Wild ramification in complex algebraic geometry
Transcript of Wild ramification in complex algebraic geometry
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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
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“Tame” complex algebraic geometry
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“Tame” complex algebraic geometry
Underlying space: complex algebraic variety (or cplxanalytic space)
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“Tame” complex algebraic geometry
Underlying space: complex algebraic variety (or cplxanalytic space)
Monodromy←→ local systems, coefficients ink = Q,R,C.
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“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.
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“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:
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“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:Regular holon. D-modules←→ C-perverse sheaves
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“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:Regular holon. D-modules←→ C-perverse sheaves
Hodge Theory in the singular setting:
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“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:Regular holon. D-modules←→ C-perverse sheaves
Hodge Theory in the singular setting:Deligne’s Mixed Hodge complexes
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“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:Regular holon. D-modules←→ C-perverse sheaves
Hodge Theory in the singular setting:Deligne’s Mixed Hodge complexesSaito’s Hodge D-modules
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“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:Regular holon. D-modules←→ C-perverse sheaves
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)
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“Wild” complex algebraic geometry
Wild ramification in complex algebraic geometry – p. 3/17
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“Wild” complex algebraic geometryProblem: Extend these properties to diff. eqns. withirregular sing. (i.e., general holon. D-modules).
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“Wild” complex algebraic geometryProblem: Extend these properties to diff. eqns. withirregular sing. (i.e., general holon. D-modules).
What for?
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“Wild” complex algebraic geometryProblem: Extend these properties to diff. eqns. withirregular sing. (i.e., general holon. D-modules).
What for?
Exp. integrals: f : X −→ A1,∫
Γefω,
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“Wild” complex algebraic geometryProblem: Extend these properties to diff. eqns. withirregular sing. (i.e., general holon. D-modules).
What for?
Exp. integrals: f : X −→ A1,∫
Γefω,
ω: alg. diff. form on X of degree d,Γ: locally closed cycle of dim. d
Wild ramification in complex algebraic geometry – p. 3/17
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“Wild” complex algebraic geometryProblem: Extend these properties to diff. eqns. withirregular sing. (i.e., general holon. D-modules).
What for?
Exp. integrals: f : X −→ A1,∫
Γefω,
ω: alg. diff. form on X of degree d,Γ: locally closed cycle of dim. d
−→ cohom. H∗(X, f) of the complex (Ω•
X , d + df∧).
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“Wild” complex algebraic geometryProblem: Extend these properties to diff. eqns. withirregular sing. (i.e., general holon. D-modules).
What for?
Exp. integrals: f : X −→ A1,∫
Γefω,
ω: alg. diff. form on X of degree d,Γ: locally closed cycle of dim. d
−→ cohom. H∗(X, f) of the complex (Ω•
X , d + df∧).
Deligne:∫
Re−x
2
dx = π1/2↔
H1(A1,−x2) should have Hodge filtr. 1/2.
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“Wild” complex algebraic geometryProblem: Extend these properties to diff. eqns. withirregular sing. (i.e., general holon. D-modules).
What for?
Exp. integrals: f : X −→ A1,∫
Γefω,
ω: alg. diff. form on X of degree d,Γ: locally closed cycle of dim. d
−→ cohom. H∗(X, f) of the complex (Ω•
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.
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“Wild” complex algebraic geometryProblem: Extend these properties to diff. eqns. withirregular sing. (i.e., general holon. D-modules).
What for?
Exp. integrals: f : X −→ A1,∫
Γefω,
ω: alg. diff. form on X of degree d,Γ: locally closed cycle of dim. d
−→ cohom. H∗(X, f) of the complex (Ω•
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 .
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“Wild” complex algebraic geometryProblem: Extend these properties to diff. eqns. withirregular sing. (i.e., general holon. D-modules).
What for?
Exp. integrals: f : X −→ A1,∫
Γefω,
ω: alg. diff. form on X of degree d,Γ: locally closed cycle of dim. d
−→ cohom. H∗(X, f) of the complex (Ω•
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.
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A panorama of “wild” results
Wild ramification in complex algebraic geometry – p. 4/17
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A panorama of “wild” results
THEOREM (C.S.):
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A panorama of “wild” results
THEOREM (C.S.): A var. of polarized Q-Hodge struct.on A1 r p1, . . . , pr;
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A panorama of “wild” results
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.
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A panorama of “wild” results
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).
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A panorama of “wild” results
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):
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A panorama of “wild” results
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): X smooth projective /C,M a simple holon. DX -mod. =⇒ Hard Lefschetzholds for H∗(X,DR M ).
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A panorama of “wild” results
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): 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.):
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A panorama of “wild” results
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): 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.): Quantum cohom. of Fano toric varietiesunderlies a var. of polarized nc. Q-Hodge structureon a Zariski dense open set of the Kähler modulispace.
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Irregular singularities on curves
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Irregular singularities on curves
X: smooth curve /C, M : holonomic DX -moduleD ⊂ X: singular set of M .
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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.:
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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.:DR : M 7−→ DR M is an equiv:Modhol-reg(DX)
∼−→ PervC(X).
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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.:DR : M 7−→ DR M is an equiv:Modhol-reg(DX)
∼−→ PervC(X).
If M is a vect. bdle with connection (D = ∅),possible irreg. sing. at∞:
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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.:DR : M 7−→ DR M is an equiv:Modhol-reg(DX)
∼−→ PervC(X).
If M is a vect. bdle with connection (D = ∅),possible irreg. sing. at∞: −→ Stokes phenom.
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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.:DR : M 7−→ DR M is an equiv:Modhol-reg(DX)
∼−→ PervC(X).
If M is a vect. bdle with connection (D = ∅),possible irreg. sing. at∞: −→ Stokes phenom.
Formal decomposition (Levelt-Turrittin) + Stokesmatrices.
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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.:DR : M 7−→ DR M is an equiv:Modhol-reg(DX)
∼−→ PervC(X).
If M is a vect. bdle with connection (D = ∅),possible irreg. sing. at∞: −→ Stokes phenom.
Formal decomposition (Levelt-Turrittin) + Stokesmatrices. Well-suited to Diff. Galois theory.
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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.:DR : M 7−→ DR M is an equiv:Modhol-reg(DX)
∼−→ PervC(X).
If M is a vect. bdle with connection (D = ∅),possible irreg. sing. at∞: −→ Stokes phenom.
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.
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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.:DR : M 7−→ DR M is an equiv:Modhol-reg(DX)
∼−→ PervC(X).
If M is a vect. bdle with connection (D = ∅),possible irreg. sing. at∞: −→ Stokes phenom.
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)
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Stokes-filtered local systems (dim. one)
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Stokes-filtered local systems (dim. one)M = C(x)-vect. space with conn. ∇.
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Stokes-filtered local systems (dim. one)M = C(x)-vect. space with conn. ∇.M = C((x))⊗M with conn. ∇
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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)
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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∈Φ⊂x−1C[x−1],
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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∈Φ⊂x−1C[x−1], E η = (C((x)), d + dη),
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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∈Φ⊂x−1C[x−1], E η = (C((x)), d + dη), Rη reg. sing.
Wild ramification in complex algebraic geometry – p. 6/17
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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∈Φ⊂x−1C[x−1], E η = (C((x)), d + dη), Rη reg. sing.
monodromy
formalmonodromy
Stokesphenomenon
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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∈Φ⊂x−1C[x−1], E η = (C((x)), d + dη), Rη reg. sing.
monodromy
formalmonodromy
Stokesphenomenon
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Stokes-filtered local systems (dim. one)
Deligne (1978): Stokes filtration
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Stokes-filtered local systems (dim. one)
Deligne (1978): Stokes filtration
Polar coord. x = |x|eiθ, S1 = |x| = 0
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Stokes-filtered local systems (dim. one)
Deligne (1978): Stokes filtration
Polar coord. x = |x|eiθ, S1 = |x| = 0
Order on x−1C[x−1] depending on eiθ ∈ S1:
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Stokes-filtered local systems (dim. one)
Deligne (1978): Stokes filtration
Polar coord. x = |x|eiθ, S1 = |x| = 0
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.
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Stokes-filtered local systems (dim. one)
Deligne (1978): Stokes filtration
Polar coord. x = |x|eiθ, S1 = |x| = 0
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.
η not comparable to ψ
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Stokes-filtered local systems (dim. one)
Deligne (1978): Stokes filtration
Polar coord. x = |x|eiθ, S1 = |x| = 0
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.
η not comparable to ψ
η not comparable to ϕ
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Stokes-filtered local systems (dim. one)
Deligne (1978): Stokes filtration
Polar coord. x = |x|eiθ, S1 = |x| = 0
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.
η not comparable to ψ
η not comparable to ϕ
η not comparable to ω
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Stokes-filtered local systems (dim. one)
Deligne (1978): Stokes filtration
Polar coord. x = |x|eiθ, S1 = |x| = 0
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.
L local syst. on S1 (e.g., L = (M )∇)
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Stokes-filtered local systems (dim. one)
Deligne (1978): Stokes filtration
Polar coord. x = |x|eiθ, S1 = |x| = 0
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.
L local syst. on S1 (e.g., L = (M )∇)
For each η ∈ x−1C[x−1], R-constr. subsheafL6η ⊂ L , s.t.
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Stokes-filtered local systems (dim. one)
Deligne (1978): Stokes filtration
Polar coord. x = |x|eiθ, S1 = |x| = 0
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.
L local syst. on S1 (e.g., L = (M )∇)
For each η ∈ x−1C[x−1], R-constr. subsheafL6η ⊂ L , s.t.η 6
θoψ =⇒ L6η,θo
⊂ L6ψ,θo(filtr. cond. )
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Stokes-filtered local systems (dim. one)
Deligne (1978): Stokes filtration
Polar coord. x = |x|eiθ, S1 = |x| = 0
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.
L local syst. on S1 (e.g., L = (M )∇)
For each η ∈ x−1C[x−1], R-constr. subsheafL6η ⊂ L , s.t.η 6
θoψ =⇒ L6η,θo
⊂ L6ψ,θo(filtr. cond. )
Set L<η =∑ψ<η L6ψ,
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Stokes-filtered local systems (dim. one)
Deligne (1978): Stokes filtration
Polar coord. x = |x|eiθ, S1 = |x| = 0
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.
L local syst. on S1 (e.g., L = (M )∇)
For each η ∈ x−1C[x−1], R-constr. subsheafL6η ⊂ L , s.t.η 6
θoψ =⇒ L6η,θo
⊂ L6ψ,θo(filtr. cond. )
Set L<η =∑ψ<η L6ψ, grη L = L6η/L<ψ.
Then
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Stokes-filtered local systems (dim. one)
Deligne (1978): Stokes filtration
Polar coord. x = |x|eiθ, S1 = |x| = 0
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.
L local syst. on S1 (e.g., L = (M )∇)
For each η ∈ x−1C[x−1], R-constr. subsheafL6η ⊂ L , s.t.η 6
θoψ =⇒ L6η,θo
⊂ L6ψ,θo(filtr. cond. )
Set L<η =∑ψ<η L6ψ, grη L = L6η/L<ψ.
Then loc. on S1, L ≃⊕η grη L .
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Stokes-filtered local systems (dim. one)
Deligne (1978): Stokes filtration
Polar coord. x = |x|eiθ, S1 = |x| = 0
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.
L local syst. on S1 (e.g., L = (M )∇)
For each η ∈ x−1C[x−1], R-constr. subsheafL6η ⊂ L , s.t.η 6
θoψ =⇒ L6η,θo
⊂ L6ψ,θo(filtr. cond. )
Set L<η =∑ψ<η L6ψ, grη L = L6η/L<ψ.
Then loc. on S1, L ≃⊕η grη L . (=⇒ each
grη L is a loc. syst. on S1)
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Stokes-filtered local systems (dim. one)
Deligne (1978): Stokes filtration
Polar coord. x = |x|eiθ, S1 = |x| = 0
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.
L local syst. on S1 (e.g., L = (M )∇)
For each η ∈ x−1C[x−1], R-constr. subsheafL6η ⊂ L , s.t.η 6
θoψ =⇒ L6η,θo
⊂ L6ψ,θo(filtr. cond. )
Set L<η =∑ψ<η L6ψ, grη L = L6η/L<ψ.
Then loc. on S1, L ≃⊕η grη L . (=⇒ each
grη L is a loc. syst. on S1) (loc. grad. cond. )
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Riemann-Hilbert corr. (local case)
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Riemann-Hilbert corr. (local case)
(M ,∇): C(x)-vect. space with conn.
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Riemann-Hilbert corr. (local case)
(M ,∇): C(x)-vect. space with conn.
(L ,L•): Stokes-filtered C-loc. syst.
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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
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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. ,
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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. ,loc. grad. cond.
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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. ,loc. grad. cond.
THEOREM (Deligne): ∃ a R-H corr. (equivalence)
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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. ,loc. grad. cond.
THEOREM (Deligne): ∃ a R-H corr. (equivalence)
(M ,∇) ∼
C((x))⊗C(x)
(L ,L•)
gr
(M , ∇)∼ (gr L , gr L
•)
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Stokes-perverse sheaves (dim. one)
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Stokes-perverse sheaves (dim. one)
(X,D) smooth proj. curve /C with div. D.
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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
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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
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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(should take “ramified polar parts” instead)
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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: sheaf of ordered groups
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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: sheaf of ordered groups
Iét: étale space of I (not Hausdorff over D)
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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: sheaf of ordered groups
Iét: étale space of I (not Hausdorff over D):
point of Iét = germ η
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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: sheaf of ordered groups
Iét: étale space of I (not Hausdorff over D):
point of Iét = germ η
k a field (coeffs)
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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: sheaf of ordered groups
Iét: étale space of I (not Hausdorff over D):
point of Iét = germ η
k a field (coeffs)DEFINITION:
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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: sheaf of ordered groups
Iét: étale space of I (not Hausdorff over D):
point of Iét = germ η
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.
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Riemann-Hilbert corr. (global case)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.
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Riemann-Hilbert corr. (global case)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.
PROPOSITION: Abelian category, morphisms strictlyfiltered.
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Riemann-Hilbert corr. (global case)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.
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”.
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Riemann-Hilbert corr. (global case)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.
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).
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Riemann-Hilbert corr. (global case)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.
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)”.
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Riemann-Hilbert corr. (global case)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.
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)”. Compatible with duality.
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Levelt-Turrittin in dim. > 2
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Levelt-Turrittin in dim. > 2QUESTION (2 variables):
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Levelt-Turrittin in dim. > 2QUESTION (2 variables):(M , ∇): free C[[x1, x2]][x
−11 , x−1
2 ]-module with flat connect.
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Levelt-Turrittin in dim. > 2QUESTION (2 variables):(M , ∇): free C[[x1, x2]][x
−11 , x−1
2 ]-module with flat connect.∃? (maybe after ramif. x1 = yr11 , x2 = yr22 ) a decomp.
(M , ∇) ≃⊕η(E
η ⊗ Rη)
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Levelt-Turrittin in dim. > 2QUESTION (2 variables):(M , ∇): free C[[x1, x2]][x
−11 , x−1
2 ]-module with flat connect.∃? (maybe after ramif. x1 = yr11 , x2 = yr22 ) a decomp.
(M , ∇) ≃⊕η(E
η ⊗ Rη)
η∈Φ⊂C[[x1, x2]][x−11 , x−1
2 ]/C[[x1, x2]],
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Levelt-Turrittin in dim. > 2QUESTION (2 variables):(M , ∇): free C[[x1, x2]][x
−11 , x−1
2 ]-module with flat connect.∃? (maybe after ramif. x1 = yr11 , x2 = yr22 ) a decomp.
(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),
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Levelt-Turrittin in dim. > 2QUESTION (2 variables):(M , ∇): free C[[x1, x2]][x
−11 , x−1
2 ]-module with flat connect.∃? (maybe after ramif. x1 = yr11 , x2 = yr22 ) a decomp.
(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.
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Levelt-Turrittin in dim. > 2QUESTION (2 variables):(M , ∇): free C[[x1, x2]][x
−11 , x−1
2 ]-module with flat connect.∃? (maybe after ramif. x1 = yr11 , x2 = yr22 ) a decomp.
(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.
ANSWER: no , ∃ counter-ex.
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Levelt-Turrittin in dim. > 2QUESTION (2 variables):(M , ∇): free C[[x1, x2]][x
−11 , x−1
2 ]-module with flat connect.∃? (maybe after ramif. x1 = yr11 , x2 = yr22 ) a decomp.
(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.
ANSWER: no , ∃ counter-ex.
TENTATIVE STATEMENT (C.S., 1993): Local formalexistence after a sequence of blowing-up ?
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Levelt-Turrittin in dim. > 2QUESTION (2 variables):(M , ∇): free C[[x1, x2]][x
−11 , x−1
2 ]-module with flat connect.∃? (maybe after ramif. x1 = yr11 , x2 = yr22 ) a decomp.
(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.
ANSWER: no , ∃ counter-ex.
TENTATIVE STATEMENT (C.S., 1993): Local formalexistence after a sequence of blowing-up ?
REFINED TENTATIVE STATEMENT (C.S., 1993): Add a“goodness” condition in order to avoid e.g. η = x1/x2.
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Levelt-Turrittin in dim. > 2QUESTION (2 variables):(M , ∇): free C[[x1, x2]][x
−11 , x−1
2 ]-module with flat connect.∃? (maybe after ramif. x1 = yr11 , x2 = yr22 ) a decomp.
(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.
ANSWER: no , ∃ counter-ex.
TENTATIVE STATEMENT (C.S., 1993): Local formalexistence after a sequence of blowing-up ?
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 )
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Levelt-Turrittin in dim. > 2
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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.
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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.
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).
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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.
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).
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.
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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.
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).
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.
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Stokes-filtered loc. syst. (dim.> 2)
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Stokes-filtered loc. syst. (dim.> 2)
(X,D) smooth variety /C with nc div. D.
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Stokes-filtered loc. syst. (dim.> 2)
(X,D) smooth variety /C with nc div. D.
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Stokes-filtered loc. syst. (dim.> 2)
(X,D) smooth variety /C with nc div. D.
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Stokes-filtered loc. syst. (dim.> 2)
(X,D) smooth variety /C with nc div. D.
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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)
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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
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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(should take “ramified polar parts” instead)
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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: sheaf of ordered groups
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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: sheaf of ordered groups
Iét: étale space of I (not Hausdorff over D)
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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: sheaf of ordered groups
Iét: étale space of I (not Hausdorff over D):
point of Iét = germ η
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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: sheaf of ordered groups
Iét: étale space of I (not Hausdorff over D):
point of Iét = germ η
DEFINITION:
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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: sheaf of ordered groups
Iét: étale space of I (not Hausdorff over D):
point of Iét = germ η
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.
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Stokes-filtered loc. syst. (dim.> 2)
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.
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Stokes-filtered loc. syst. (dim.> 2)
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.
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Stokes-filtered loc. syst. (dim.> 2)
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.
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Stokes-filtered loc. syst. (dim.> 2)
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).
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Stokes-filtered loc. syst. (dim.> 2)
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).
THEOREM: Fix a good Σ ⊂ Iét. Then the category of F6
with support⊂ Σ is abelian and every morphism is strict .
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Riemann-Hilbert corr. (global case)
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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”.
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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”.
QUESTION: General notion of Stokes-perverse sheaf?
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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”.
QUESTION: General notion of Stokes-perverse sheaf?
PROBLEM: Behaviour of the Stokes filtration by properpush-forward?
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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”.
QUESTION: General notion of Stokes-perverse sheaf?
PROBLEM: Behaviour of the Stokes filtration by properpush-forward?
OTHER ANSWER (T. Mochizuki): Embed the Stokesfiltration inside a hol. D-module −→ hol. D-modulewith k-Betti structure .
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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”.
QUESTION: General notion of Stokes-perverse sheaf?
PROBLEM: Behaviour of the Stokes filtration by properpush-forward?
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
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Example
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Example
x
0
P1
∞
P1
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Example
x
z
0
P1
∞A1
P1 × A1
Wild ramification in complex algebraic geometry – p. 16/17
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Example
x
z
0
p
P1
z
0
∞
A1
A1
P1 × A1 p−→ A1
Wild ramification in complex algebraic geometry – p. 16/17
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Example
x
z
0
p
P1
z
0
∞
A1
A1
P1 × A1 p−→ A1
M = reg. hol. DP1×A1-mod.
Wild ramification in complex algebraic geometry – p. 16/17
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Example
x
z
0
Si
p
P1
z
0
∞
A1
A1
P1 × A1 p−→ A1
M = reg. hol. DP1×A1-mod.
S =⋃j Sj = sing. set of M
Wild ramification in complex algebraic geometry – p. 16/17
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Example
x
z
0
Si
p
P1
z
0
∞
A1
A1
P1 × A1 p−→ A1
M = reg. hol. DP1×A1-mod.
S =⋃j Sj = sing. set of M
Pb: Levelt-Turrittin ofN := p∗(E
x ⊗M )
i.e. diff. eqn for∫
γz
f(x, z)ex dx
f : sol. of M
Wild ramification in complex algebraic geometry – p. 16/17
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Example
x
0
Si
p∆
P1
z
0
∞ z1/q
1/q
P1 × A1 p−→ A1
M = reg. hol. DP1×A1-mod.
S =⋃j Sj = sing. set of M
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
Wild ramification in complex algebraic geometry – p. 16/17
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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.
Wild ramification in complex algebraic geometry – p. 16/17
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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
Wild ramification in complex algebraic geometry – p. 17/17
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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 ).
Wild ramification in complex algebraic geometry – p. 17/17
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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
Wild ramification in complex algebraic geometry – p. 17/17
![Page 142: Wild ramification in complex algebraic geometry](https://reader030.fdocuments.in/reader030/viewer/2022012604/61997f98716332085d0936a4/html5/thumbnails/142.jpg)
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
Wild ramification in complex algebraic geometry – p. 17/17
![Page 143: Wild ramification in complex algebraic geometry](https://reader030.fdocuments.in/reader030/viewer/2022012604/61997f98716332085d0936a4/html5/thumbnails/143.jpg)
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
Wild ramification in complex algebraic geometry – p. 17/17