Upload
others
View
7
Download
0
Embed Size (px)
Citation preview
Wild twistor D-modulesClaude Sabbah
Centre de Mathematiques Laurent Schwartz
UMR 7640 du CNRS
Ecole polytechnique, Palaiseau, France
Wild twistor D-modules – p. 1/28
Introduction
(V,∇) an alg. vect. bundle with connection on A1rP ,P = a finite set of points.
Wild twistor D-modules – p. 2/28
Introduction
(V,∇) an alg. vect. bundle with connection on A1rP ,P = a finite set of points.
→ a holonomic module M on C[t]〈∂t〉.
Wild twistor D-modules – p. 2/28
Introduction
(V,∇) an alg. vect. bundle with connection on A1rP ,P = a finite set of points.
→ a holonomic module M on C[t]〈∂t〉.
Fourier-Laplace transform of M : a C[τ ]〈∂τ 〉-module,τ = ∂t, ∂τ = −t.
Wild twistor D-modules – p. 2/28
Introduction
(V,∇) an alg. vect. bundle with connection on A1rP ,P = a finite set of points.
→ a holonomic module M on C[t]〈∂t〉.
Fourier-Laplace transform of M : a C[τ ]〈∂τ 〉-module,τ = ∂t, ∂τ = −t.
Assume that (V,∇) underlies a variation of polarizedHodge structure on A1 r P .
Wild twistor D-modules – p. 2/28
Introduction
(V,∇) an alg. vect. bundle with connection on A1rP ,P = a finite set of points.
→ a holonomic module M on C[t]〈∂t〉.
Fourier-Laplace transform of M : a C[τ ]〈∂τ 〉-module,τ = ∂t, ∂τ = −t.
Assume that (V,∇) underlies a variation of polarizedHodge structure on A1 r P .
Question:
Wild twistor D-modules – p. 2/28
Introduction
(V,∇) an alg. vect. bundle with connection on A1rP ,P = a finite set of points.
→ a holonomic module M on C[t]〈∂t〉.
Fourier-Laplace transform of M : a C[τ ]〈∂τ 〉-module,τ = ∂t, ∂τ = −t.
Assume that (V,∇) underlies a variation of polarizedHodge structure on A1 r P .
Question: What kind of a structure does M underlie?
Wild twistor D-modules – p. 2/28
Introduction
(V,∇) an alg. vect. bundle with connection on A1rP ,P = a finite set of points.
→ a holonomic module M on C[t]〈∂t〉.
Fourier-Laplace transform of M : a C[τ ]〈∂τ 〉-module,τ = ∂t, ∂τ = −t.
Assume that (V,∇) underlies a variation of polarizedHodge structure on A1 r P .
Question: What kind of a structure does M underlie?Problem:
Wild twistor D-modules – p. 2/28
Introduction
(V,∇) an alg. vect. bundle with connection on A1rP ,P = a finite set of points.
→ a holonomic module M on C[t]〈∂t〉.
Fourier-Laplace transform of M : a C[τ ]〈∂τ 〉-module,τ = ∂t, ∂τ = −t.
Assume that (V,∇) underlies a variation of polarizedHodge structure on A1 r P .
Question: What kind of a structure does M underlie?Problem: Irregularity of M at τ =∞.
Wild twistor D-modules – p. 2/28
Introduction
(V,∇) an alg. vect. bundle with connection on A1rP ,P = a finite set of points.
→ a holonomic module M on C[t]〈∂t〉.
Fourier-Laplace transform of M : a C[τ ]〈∂τ 〉-module,τ = ∂t, ∂τ = −t.
Assume that (V,∇) underlies a variation of polarizedHodge structure on A1 r P .
Question: What kind of a structure does M underlie?Problem: Irregularity of M at τ =∞.Analogue for ℓ-adic sheaves: Results of Deligne andLaumon.
Wild twistor D-modules – p. 2/28
Introduction
(V,∇) an alg. vect. bundle with connection on A1rP ,P = a finite set of points.
→ a holonomic module M on C[t]〈∂t〉.
Fourier-Laplace transform of M : a C[τ ]〈∂τ 〉-module,τ = ∂t, ∂τ = −t.
Assume that (V,∇) underlies a variation of polarizedHodge structure on A1 r P .
Question: What kind of a structure does M underlie?Problem: Irregularity of M at τ =∞.Analogue for ℓ-adic sheaves: Results of Deligne andLaumon.General problem:
Wild twistor D-modules – p. 2/28
Introduction
(V,∇) an alg. vect. bundle with connection on A1rP ,P = a finite set of points.
→ a holonomic module M on C[t]〈∂t〉.
Fourier-Laplace transform of M : a C[τ ]〈∂τ 〉-module,τ = ∂t, ∂τ = −t.
Assume that (V,∇) underlies a variation of polarizedHodge structure on A1 r P .
Question: What kind of a structure does M underlie?Problem: Irregularity of M at τ =∞.Analogue for ℓ-adic sheaves: Results of Deligne andLaumon.General problem: Hodge theory in presence ofirregular singularities.
Wild twistor D-modules – p. 2/28
Introduction
(V,∇) an alg. vect. bundle with connection on A1rP ,P = a finite set of points.
→ a holonomic module M on C[t]〈∂t〉.
Fourier-Laplace transform of M : a C[τ ]〈∂τ 〉-module,τ = ∂t, ∂τ = −t.
Assume that (V,∇) underlies a variation of polarizedHodge structure on A1 r P .
Question: What kind of a structure does M underlie?Problem: Irregularity of M at τ =∞.Analogue for ℓ-adic sheaves: Results of Deligne andLaumon.General problem: Hodge theory in presence ofirregular singularities. Cf. Deligne’s notes (1984, 2006).
Wild twistor D-modules – p. 2/28
Conjecture of Kashiwara (regular case)
f : X −→ Y : a morphism between smooth complexprojective varieties.
Wild twistor D-modules – p. 3/28
Conjecture of Kashiwara (regular case)
f : X −→ Y : a morphism between smooth complexprojective varieties.
Semi-simple perverse
sheaf onX
Wild twistor D-modules – p. 3/28
Conjecture of Kashiwara (regular case)
f : X −→ Y : a morphism between smooth complexprojective varieties.
Semi-simple perverse
sheaf on Y
Semi-simple perverse
sheaf onX
Conjecture of Kashiwara
regular case
Wild twistor D-modules – p. 3/28
Conjecture of Kashiwara (regular case)
f : X −→ Y : a morphism between smooth complexprojective varieties.Sketch of the analytic proof :
Semi-simple perverse
sheaf on Y
Semi-simple perverse
sheaf onX
Wild twistor D-modules – p. 3/28
Conjecture of Kashiwara (regular case)
f : X −→ Y : a morphism between smooth complexprojective varieties.Sketch of the analytic proof :
Semi-simple perverse
sheaf on Y
Polarized regular twistor
D-module onX
Semi-simple perverse
sheaf onX
Wild twistor D-modules – p. 3/28
Conjecture of Kashiwara (regular case)
f : X −→ Y : a morphism between smooth complexprojective varieties.Sketch of the analytic proof :
Polarized regular twistor
D-module on Y
Semi-simple perverse
sheaf on Y
decomposition
theorem (C.S.)
Polarized regular twistor
D-module onX
Semi-simple perverse
sheaf onX
Wild twistor D-modules – p. 3/28
Conjecture of Kashiwara (regular case)
f : X −→ Y : a morphism between smooth complexprojective varieties.Sketch of the analytic proof :
Polarized regular twistor
D-module on Y
Semi-simple perverse
sheaf on Y
decomposition
theorem (C.S.)
Simpson
+ Hamm-Le D.T.
Polarized regular twistor
D-module onX
Semi-simple perverse
sheaf onX
Wild twistor D-modules – p. 3/28
Conjecture of Kashiwara (regular case)
f : X −→ Y : a morphism between smooth complexprojective varieties.Sketch of the analytic proof :
Polarized regular twistor
D-module on Y
Semi-simple perverse
sheaf on Y
decomposition
theorem (C.S.)
Simpson
+ Hamm-Le D.T.
Corlette
+ Simpson
(smooth case)
Polarized smooth twistor
D-module onX
Semi-simple local
system onX
Wild twistor D-modules – p. 3/28
Conjecture of Kashiwara (regular case)
f : X −→ Y : a morphism between smooth complexprojective varieties.Sketch of the analytic proof :
Polarized regular twistor
D-module on Y
Semi-simple perverse
sheaf on Y
decomposition
theorem (C.S.)
Simpson
+ Hamm-Le D.T.
Polarized regular twistor
D-module onXT. Mochizuki
(NCD)
Semi-simple perverse
sheaf onX
Wild twistor D-modules – p. 3/28
Conjecture of Kashiwara (regular case)
f : X −→ Y : a morphism between smooth complexprojective varieties.Sketch of the analytic proof :
Polarized regular twistor
D-module on Y
Semi-simple perverse
sheaf on Y
decomposition
theorem (C.S.)
Simpson
+ Hamm-Le D.T.
Polarized regular twistor
D-module onXT. Mochizuki
(NCD)
Semi-simple perverse
sheaf onX
Conjecture of Kashiwara
regular case
Wild twistor D-modules – p. 3/28
Twistor structures(C. Simpson)
Hodge structures Twistor structuresFiltered vect. sp. (H,F •H) Holom. vect. bundle on P1
Wild twistor D-modules – p. 4/28
Twistor structures(C. Simpson)
Hodge structures Twistor structuresFiltered vect. sp. (H,F •H) Holom. vect. bundle on P1
Conjugation H→H Twistor conjugation
Wild twistor D-modules – p. 4/28
Twistor structures(C. Simpson)
Hodge structures Twistor structuresFiltered vect. sp. (H,F •H) Holom. vect. bundle on P1
Conjugation H→H Twistor conjugationH→H = σ∗H
Wild twistor D-modules – p. 4/28
Twistor structures(C. Simpson)
Hodge structures Twistor structuresFiltered vect. sp. (H,F •H) Holom. vect. bundle on P1
Conjugation H→H Twistor conjugationH→H = σ∗Hσ : z 7→−1/z
Wild twistor D-modules – p. 4/28
Twistor structures(C. Simpson)
Hodge structures Twistor structuresFiltered vect. sp. (H,F •H) Holom. vect. bundle on P1
Conjugation H→H Twistor conjugationH→H = σ∗Hσ : z 7→−1/z (z=−1/z)
Wild twistor D-modules – p. 4/28
Twistor structures(C. Simpson)
Hodge structures Twistor structuresFiltered vect. sp. (H,F •H) Holom. vect. bundle on P1
Conjugation H→H Twistor conjugationH→H = σ∗Hσ : z 7→−1/z (z=−1/z)
Pure Hodge structurew=0 H ≃ OdP1
Wild twistor D-modules – p. 4/28
Twistor structures(C. Simpson)
Hodge structures Twistor structuresFiltered vect. sp. (H,F •H) Holom. vect. bundle on P1
Conjugation H→H Twistor conjugationH→H = σ∗Hσ : z 7→−1/z (z=−1/z)
Pure Hodge structurew=0 H ≃ OdP1
Vector space H Γ(P1,H )
Wild twistor D-modules – p. 4/28
Twistor structures(C. Simpson)
Hodge structures Twistor structuresFiltered vect. sp. (H,F •H) Holom. vect. bundle on P1
Conjugation H→H Twistor conjugationH→H = σ∗Hσ : z 7→−1/z (z=−1/z)
Pure Hodge structurew=0 H ≃ OdP1
Vector space H Γ(P1,H )
H ≃ H∗ H ≃ H ∗ := H ∨
Wild twistor D-modules – p. 4/28
Twistor structures(C. Simpson)
Hodge structures Twistor structuresFiltered vect. sp. (H,F •H) Holom. vect. bundle on P1
Conjugation H→H Twistor conjugationH→H = σ∗Hσ : z 7→−1/z (z=−1/z)
Pure Hodge structurew=0 H ≃ OdP1
Vector space H Γ(P1,H )
H ≃ H∗ H ≃ H ∗ := H ∨
→ Γ(P1,H ) ≃ Γ(P1,H )∗
Wild twistor D-modules – p. 4/28
Twistor structures(C. Simpson)
Hodge structures Twistor structuresFiltered vect. sp. (H,F •H) Holom. vect. bundle on P1
Conjugation H→H Twistor conjugationH→H = σ∗Hσ : z 7→−1/z (z=−1/z)
Pure Hodge structurew=0 H ≃ OdP1
Vector space H Γ(P1,H )
H ≃ H∗ H ≃ H ∗ := H ∨
→ Γ(P1,H ) ≃ Γ(P1,H )∗
Positivity Positivity on Γ(P1,H )
Wild twistor D-modules – p. 4/28
Twistor structures(C. Simpson)
Hodge structures Twistor structuresFiltered vect. sp. (H,F •H) Holom. vect. bundle on P1
Conjugation H→H Twistor conjugationH→H = σ∗Hσ : z 7→−1/z (z=−1/z)
Pure Hodge structurew=0 H ≃ OdP1
Vector space H Γ(P1,H )
H ≃ H∗ H ≃ H ∗ := H ∨
→ Γ(P1,H ) ≃ Γ(P1,H )∗
Positivity Positivity on Γ(P1,H )
Tate twist (k), k ∈ Z ⊗ OP1(−2k) (k ∈ 12Z)
Wild twistor D-modules – p. 4/28
Twistor structures(C. Simpson)
H ′,H ′′ holomorphic on A1, “gluing”:C : H ′
|S ⊗OSH′′|S −→ OS , S = |z| = 1.
Wild twistor D-modules – p. 6/28
Twistor structures(C. Simpson)
H ′,H ′′ holomorphic on A1, “gluing”:C : H ′
|S ⊗OSH′′|S −→ OS , S = |z| = 1.
Twistor adjoint H ∗ = H ∨
Wild twistor D-modules – p. 6/28
Twistor structures(C. Simpson)
H ′,H ′′ holomorphic on A1, “gluing”:C : H ′
|S ⊗OSH′′|S −→ OS , S = |z| = 1.
Twistor adjoint (H ′,H ′′, C)∗ = (H ′′,H ′, C∗),C∗(v,u) := C(u,v).
Wild twistor D-modules – p. 6/28
Twistor structures(C. Simpson)
H ′,H ′′ holomorphic on A1, “gluing”:C : H ′
|S ⊗OSH′′|S −→ OS , S = |z| = 1.
Twistor adjoint (H ′,H ′′, C)∗ = (H ′′,H ′, C∗),C∗(v,u) := C(u,v).
Tate twist H (k) := H ⊗ OP1(−2k)
Wild twistor D-modules – p. 6/28
Twistor structures(C. Simpson)
H ′,H ′′ holomorphic on A1, “gluing”:C : H ′
|S ⊗OSH′′|S −→ OS , S = |z| = 1.
Twistor adjoint (H ′,H ′′, C)∗ = (H ′′,H ′, C∗),C∗(v,u) := C(u,v).
Tate twist (H ′,H ′′, C)(k) = (H ′,H ′′, (iz)−2kC).
Wild twistor D-modules – p. 6/28
Twistor structures(C. Simpson)
H ′,H ′′ holomorphic on A1, “gluing”:C : H ′
|S ⊗OSH′′|S −→ OS , S = |z| = 1.
Twistor adjoint (H ′,H ′′, C)∗ = (H ′′,H ′, C∗),C∗(v,u) := C(u,v).
Tate twist (H ′,H ′′, C)(k) = (H ′,H ′′, (iz)−2kC).
Polarization in weight 0
Wild twistor D-modules – p. 6/28
Twistor structures(C. Simpson)
H ′,H ′′ holomorphic on A1, “gluing”:C : H ′
|S ⊗OSH′′|S −→ OS , S = |z| = 1.
Twistor adjoint (H ′,H ′′, C)∗ = (H ′′,H ′, C∗),C∗(v,u) := C(u,v).
Tate twist (H ′,H ′′, C)(k) = (H ′,H ′′, (iz)−2kC).
H ′ = H ′′ and existence of a global frame ε of H ′
such that, on S, C(εi,εj) = δij .
Wild twistor D-modules – p. 6/28
Variation of twistor structures(C. Simpson)
X: complex manifold, X = X × A1.
Wild twistor D-modules – p. 7/28
Variation of twistor structures(C. Simpson)
X: complex manifold, X = X × A1.
Twistor conjugation: ordinary conjugation on X andtwistor conjugation on P1.
Wild twistor D-modules – p. 8/28
Variation of twistor structures(C. Simpson)
X: complex manifold, X = X × A1.
Twistor conjugation: ordinary conjugation on X andtwistor conjugation on P1.
H ′,H ′′ holomorphic bdles on X × A1, “gluing”:C : H ′
|X×S ⊗OSH′′|X×S −→ C
∞,anX×S .
Wild twistor D-modules – p. 8/28
Variation of twistor structures(C. Simpson)
X: complex manifold, X = X × A1.
Twistor conjugation: ordinary conjugation on X andtwistor conjugation on P1.
H ′,H ′′ holomorphic bdles on X × A1, “gluing”:C : H ′
|X×S ⊗OSH′′|X×S −→ C
∞,anX×S .
Flat holomorphic relative connections∇′,∇′′:
H ′(′′) −→1
zΩ1
X /A1 ⊗H ′(′′), compatibility with C:
d′XC(u,v)=C(∇′u,v), d′′XC(u,v)=C(u,∇′′v).
Wild twistor D-modules – p. 8/28
Variation of twistor structures(C. Simpson)
X: complex manifold, X = X × A1.
Twistor conjugation: ordinary conjugation on X andtwistor conjugation on P1.
H ′,H ′′ holomorphic bdles on X × A1, “gluing”:C : H ′
|X×S ⊗OSH′′|X×S −→ C
∞,anX×S .
Flat holomorphic relative connections∇′,∇′′:
H ′(′′) −→1
zΩ1
X /A1 ⊗H ′(′′), compatibility with C:
d′XC(u,v)=C(∇′u,v), d′′XC(u,v)=C(u,∇′′v).
Twistor adjoint (H ′,H ′′, C)∗ = (H ′′,H ′, C∗),C∗(v,u) := C(u,v).
Wild twistor D-modules – p. 8/28
Variation of twistor structures(C. Simpson)
X: complex manifold, X = X × A1.
Tate twist (H ′,H ′′, C)(k) = (H ′,H ′′, (iz)−2kC).
Wild twistor D-modules – p. 9/28
Variation of twistor structures(C. Simpson)
X: complex manifold, X = X × A1.
Tate twist (H ′,H ′′, C)(k) = (H ′,H ′′, (iz)−2kC).
Hermitian pairing in weight 0:(H ′,H ′′, C) ≃ (H ′,H ′′, C)∗.
Wild twistor D-modules – p. 9/28
Variation of twistor structures(C. Simpson)
X: complex manifold, X = X × A1.
Tate twist (H ′,H ′′, C)(k) = (H ′,H ′′, (iz)−2kC).
Hermitian pairing in weight 0:(H ′,H ′′, C) ≃ (H ′,H ′′, C)∗.
Polarization in weight 0: the restriction to any xo ∈ Xof the Hermitian pairing is a polarization of therestricted twistor structure.
Wild twistor D-modules – p. 9/28
Variation of twistor structures(C. Simpson)
X: complex manifold, X = X × A1.
Tate twist (H ′,H ′′, C)(k) = (H ′,H ′′, (iz)−2kC).
Hermitian pairing in weight 0:(H ′,H ′′, C) ≃ (H ′,H ′′, C)∗.
Polarization in weight 0: the restriction to any xo ∈ Xof the Hermitian pairing is a polarization of therestricted twistor structure.
C. Simpson: Variations of pol. twistor struct. of weight 0
Wild twistor D-modules – p. 9/28
Variation of twistor structures(C. Simpson)
X: complex manifold, X = X × A1.
Tate twist (H ′,H ′′, C)(k) = (H ′,H ′′, (iz)−2kC).
Hermitian pairing in weight 0:(H ′,H ′′, C) ≃ (H ′,H ′′, C)∗.
Polarization in weight 0: the restriction to any xo ∈ Xof the Hermitian pairing is a polarization of therestricted twistor structure.
C. Simpson: Variations of pol. twistor struct. of weight 0z=1←→ holom. vector bundle on X with flat connection∇and Hermitian metric h which is harmonic
Wild twistor D-modules – p. 9/28
Variation of twistor structures(C. Simpson)
X: complex manifold, X = X × A1.
Tate twist (H ′,H ′′, C)(k) = (H ′,H ′′, (iz)−2kC).
Hermitian pairing in weight 0:(H ′,H ′′, C) ≃ (H ′,H ′′, C)∗.
Polarization in weight 0: the restriction to any xo ∈ Xof the Hermitian pairing is a polarization of therestricted twistor structure.
C. Simpson: Variations of pol. twistor struct. of weight 0z=1←→ holom. vector bundle on X with flat connection∇and Hermitian metric h which is harmonicz=0←→ holom. vector bundle on X with a Higgs field θ, andHermitian metric h which is harmonic .
Wild twistor D-modules – p. 9/28
Twistor D-modules
X: complex manifold, X = X × A1.
H ′,H ′′ holomorphic on X × A1 with flatholomorphic relative connections∇′,∇′′:
H −→1
zΩ1
X /A1 ⊗H (H = H ′,H ′′).
Wild twistor D-modules – p. 10/28
Twistor D-modules
X: complex manifold, X = X × A1.
M ′,M ′′ holonomic RX -modules,RX = OX 〈ðx1
, . . . , ðxn〉, ðxi
= z∂xi.
Wild twistor D-modules – p. 10/28
Twistor D-modules
X: complex manifold, X = X × A1.
M ′,M ′′ holonomic RX -modules,RX = OX 〈ðx1
, . . . , ðxn〉, ðxi
= z∂xi.
“gluing”: C : H ′|X×S ⊗OS
H′′|X×S −→ C
∞,anX×S
compatible with ∇′,∇′′:d′XC(u,v)=C(∇′u,v), d′′XC(u,v)=C(u,∇′′v).
Wild twistor D-modules – p. 10/28
Twistor D-modules
X: complex manifold, X = X × A1.
M ′,M ′′ holonomic RX -modules,RX = OX 〈ðx1
, . . . , ðxn〉, ðxi
= z∂xi.
“gluing”: C : M ′|X×S ⊗OS
M′′|X×S −→ DbX×S/S
compatible with the RX ⊗RX -action.
Wild twistor D-modules – p. 10/28
Twistor D-modules
X: complex manifold, X = X × A1.
M ′,M ′′ holonomic RX -modules,RX = OX 〈ðx1
, . . . , ðxn〉, ðxi
= z∂xi.
“gluing”: C : M ′|X×S ⊗OS
M′′|X×S −→ DbX×S/S
compatible with the RX ⊗RX -action.
Twistor adjoint (M ′,M ′′, C)∗ = (M ′′,M ′, C∗).
Wild twistor D-modules – p. 10/28
Twistor D-modules
X: complex manifold, X = X × A1.
M ′,M ′′ holonomic RX -modules,RX = OX 〈ðx1
, . . . , ðxn〉, ðxi
= z∂xi.
“gluing”: C : M ′|X×S ⊗OS
M′′|X×S −→ DbX×S/S
compatible with the RX ⊗RX -action.
Twistor adjoint (M ′,M ′′, C)∗ = (M ′′,M ′, C∗).
. . . Strictness
Wild twistor D-modules – p. 10/28
Twistor D-modules
X: complex manifold, X = X × A1.
M ′,M ′′ holonomic RX -modules,RX = OX 〈ðx1
, . . . , ðxn〉, ðxi
= z∂xi.
“gluing”: C : M ′|X×S ⊗OS
M′′|X×S −→ DbX×S/S
compatible with the RX ⊗RX -action.
Twistor adjoint (M ′,M ′′, C)∗ = (M ′′,M ′, C∗).
. . . Strictness
Problem:
Wild twistor D-modules – p. 10/28
Twistor D-modules
X: complex manifold, X = X × A1.
M ′,M ′′ holonomic RX -modules,RX = OX 〈ðx1
, . . . , ðxn〉, ðxi
= z∂xi.
“gluing”: C : M ′|X×S ⊗OS
M′′|X×S −→ DbX×S/S
compatible with the RX ⊗RX -action.
Twistor adjoint (M ′,M ′′, C)∗ = (M ′′,M ′, C∗).
. . . Strictness
Problem: How to define the restriction to xo ∈ X?
Wild twistor D-modules – p. 10/28
Twistor D-modules
X: complex manifold, X = X × A1.
M ′,M ′′ holonomic RX -modules,RX = OX 〈ðx1
, . . . , ðxn〉, ðxi
= z∂xi.
“gluing”: C : M ′|X×S ⊗OS
M′′|X×S −→ DbX×S/S
compatible with the RX ⊗RX -action.
Twistor adjoint (M ′,M ′′, C)∗ = (M ′′,M ′, C∗).
. . . Strictness
Problem: How to define the restriction to xo ∈ X?Answer:
Wild twistor D-modules – p. 10/28
Twistor D-modules
X: complex manifold, X = X × A1.
M ′,M ′′ holonomic RX -modules,RX = OX 〈ðx1
, . . . , ðxn〉, ðxi
= z∂xi.
“gluing”: C : M ′|X×S ⊗OS
M′′|X×S −→ DbX×S/S
compatible with the RX ⊗RX -action.
Twistor adjoint (M ′,M ′′, C)∗ = (M ′′,M ′, C∗).
. . . Strictness
Problem: How to define the restriction to xo ∈ X?Answer: Use iterated nearby cycle functors.
Wild twistor D-modules – p. 10/28
Twistor D-modules
X: complex manifold, X = X × A1.
M ′,M ′′ holonomic RX -modules,RX = OX 〈ðx1
, . . . , ðxn〉, ðxi
= z∂xi.
“gluing”: C : M ′|X×S ⊗OS
M′′|X×S −→ DbX×S/S
compatible with the RX ⊗RX -action.
Twistor adjoint (M ′,M ′′, C)∗ = (M ′′,M ′, C∗).
. . . Strictness
Problem: How to define the restriction to xo ∈ X?Answer: Use iterated nearby cycle functors.
Kashiwara-Malgrange V -filtration for M ′,M ′′ → ψfM .
Wild twistor D-modules – p. 10/28
Twistor D-modules
X: complex manifold, X = X × A1.
M ′,M ′′ holonomic RX -modules,RX = OX 〈ðx1
, . . . , ðxn〉, ðxi
= z∂xi.
“gluing”: C : M ′|X×S ⊗OS
M′′|X×S −→ DbX×S/S
compatible with the RX ⊗RX -action.
Twistor adjoint (M ′,M ′′, C)∗ = (M ′′,M ′, C∗).
. . . Strictness
Problem: How to define the restriction to xo ∈ X?Answer: Use iterated nearby cycle functors.
Kashiwara-Malgrange V -filtration for M ′,M ′′ → ψfM .Barlet Hermitian form on nearby cycles→ ψfC.
Wild twistor D-modules – p. 10/28
Polarized pure twistor D-modules
Hodge Modules(M. Saito)
Twistor D-modules
Wild twistor D-modules – p. 11/28
Polarized pure twistor D-modules
Hodge Modules(M. Saito)
Twistor D-modules
Filtered D-mod. (M,F•M) Triple T = (M ′,M ′′, C)
Wild twistor D-modules – p. 11/28
Polarized pure twistor D-modules
Hodge Modules(M. Saito)
Twistor D-modules
Filtered D-mod. (M,F•M) Triple T = (M ′,M ′′, C)Q-structure: perverse FQ
Wild twistor D-modules – p. 11/28
Polarized pure twistor D-modules
Hodge Modules(M. Saito)
Twistor D-modules
Filtered D-mod. (M,F•M) Triple T = (M ′,M ′′, C)Q-structure: perverse FQ
C⊗Q FQ ≃ DRM Hermitian pairing T ≃ T ∗
Wild twistor D-modules – p. 11/28
Polarized pure twistor D-modules
Hodge Modules(M. Saito)
Twistor D-modules
Filtered D-mod. (M,F•M) Triple T = (M ′,M ′′, C)Q-structure: perverse FQ
C⊗Q FQ ≃ DRM Hermitian pairing T ≃ T ∗
Category MH6d(X,w) Category MT6d(X,w)
Wild twistor D-modules – p. 11/28
Polarized pure twistor D-modules
Hodge Modules(M. Saito)
Twistor D-modules
Filtered D-mod. (M,F•M) Triple T = (M ′,M ′′, C)Q-structure: perverse FQ
C⊗Q FQ ≃ DRM Hermitian pairing T ≃ T ∗
Category MH6d(X,w) Category MT6d(X,w)
∀ holom. germ f : (X,xo) −→ C, constraint onψf (M,F•M,FQ) ψfT
Wild twistor D-modules – p. 11/28
Polarized pure twistor D-modules
Hodge Modules(M. Saito)
Twistor D-modules
Filtered D-mod. (M,F•M) Triple T = (M ′,M ′′, C)Q-structure: perverse FQ
C⊗Q FQ ≃ DRM Hermitian pairing T ≃ T ∗
Category MH6d(X,w) Category MT6d(X,w)
∀ holom. germ f : (X,xo) −→ C, constraint onψf (M,F•M,FQ) ψfT
∀ℓ ∈ Z, grMℓ ψf (M,F•M,FQ) ∈ MH6d−1(X,w)
Wild twistor D-modules – p. 11/28
Polarized pure twistor D-modules
Hodge Modules(M. Saito)
Twistor D-modules
Filtered D-mod. (M,F•M) Triple T = (M ′,M ′′, C)Q-structure: perverse FQ
C⊗Q FQ ≃ DRM Hermitian pairing T ≃ T ∗
Category MH6d(X,w) Category MT6d(X,w)
∀ holom. germ f : (X,xo) −→ C, constraint onψf (M,F•M,FQ) ψfT
∀ℓ ∈ Z, grMℓ ψfT ∈MT6d−1(X,w)
Wild twistor D-modules – p. 11/28
Polarized pure twistor D-modules
Hodge Modules(M. Saito)
Twistor D-modules
Filtered D-mod. (M,F•M) Triple T = (M ′,M ′′, C)Q-structure: perverse FQ
C⊗Q FQ ≃ DRM Hermitian pairing T ≃ T ∗
Category MH6d(X,w) Category MT6d(X,w)
∀ holom. germ f : (X,xo) −→ C, constraint onψf (M,F•M,FQ) ψfT
∀ℓ ∈ Z, grMℓ ψfT ∈MT6d−1(X,w)
Polarization condition by “restriction” to any xo
Wild twistor D-modules – p. 11/28
Polarized pure twistor D-modules
Hodge Modules(M. Saito)
Twistor D-modules
Filtered D-mod. (M,F•M) Triple T = (M ′,M ′′, C)Q-structure: perverse FQ
C⊗Q FQ ≃ DRM Hermitian pairing T ≃ T ∗
Category MH6d(X,w) Category MT6d(X,w)
∀ holom. germ f : (X,xo) −→ C, constraint onψf (M,F•M,FQ) ψfT
∀ℓ ∈ Z, grMℓ ψfT ∈MT6d−1(X,w)
Polarization condition by “restriction” to any xo
Category MH6d(X,w)(p) Category MT6d(X,w)(p)
Wild twistor D-modules – p. 11/28
Regularity (tameness)
Hodge D-modules are regular holonomic D-modules.
Wild twistor D-modules – p. 12/28
Regularity (tameness)
Hodge D-modules are regular holonomic D-modules.
Twistor D-modules need not be regular, in the sensethat the associated D-module M = M/(z − 1)M ,which is holonomic, need not be regular.
Wild twistor D-modules – p. 12/28
Regularity (tameness)
Hodge D-modules are regular holonomic D-modules.
Twistor D-modules need not be regular, in the sensethat the associated D-module M = M/(z − 1)M ,which is holonomic, need not be regular.
This is an advantage of this generalized framework
Wild twistor D-modules – p. 12/28
Regularity (tameness)
Hodge D-modules are regular holonomic D-modules.
Twistor D-modules need not be regular, in the sensethat the associated D-module M = M/(z − 1)M ,which is holonomic, need not be regular.
This is an advantage of this generalized frameworkcf. the original problemGeneral problem: Hodge theory in presence ofirregular singularities.
Wild twistor D-modules – p. 12/28
Regularity (tameness)
Hodge D-modules are regular holonomic D-modules.
Twistor D-modules need not be regular, in the sensethat the associated D-module M = M/(z − 1)M ,which is holonomic, need not be regular.
This is an advantage of this generalized frameworkcf. the original problemGeneral problem: Hodge theory in presence ofirregular singularities.
Nevertheless, the first results for twistor D-modulesare obtained in the regular case.
Wild twistor D-modules – p. 12/28
The decomposition theorem
f : X −→ Y : a morphism between smooth complexprojective varieties,T : a pol. regular twistor D-module on X, weight w.
Wild twistor D-modules – p. 13/28
The decomposition theorem
f : X −→ Y : a morphism between smooth complexprojective varieties,T : a pol. regular twistor D-module on X, weight w.
Polarized regular twistor
D-module on Y
Semi-simple perverse
sheaf on Y
decomposition
theorem (C.S.)
Polarized regular twistor
D-module onX
Semi-simple perverse
sheaf onX
Wild twistor D-modules – p. 13/28
The decomposition theorem
f : X −→ Y : a morphism between smooth complexprojective varieties,T : a pol. regular twistor D-module on X, weight w.
The categories MT(r)(X,w)(p), MT(r)(Y,w′)(p) aresemi-simple,
Wild twistor D-modules – p. 14/28
The decomposition theorem
f : X −→ Y : a morphism between smooth complexprojective varieties,T : a pol. regular twistor D-module on X, weight w.
The categories MT(r)(X,w)(p), MT(r)(Y,w′)(p) aresemi-simple,
each fk+T is a polarizable twistor D-module on Y ofweight w + k,
Wild twistor D-modules – p. 14/28
The decomposition theorem
f : X −→ Y : a morphism between smooth complexprojective varieties,T : a pol. regular twistor D-module on X, weight w.
The categories MT(r)(X,w)(p), MT(r)(Y,w′)(p) aresemi-simple,
each fk+T is a polarizable twistor D-module on Y ofweight w + k,
f+T ≃⊕k f
k+T [−k]
Wild twistor D-modules – p. 14/28
The decomposition theorem
f : X −→ Y : a morphism between smooth complexprojective varieties,T : a pol. regular twistor D-module on X, weight w.
The categories MT(r)(X,w)(p), MT(r)(Y,w′)(p) aresemi-simple,
each fk+T is a polarizable twistor D-module on Y ofweight w + k,
f+T ≃⊕k f
k+T [−k] (⇐= rel. Hard Lefschetz).
Wild twistor D-modules – p. 14/28
Conjecture of Kashiwara (general case)?
f : X −→ Y : a morphism between smooth complexprojective varieties.
Wild twistor D-modules – p. 15/28
Conjecture of Kashiwara (general case)?
f : X −→ Y : a morphism between smooth complexprojective varieties.Semi-simple holonomic
D-module onX
Conjecture of Kashiwara
general case ??
Semi-simple holonomic
D-module on Y
Wild twistor D-modules – p. 15/28
Conjecture of Kashiwara (general case)?
f : X −→ Y : a morphism between smooth complexprojective varieties.Semi-simple holonomic
D-module onX
Polarized wild twistor
D-module onX
decomposition
theorem ??
Conjecture of Kashiwara
general case ??
Polarized wild twistor
D-module on Y??
Semi-simple holonomic
D-module on Y
??
Wild twistor D-modules – p. 15/28
Holonomic D-modules on curves
D: disc with coordinate t
M : holonomic D-module on D
Wild twistor D-modules – p. 16/28
Holonomic D-modules on curves
D: disc with coordinate t
M : holonomic D-module on DM = C[[t]]⊗CtM the formalized module.
Wild twistor D-modules – p. 16/28
Holonomic D-modules on curves
D: disc with coordinate t
M : holonomic D-module on DM = C[[t]]⊗CtM the formalized module.
Turrittin-Levelt: M = Mreg ⊕ Mirr
Wild twistor D-modules – p. 16/28
Holonomic D-modules on curves
D: disc with coordinate t
M : holonomic D-module on DM = C[[t]]⊗CtM the formalized module.
Turrittin-Levelt: M = Mreg ⊕ Mirr and
∃ ρ : D′t′→7→
Dt=t′q
, ρ+Mirr ≃⊕
ϕ∈t′−1C[t′−1]
(E ϕ ⊗Rϕ),
Wild twistor D-modules – p. 16/28
Holonomic D-modules on curves
D: disc with coordinate t
M : holonomic D-module on DM = C[[t]]⊗CtM the formalized module.
Turrittin-Levelt: M = Mreg ⊕ Mirr and
∃ ρ : D′t′→7→
Dt=t′q
, ρ+Mirr ≃⊕
ϕ∈t′−1C[t′−1]
(E ϕ ⊗Rϕ),
Rϕ regular, E ϕ = (C((t′)), d+ dϕ).
Wild twistor D-modules – p. 16/28
Holonomic D-modules on curves
D: disc with coordinate t
M : holonomic D-module on DM = C[[t]]⊗CtM the formalized module.
Turrittin-Levelt: M = Mreg ⊕ Mirr and
∃ ρ : D′t′→7→
Dt=t′q
, ρ+Mirr ≃⊕
ϕ∈t′−1C[t′−1]
(E ϕ ⊗Rϕ),
Rϕ regular, E ϕ = (C((t′)), d+ dϕ).
ψtM = ψtM = ψtMreg←→ Mreg,
Wild twistor D-modules – p. 16/28
Holonomic D-modules on curves
D: disc with coordinate t
M : holonomic D-module on DM = C[[t]]⊗CtM the formalized module.
Turrittin-Levelt: M = Mreg ⊕ Mirr and
∃ ρ : D′t′→7→
Dt=t′q
, ρ+Mirr ≃⊕
ϕ∈t′−1C[t′−1]
(E ϕ ⊗Rϕ),
Rϕ regular, E ϕ = (C((t′)), d+ dϕ).
ψtM = ψtM = ψtMreg←→ Mreg, and
ψtMirr = 0.
Wild twistor D-modules – p. 16/28
Holonomic D-modules on curves
D: disc with coordinate t
M : holonomic D-module on DM = C[[t]]⊗CtM the formalized module.
Turrittin-Levelt: M = Mreg ⊕ Mirr and
∃ ρ : D′t′→7→
Dt=t′q
, ρ+Mirr ≃⊕
ϕ∈t′−1C[t′−1]
(E ϕ ⊗Rϕ),
Rϕ regular, E ϕ = (C((t′)), d+ dϕ).
ψtM = ψtM = ψtMreg←→ Mreg, and
ψtMirr = 0.
M ←→ ψt′(E−η ⊗ ρ+M), q ≫ 0,
∀ η ∈ t′−1C[t′−1].
Wild twistor D-modules – p. 16/28
Holonomic D-modules on curves
D: disc with coordinate t
M : holonomic D-module on DM = C[[t]]⊗CtM the formalized module.
Turrittin-Levelt: M = Mreg ⊕ Mirr and
∃ ρ : D′t′→7→
Dt=t′q
, ρ+Mirr ≃⊕
ϕ∈t′−1C[t′−1]
(E ϕ ⊗Rϕ),
Rϕ regular, E ϕ = (C((t′)), d+ dϕ).
ψtM = ψtM = ψtMreg←→ Mreg, and
ψtMirr = 0.
M ←→ ψt′(E−η ⊗ ρ+M), q ≫ 0,
∀ η ∈ t′−1C[t′−1]. = ψt′Rη
Wild twistor D-modules – p. 16/28
Holonomic D-modules on curves
D: disc with coordinate t
M : holonomic D-module on DM = C[[t]]⊗CtM the formalized module.
Turrittin-Levelt: M = Mreg ⊕ Mirr and
∃ ρ : D′t′→7→
Dt=t′q
, ρ+Mirr ≃⊕
ϕ∈t′−1C[t′−1]
(E ϕ ⊗Rϕ),
Rϕ regular, E ϕ = (C((t′)), d+ dϕ).
ψtM = ψtM = ψtMreg←→ Mreg, and
ψtMirr = 0.
M ←→ ψt′(E−η ⊗ ρ+M), q ≫ 0,
∀ η ∈ t′−1C[t′−1]. = ψt′Rη
Wild twistor D-modules – p. 16/28
Holonomic D-modules on curves
D: disc with coordinate t
M : holonomic D-module on DM = C[[t]]⊗CtM the formalized module.
Turrittin-Levelt: M = Mreg ⊕ Mirr and
∃ ρ : D′t′→7→
Dt=t′q
, ρ+Mirr ≃⊕
ϕ∈t′−1C[t′−1]
(E ϕ ⊗Rϕ),
Rϕ regular, E ϕ = (C((t′)), d+ dϕ).
ψtM = ψtM = ψtMreg←→ Mreg, and
ψtMirr = 0.
M ←→ ψt′(E−η ⊗ ρ+M), q ≫ 0,
∀ η ∈ t′−1C[t′−1]. = ψt′Rη
M ←→ M + Stokes structure.Wild twistor D-modules – p. 16/28
Meromorphic Higgs bundles on curves
M : a free C(t)-module of finite rank
Wild twistor D-modules – p. 17/28
Meromorphic Higgs bundles on curves
M : a free C(t)-module of finite rank
Higgs field: θ = Θdt
t, Θ ∈ EndC(t)(M).
Wild twistor D-modules – p. 17/28
Meromorphic Higgs bundles on curves
M : a free C(t)-module of finite rank
Higgs field: θ = Θdt
t, Θ ∈ EndC(t)(M).
M = Mreg ⊕Mirr holds over C(t).
Wild twistor D-modules – p. 17/28
Meromorphic Higgs bundles on curves
M : a free C(t)-module of finite rank
Higgs field: θ = Θdt
t, Θ ∈ EndC(t)(M).
M = Mreg ⊕Mirr holds over C(t).
The decomposition ρ+Mirr ≃⊕
ϕ∈t′−1C[t′−1]
(E ϕ ⊗Rϕ)
holds over C(t′),
Wild twistor D-modules – p. 17/28
Meromorphic Higgs bundles on curves
M : a free C(t)-module of finite rank
Higgs field: θ = Θdt
t, Θ ∈ EndC(t)(M).
M = Mreg ⊕Mirr holds over C(t).
The decomposition ρ+Mirr ≃⊕
ϕ∈t′−1C[t′−1]
(E ϕ ⊗Rϕ)
holds over C(t′),Rϕ regular (ΘRϕ
is holomorphic),
Wild twistor D-modules – p. 17/28
Meromorphic Higgs bundles on curves
M : a free C(t)-module of finite rank
Higgs field: θ = Θdt
t, Θ ∈ EndC(t)(M).
M = Mreg ⊕Mirr holds over C(t).
The decomposition ρ+Mirr ≃⊕
ϕ∈t′−1C[t′−1]
(E ϕ ⊗Rϕ)
holds over C(t′),Rϕ regular (ΘRϕ
is holomorphic),E ϕ = (C(t′), dϕ).
Wild twistor D-modules – p. 17/28
Meromorphic Higgs bundles on curves
M : a free C(t)-module of finite rank
Higgs field: θ = Θdt
t, Θ ∈ EndC(t)(M).
M = Mreg ⊕Mirr holds over C(t).
The decomposition ρ+Mirr ≃⊕
ϕ∈t′−1C[t′−1]
(E ϕ ⊗Rϕ)
holds over C(t′),Rϕ regular (ΘRϕ
is holomorphic),E ϕ = (C(t′), dϕ).
No Stokes phenomenon .
Wild twistor D-modules – p. 17/28
Wild twistor D-modules on curves
DEFINITION: T = (M ′,M ′′, C) on X = D is a wildtwistor D-module of weight w at t = 0 if
Wild twistor D-modules – p. 18/28
Wild twistor D-modules on curves
DEFINITION: T = (M ′,M ′′, C) on X = D is a wildtwistor D-module of weight w at t = 0 if ∀ q,∀ϕ ∈ t′−1C[t′−1], ψt′(E−ϕ/z ⊗ ρ+T ) is well-defined
Wild twistor D-modules – p. 18/28
Wild twistor D-modules on curves
DEFINITION: T = (M ′,M ′′, C) on X = D is a wildtwistor D-module of weight w at t = 0 if ∀ q,∀ϕ ∈ t′−1C[t′−1], ψt′(E−ϕ/z ⊗ ρ+T ) is well-definedand
∀ ℓ ∈ Z, grMℓ ψt′(E
−ϕ/z ⊗ ρ+T )
is a pure twistor structure of weight w + ℓ.
Wild twistor D-modules – p. 18/28
Wild twistor D-modules on curves
DEFINITION: T = (M ′,M ′′, C) on X = D is a wildtwistor D-module of weight w at t = 0 if ∀ q,∀ϕ ∈ t′−1C[t′−1], ψt′(E−ϕ/z ⊗ ρ+T ) is well-definedand
∀ ℓ ∈ Z, grMℓ ψt′(E
−ϕ/z ⊗ ρ+T )
is a pure twistor structure of weight w + ℓ.Problem:
Wild twistor D-modules – p. 18/28
Wild twistor D-modules on curves
DEFINITION: T = (M ′,M ′′, C) on X = D is a wildtwistor D-module of weight w at t = 0 if ∀ q,∀ϕ ∈ t′−1C[t′−1], ψt′(E−ϕ/z ⊗ ρ+T ) is well-definedand
∀ ℓ ∈ Z, grMℓ ψt′(E
−ϕ/z ⊗ ρ+T )
is a pure twistor structure of weight w + ℓ.Problem:How to define the “C part” Cϕ of E−ϕ(t)/z ⊗ T ?
Wild twistor D-modules – p. 18/28
Wild twistor D-modules on curves
DEFINITION: T = (M ′,M ′′, C) on X = D is a wildtwistor D-module of weight w at t = 0 if ∀ q,∀ϕ ∈ t′−1C[t′−1], ψt′(E−ϕ/z ⊗ ρ+T ) is well-definedand
∀ ℓ ∈ Z, grMℓ ψt′(E
−ϕ/z ⊗ ρ+T )
is a pure twistor structure of weight w + ℓ.Problem:How to define the “C part” Cϕ of E−ϕ(t)/z ⊗ T ?Answer:
Wild twistor D-modules – p. 18/28
Wild twistor D-modules on curves
DEFINITION: T = (M ′,M ′′, C) on X = D is a wildtwistor D-module of weight w at t = 0 if ∀ q,∀ϕ ∈ t′−1C[t′−1], ψt′(E−ϕ/z ⊗ ρ+T ) is well-definedand
∀ ℓ ∈ Z, grMℓ ψt′(E
−ϕ/z ⊗ ρ+T )
is a pure twistor structure of weight w + ℓ.Problem:How to define the “C part” Cϕ of E−ϕ(t)/z ⊗ T ?Answer:Cϕ(u,v) = C(e−ϕ/zu,e−ϕ/zv) = e−ϕ/z+zϕC(u,v).
Wild twistor D-modules – p. 18/28
Wild twistor D-modules on curves
DEFINITION: T = (M ′,M ′′, C) on X = D is a wildtwistor D-module of weight w at t = 0 if ∀ q,∀ϕ ∈ t′−1C[t′−1], ψt′(E−ϕ/z ⊗ ρ+T ) is well-definedand
∀ ℓ ∈ Z, grMℓ ψt′(E
−ϕ/z ⊗ ρ+T )
is a pure twistor structure of weight w + ℓ.Problem:How to define the “C part” Cϕ of E−ϕ(t)/z ⊗ T ?Answer:Cϕ(u,v) = C(e−ϕ/zu,e−ϕ/zv) = e−ϕ/z+zϕC(u,v).Is this a distribution?
Wild twistor D-modules – p. 18/28
Wild twistor D-modules on curves
DEFINITION: T = (M ′,M ′′, C) on X = D is a wildtwistor D-module of weight w at t = 0 if ∀ q,∀ϕ ∈ t′−1C[t′−1], ψt′(E−ϕ/z ⊗ ρ+T ) is well-definedand
∀ ℓ ∈ Z, grMℓ ψt′(E
−ϕ/z ⊗ ρ+T )
is a pure twistor structure of weight w + ℓ.Problem:How to define the “C part” Cϕ of E−ϕ(t)/z ⊗ T ?Answer:Cϕ(u,v) = C(e−ϕ/zu,e−ϕ/zv) = e−ϕ/z+zϕC(u,v).Is this a distribution?z ∈ S, hence −1/z = −z, hence−ϕ/z + zϕ = 2i Im(zϕ).
Wild twistor D-modules – p. 18/28
Wild twistor D-modules on curves
DEFINITION: T = (M ′,M ′′, C) on X = D is a wildtwistor D-module of weight w at t = 0 if ∀ q,∀ϕ ∈ t′−1C[t′−1], ψt′(E−ϕ/z ⊗ ρ+T ) is well-definedand
∀ ℓ ∈ Z, grMℓ ψt′(E
−ϕ/z ⊗ ρ+T )
is a pure twistor structure of weight w + ℓ.Problem:How to define the “C part” Cϕ of E−ϕ(t)/z ⊗ T ?Answer:Cϕ(u,v) = C(e−ϕ/zu,e−ϕ/zv) = e−ϕ/z+zϕC(u,v).Is this a distribution?z ∈ S, hence −1/z = −z, hence−ϕ/z + zϕ = 2i Im(zϕ). OK
Wild twistor D-modules – p. 18/28
Wild twistor D-modules on curves
THEOREM 1: Assume T is a polarized wild twistorD-module on D at t = 0.
Wild twistor D-modules – p. 19/28
Wild twistor D-modules on curves
THEOREM 1: Assume T is a polarized wild twistorD-module on D at t = 0. Then it is so at any to in someneighbourhood of t = 0.
Wild twistor D-modules – p. 19/28
Wild twistor D-modules on curves
THEOREM 1: Assume T is a polarized wild twistorD-module on D at t = 0. Then it is so at any to in someneighbourhood of t = 0.
REMARK: This is analogous to part of the Nilpotent OrbitTheorem (Schmid).
Wild twistor D-modules – p. 19/28
Wild twistor D-modules on curves
T : a polarized regular twistor D-module of weight won P1.
Wild twistor D-modules – p. 21/28
Wild twistor D-modules on curves
T : a polarized regular twistor D-module of weight won P1. f : P1 −→ Spec C the constant map.
Wild twistor D-modules – p. 21/28
Wild twistor D-modules on curves
T : a polarized regular twistor D-module of weight won P1. f : P1 −→ Spec C the constant map.
THEOREM 2: fk+(E t/z ⊗ T ) is a polarized pure twistorstructure of weight w + k.
Wild twistor D-modules – p. 21/28
Wild twistor D-modules on curves
T : a polarized regular twistor D-module of weight won P1. f : P1 −→ Spec C the constant map.
THEOREM 2: fk+(E t/z ⊗ T ) is a polarized pure twistorstructure of weight w + k.
REMARK:
Wild twistor D-modules – p. 21/28
Wild twistor D-modules on curves
T : a polarized regular twistor D-module of weight won P1. f : P1 −→ Spec C the constant map.
THEOREM 2: fk+(E t/z ⊗ T ) is a polarized pure twistorstructure of weight w + k.
REMARK:
can define the Fourier-Laplace transformT = f0
+(E tτ /z ⊗ T ).
Wild twistor D-modules – p. 21/28
Wild twistor D-modules on curves
T : a polarized regular twistor D-module of weight won P1. f : P1 −→ Spec C the constant map.
THEOREM 2: fk+(E t/z ⊗ T ) is a polarized pure twistorstructure of weight w + k.
REMARK:
can define the Fourier-Laplace transformT = f0
+(E tτ /z ⊗ T ).
T is a pure twistor D-module on Cτ which is regularat τ = 0.
Wild twistor D-modules – p. 21/28
Wild twistor D-modules on curves
T : a polarized regular twistor D-module of weight won P1. f : P1 −→ Spec C the constant map.
THEOREM 2: fk+(E t/z ⊗ T ) is a polarized pure twistorstructure of weight w + k.
REMARK:
can define the Fourier-Laplace transformT = f0
+(E tτ /z ⊗ T ).
T is a pure twistor D-module on Cτ which is regularat τ = 0.
At τ =∞, one expects that T is a wild twistorD-module
Wild twistor D-modules – p. 21/28
Wild twistor D-modules on curves
T : a polarized regular twistor D-module of weight won P1. f : P1 −→ Spec C the constant map.
THEOREM 2: fk+(E t/z ⊗ T ) is a polarized pure twistorstructure of weight w + k.
REMARK:
can define the Fourier-Laplace transformT = f0
+(E tτ /z ⊗ T ).
T is a pure twistor D-module on Cτ which is regularat τ = 0.
At τ =∞, one expects that T is a wild twistorD-module (cf. the work of S. Szabo, 2005).
Wild twistor D-modules – p. 21/28
Irregular nearby cycles, after Deligne
X complex manifold, f : X −→ C.
Wild twistor D-modules – p. 22/28
Irregular nearby cycles, after Deligne
X complex manifold, f : X −→ C.
M a holonomic DX -module.
Wild twistor D-modules – p. 22/28
Irregular nearby cycles, after Deligne
X complex manifold, f : X −→ C.
M a holonomic DX -module.
Deligne (Letter to Malgrange, 1983):
ψDelf M :=
⊕
N
ψf(f+N ⊗M
),
Wild twistor D-modules – p. 22/28
Irregular nearby cycles, after Deligne
X complex manifold, f : X −→ C.
M a holonomic DX -module.
Deligne (Letter to Malgrange, 1983):
ψDelf M :=
⊕
N
ψf(f+N ⊗M
),
N such that N is an irred. C((t))-module withconnection.
Wild twistor D-modules – p. 22/28
Irregular nearby cycles, after Deligne
Problem:
X = A2, coordinates x, y,
Wild twistor D-modules – p. 23/28
Irregular nearby cycles, after Deligne
Problem:
X = A2, coordinates x, y,
f : X −→ A1, (x, y) 7−→ y,
Wild twistor D-modules – p. 23/28
Irregular nearby cycles, after Deligne
Problem:
X = A2, coordinates x, y,
f : X −→ A1, (x, y) 7−→ y,
M = OA2[1/y],∇ = d+ d(x/y) = d+dx
y−xdy
y2.
Wild twistor D-modules – p. 23/28
Irregular nearby cycles, after Deligne
Problem:
X = A2, coordinates x, y,
f : X −→ A1, (x, y) 7−→ y,
M = OA2[1/y],∇ = d+ d(x/y) = d+dx
y−xdy
y2.
Then ψDelf M = 0 (M = E x/y).
Wild twistor D-modules – p. 23/28
Irregular nearby cycles, after Deligne
Problem:
X = A2, coordinates x, y,
f : X −→ A1, (x, y) 7−→ y,
M = OA2[1/y],∇ = d+ d(x/y) = d+dx
y−xdy
y2.
Then ψDelf M = 0 (M = E x/y).
Solution: Consider ψDelg M for various g in order to
recover information on M .
Wild twistor D-modules – p. 23/28
Irregular nearby cycles, after Deligne
Problem:
X = A2, coordinates x, y,
f : X −→ A1, (x, y) 7−→ y,
M = OA2[1/y],∇ = d+ d(x/y) = d+dx
y−xdy
y2.
Then ψDelf M = 0 (M = E x/y).
Solution: Consider ψDelg M for various g in order to
recover information on M .→MT
(s)6d(X,w) (‘s’ is for ‘sauvage’, i.e., ‘wild’).
Wild twistor D-modules – p. 23/28
Wild twistor D-modules
Semi-simple holonomic
D-module onX
Polarized wild twistor
D-module onX
decomposition
theorem ??
Conjecture of Kashiwara
general case ??
Polarized wild twistor
D-module on Y??
Semi-simple holonomic
D-module on Y
??
Wild twistor D-modules – p. 24/28
Wild twistor D-modules
Semi-simple holonomic
D-module onX
Polarized wild twistor
D-module onX
decomposition
theorem ??
Conjecture of Kashiwara
general case ??
Polarized wild twistor
D-module on Y??
Semi-simple holonomic
D-module on Y
??
Work of O. Biquard and Ph. Boalch on curves.
Wild twistor D-modules – p. 24/28
Wild twistor D-modules
Semi-simple holonomic
D-module onX
Polarized wild twistor
D-module onX
decomposition
theorem ??
Conjecture of Kashiwara
general case ??
Polarized wild twistor
D-module on Y??
Semi-simple holonomic
D-module on Y
??
Work of O. Biquard and Ph. Boalch on curves.
Recent work of T. Mochizuki.
Wild twistor D-modules – p. 24/28
Wild twistor D-modules on curves
T : a polarized regular twistor D-module of weight won P1.
Wild twistor D-modules – p. 25/28
Wild twistor D-modules on curves
T : a polarized regular twistor D-module of weight won P1. f : P1 −→ Spec C the constant map.
Wild twistor D-modules – p. 25/28
Wild twistor D-modules on curves
T : a polarized regular twistor D-module of weight won P1. f : P1 −→ Spec C the constant map.
THEOREM 2: fk+(E t/z ⊗ T ) is a polarized pure twistorstructure of weight w + k.
Wild twistor D-modules – p. 25/28
Irregular nearby cycles, after Deligne
X complex manifold, f : X −→ C.
Wild twistor D-modules – p. 26/28
Irregular nearby cycles, after Deligne
X complex manifold, f : X −→ C.
M a holonomic DX -module.
Wild twistor D-modules – p. 26/28
Irregular nearby cycles, after Deligne
X complex manifold, f : X −→ C.
M a holonomic DX -module.
Deligne (Letter to Malgrange, 1983):
ψDelf M :=
⊕
N
ψf(f+N ⊗M
),
Wild twistor D-modules – p. 26/28
Irregular nearby cycles, after Deligne
X complex manifold, f : X −→ C.
M a holonomic DX -module.
Deligne (Letter to Malgrange, 1983):
ψDelf M :=
⊕
N
ψf(f+N ⊗M
),
N such that N is an irred. C((t))-module withconnection.
Wild twistor D-modules – p. 26/28
Irregular nearby cycles, after Deligne
Problem:
X = A2, coordinates x, y,
Wild twistor D-modules – p. 27/28
Irregular nearby cycles, after Deligne
Problem:
X = A2, coordinates x, y,
f : X −→ A1, (x, y) 7−→ y,
Wild twistor D-modules – p. 27/28
Irregular nearby cycles, after Deligne
Problem:
X = A2, coordinates x, y,
f : X −→ A1, (x, y) 7−→ y,
M = OA2[1/y],∇ = d+ d(x/y) = d+dx
y−xdy
y2.
Wild twistor D-modules – p. 27/28
Irregular nearby cycles, after Deligne
Problem:
X = A2, coordinates x, y,
f : X −→ A1, (x, y) 7−→ y,
M = OA2[1/y],∇ = d+ d(x/y) = d+dx
y−xdy
y2.
Then ψDelf M = 0 (M = E x/y).
Wild twistor D-modules – p. 27/28
Irregular nearby cycles, after Deligne
Problem:
X = A2, coordinates x, y,
f : X −→ A1, (x, y) 7−→ y,
M = OA2[1/y],∇ = d+ d(x/y) = d+dx
y−xdy
y2.
Then ψDelf M = 0 (M = E x/y).
Solution: Consider ψDelg M for various g in order to
recover information on M .
Wild twistor D-modules – p. 27/28
Irregular nearby cycles, after Deligne
Problem:
X = A2, coordinates x, y,
f : X −→ A1, (x, y) 7−→ y,
M = OA2[1/y],∇ = d+ d(x/y) = d+dx
y−xdy
y2.
Then ψDelf M = 0 (M = E x/y).
Solution: Consider ψDelg M for various g in order to
recover information on M .→MT
(s)6d(X,w) (‘s’ is for ‘sauvage’, i.e., ‘wild’).
Wild twistor D-modules – p. 27/28
Wild twistor D-modules
Semi-simple holonomic
D-module onX
Polarized wild twistor
D-module onX
decomposition
theorem ??
Conjecture of Kashiwara
general case ??
Polarized wild twistor
D-module on Y??
Semi-simple holonomic
D-module on Y
??
Wild twistor D-modules – p. 28/28
Wild twistor D-modules
Semi-simple holonomic
D-module onX
Polarized wild twistor
D-module onX
decomposition
theorem ??
Conjecture of Kashiwara
general case ??
Polarized wild twistor
D-module on Y??
Semi-simple holonomic
D-module on Y
??
Work of O. Biquard and Ph. Boalch on curves.
Wild twistor D-modules – p. 28/28
Wild twistor D-modules
Semi-simple holonomic
D-module onX
Polarized wild twistor
D-module onX
decomposition
theorem ??
Conjecture of Kashiwara
general case ??
Polarized wild twistor
D-module on Y??
Semi-simple holonomic
D-module on Y
??
Work of O. Biquard and Ph. Boalch on curves.
Recent work of T. Mochizuki.
Wild twistor D-modules – p. 28/28