x2]][x−11 , x−1

2 ]/C[[x1, x2]],E η = (C[[x1, x2]][x

−11 , x−1




REFINED TENTATIVE STATEMENT (C.S., 1993): Add a“goodness” condition in order to avoid e.g. η = x1/x2.



−11 , x−1


(M , ∇) ≃⊕η(E

η ⊗ Rη)

η∈Φ⊂C[[x1, x2]][x−11 , x−1

2 ]/C[[x1, x2]],E η = (C[[x1, x2]][x

−11 , x−1




REFINED TENTATIVE STATEMENT (C.S., 1993): Add a“goodness” condition in order to avoid e.g. η = x1/x2.(good formal structure, import. for asympt. analysis )


Levelt-Turrittin in dim. > 2


Levelt-Turrittin in dim. > 2DEFINTION: D = nc. divisor in X. A familyΦ ⊂ OX(∗D)/OX of local sect. is good if

∀η, ψ ∈ Φ, div(η − ψ) =∑

i

niDi, with ni 6 0 ∀i.



∀η, ψ ∈ Φ, div(η − ψ) =∑

i


THEOREM (K. Kedlaya, T. Mochizuki, 2008-2009): (M ,∇)coh. OX(∗D)-module with flat conn. Then ∃ a finite seq.of blow-ups e : X ′ −→ X centered in D s.t. e∗(M ,∇)

has good formal structure at each point of e−1(D).



∀η, ψ ∈ Φ, div(η − ψ) =∑

i




PREVIOUS WORK: Gérard-Sibuya (1979),Levelt-van den Essen (1982), H. Majima (1984),C.S. (2000): rkM 6 5,Y. André (2007): direct proof of Malgrange’s conj.



∀η, ψ ∈ Φ, div(η − ψ) =∑

i




PREVIOUS WORK: Gérard-Sibuya (1979),Levelt-van den Essen (1982), H. Majima (1984),C.S. (2000): rkM 6 5,Y. André (2007): direct proof of Malgrange’s conj.

APPLICATION TO ASYMPT. ANALYSIS: Y. Sibuya (70’s),H. Majima (1984), C.S. (1993, 2000): dimX = 2,T. Mochizuki (2010): dimX > 2.


Stokes-filtered loc. syst. (dim.> 2)



(X,D) smooth variety /C with nc div. D.













: X(D)→X = real oriented blow-up of X at (Di)





I = −1(OX(∗D)/OX

)sheaf on X,

zero on X∗ :=XrD





I = −1(OX(∗D)/OX

)sheaf on X,

zero on X∗ :=XrD(should take “ramified polar parts” instead)





I = −1(OX(∗D)/OX

)sheaf on X,

zero on X∗ :=XrD






I = −1(OX(∗D)/OX

)sheaf on X,

zero on X∗ :=XrD


Iét: étale space of I (not Hausdorff over D)





I = −1(OX(∗D)/OX

)sheaf on X,

zero on X∗ :=XrD








I = −1(OX(∗D)/OX

)sheaf on X,

zero on X∗ :=XrD




DEFINITION:





I = −1(OX(∗D)/OX

)sheaf on X,

zero on X∗ :=XrD




DEFINITION: k-Loc. syst. on X∗ with Stokes filtr. at D⇐⇒ sheaf F6 on I

ét s.t., on each stratumfiltr. cond. and loc. grad. cond.compatibility cond. between strata.









Loc. grad. cond. ⇒ loc. on X, F6 ≃ gr F6 =⊕η grη F6.






DEFINITION (goodness): F6 is good if locally on X, thefamily (η) is good.







Support Σ(gr F6) ⊂ Iét: stratified covering of ∂X(D).







Support Σ(gr F6) ⊂ Iét: stratified covering of ∂X(D).

THEOREM: Fix a good Σ ⊂ Iét. Then the category of F6

with support⊂ Σ is abelian and every morphism is strict .


Riemann-Hilbert corr. (global case)


Riemann-Hilbert corr. (global case)THEOREM (T. Mochizuki, C.S.): ∃ R-H equivalence“hol. bdles with connection on (X,D) with good formalstruct. ” ←→ “good Stokes filtered C-loc. syst. on I

ét”.



ét”.

QUESTION: General notion of Stokes-perverse sheaf?



ét”.


PROBLEM: Behaviour of the Stokes filtration by properpush-forward?



ét”.



OTHER ANSWER (T. Mochizuki): Embed the Stokesfiltration inside a hol. D-module −→ hol. D-modulewith k-Betti structure .



ét”.



OTHER ANSWER (T. Mochizuki): Embed the Stokesfiltration inside a hol. D-module −→ hol. D-modulewith k-Betti structure .

Deligne, 2007: “La théorie des structures de Stokesfournit une notion de structure de Betti si dim(X) = 1.On voudrait une définition en toute dimension, et unestabilité par les six opérations(Rf∗, Rf!, f

∗, Rf !,⊗L,RHom). On est loin du compte.”Wild ramification in complex algebraic geometry – p. 15/17

Example


Example

x

0

P1

∞

P1


Example

x

z

0

P1

∞A1

P1 × A1


Example

x

z

0

p

P1

z

0

∞

A1

A1

P1 × A1 p−→ A1


Example

x

z

0

p

P1

z

0

∞

A1

A1

P1 × A1 p−→ A1

M = reg. hol. DP1×A1-mod.


Example

x

z

0

Si

p

P1

z

0

∞

A1

A1

P1 × A1 p−→ A1


S =⋃j Sj = sing. set of M


Example

x

z

0

Si

p

P1

z

0

∞

A1

A1

P1 × A1 p−→ A1



Pb: Levelt-Turrittin ofN := p∗(E

x ⊗M )

i.e. diff. eqn for∫

γz

f(x, z)ex dx

f : sol. of M


Example

x

0

Si

p∆

P1

z

0

∞ z1/q

1/q

P1 × A1 p−→ A1



Pb: Levelt-Turrittin ofN := p∗(E

x ⊗M )

i.e. diff. eqn for∫

γz

f(x, z)ex dx

f : sol. of M

ramif : z1/q


Example

x

0

Si

p∆

P1

z

0

∞ z1/q

1/q

THM (C. Roucairol, 2007):N =

⊕i(E

ηi ⊗ Ri)

ηi(z) = pol. part of x(z)|Si

Ri = vanishing cyclemodule of M along Si.


Example

x

0

Si

p∆

P1

z

0

∞ z1/q

1/q

E1

E2

E3

E4

s s s

En−1

En

s s s

e


Example

x

0

Si

p∆

P1

z

0

∞ z1/q

1/q

E1

E2

E3

E4

s s s

En−1

En

s s s

e

THM (C.S.):Stokes filtr. of N =p∗(E

x⊗M )= push-forward of theStokes filtr. of e∗(E x⊗M ).


Example

x

0

Si

p∆

P1

z

0

∞ z1/q

1/q

E1

E2

E3

E4

s s s

En−1

En

s s s

e


Example

x

0

Si

p∆

P1

z

0

∞ z1/q

1/q

E1

E2

E3

E4

s s s

En−1

En

s s s

e


Example

x

0

Si

p∆

P1

z

0

∞ z1/q

1/q

E1

E2

E3

E4

s s s

En−1

En

s s s

e


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Wild ramiﬁcation in complex algebraic geometry