Wild twistor D-modules

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


Introduction

(V,∇) an alg. vect. bundle with connection on A1rP ,P = a finite set of points.


Introduction


→ a holonomic module M on C[t]〈∂t〉.


Introduction



Fourier-Laplace transform of M : a C[τ ]〈∂τ 〉-module,τ = ∂t, ∂τ = −t.


Introduction




Assume that (V,∇) underlies a variation of polarizedHodge structure on A1 r P .


Introduction





Question:


Introduction





Question: What kind of a structure does M underlie?


Introduction





Question: What kind of a structure does M underlie?Problem:


Introduction





Question: What kind of a structure does M underlie?Problem: Irregularity of M at τ =∞.


Introduction





Question: What kind of a structure does M underlie?Problem: Irregularity of M at τ =∞.Analogue for ℓ-adic sheaves: Results of Deligne andLaumon.


Introduction





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:


Introduction





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.


Introduction





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).


Conjecture of Kashiwara (regular case)



f : X −→ Y : a morphism between smooth complexprojective varieties.




Semi-simple perverse

sheaf onX





sheaf on Y


sheaf onX

Conjecture of Kashiwara

regular case



f : X −→ Y : a morphism between smooth complexprojective varieties.Sketch of the analytic proof :


sheaf on Y


sheaf onX





sheaf on Y

Polarized regular twistor

D-module onX


sheaf onX





D-module on Y


sheaf on Y

decomposition

theorem (C.S.)


D-module onX


sheaf onX





D-module on Y


sheaf on Y

decomposition

theorem (C.S.)

Simpson

+ Hamm-Le D.T.


D-module onX


sheaf onX





D-module on Y


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





D-module on Y


sheaf on Y

decomposition

theorem (C.S.)

Simpson

+ Hamm-Le D.T.


D-module onXT. Mochizuki

(NCD)


sheaf onX





D-module on Y


sheaf on Y

decomposition

theorem (C.S.)

Simpson

+ Hamm-Le D.T.


D-module onXT. Mochizuki

(NCD)


sheaf onX


regular case


Twistor structures(C. Simpson)



Hodge structures Twistor structures



Hodge structures Twistor structuresFiltered vect. sp. (H,F •H) Holom. vect. bundle on P1




Conjugation H→H Twistor conjugation




Conjugation H→H Twistor conjugationH→H = σ∗H




Conjugation H→H Twistor conjugationH→H = σ∗Hσ : z 7→−1/z




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 ∨







H ≃ H∗ H ≃ H ∗ := H ∨

→ Γ(P1,H ) ≃ Γ(P1,H )∗







H ≃ H∗ H ≃ H ∗ := H ∨

→ Γ(P1,H ) ≃ Γ(P1,H )∗

Positivity Positivity on 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)





Holomorphic vector bundle on P1



H ′,H ′′ holomorphic on A1, “gluing”:C : H ′

|S ⊗OSH′′|S −→ OS , S = |z| = 1.




|S ⊗OSH′′|S −→ OS , S = |z| = 1.

Twistor adjoint H ∗ = H ∨




|S ⊗OSH′′|S −→ OS , S = |z| = 1.

Twistor adjoint (H ′,H ′′, C)∗ = (H ′′,H ′, C∗),C∗(v,u) := C(u,v).




|S ⊗OSH′′|S −→ OS , S = |z| = 1.


Tate twist H (k) := H ⊗ OP1(−2k)




|S ⊗OSH′′|S −→ OS , S = |z| = 1.


Tate twist (H ′,H ′′, C)(k) = (H ′,H ′′, (iz)−2kC).




|S ⊗OSH′′|S −→ OS , S = |z| = 1.



Polarization in weight 0




|S ⊗OSH′′|S −→ OS , S = |z| = 1.



H ′ = H ′′ and existence of a global frame ε of H ′

such that, on S, C(εi,εj) = δij .


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 .







∞,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).







∞,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).










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







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







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 .


Twistor D-modules



Twistor D-modules


H ′,H ′′ holomorphic on X × A1 with flatholomorphic relative connections∇′,∇′′:

H −→1

zΩ1

X /A1 ⊗H (H = H ′,H ′′).


Twistor D-modules


M ′,M ′′ holonomic RX -modules,RX = OX 〈ðx1

, . . . , ðxn〉, ðxi

= z∂xi.


Twistor D-modules



, . . . , 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).


Twistor D-modules



, . . . , ðxn〉, ðxi

= z∂xi.

“gluing”: C : M ′|X×S ⊗OS

M′′|X×S −→ DbX×S/S

compatible with the RX ⊗RX -action.


Twistor D-modules



, . . . , ðxn〉, ðxi

= z∂xi.




Twistor adjoint (M ′,M ′′, C)∗ = (M ′′,M ′, C∗).


Twistor D-modules



, . . . , ðxn〉, ðxi

= z∂xi.





. . . Strictness


Twistor D-modules



, . . . , ðxn〉, ðxi

= z∂xi.





. . . Strictness

Problem:


Twistor D-modules



, . . . , ðxn〉, ðxi

= z∂xi.





. . . Strictness

Problem: How to define the restriction to xo ∈ X?


Twistor D-modules



, . . . , ðxn〉, ðxi

= z∂xi.





. . . Strictness

Problem: How to define the restriction to xo ∈ X?Answer:


Twistor D-modules



, . . . , ðxn〉, ðxi

= z∂xi.





. . . Strictness

Problem: How to define the restriction to xo ∈ X?Answer: Use iterated nearby cycle functors.


Twistor D-modules



, . . . , ðxn〉, ðxi

= z∂xi.





. . . Strictness


Kashiwara-Malgrange V -filtration for M ′,M ′′ → ψfM .


Twistor D-modules



, . . . , ðxn〉, ðxi

= z∂xi.





. . . Strictness


Kashiwara-Malgrange V -filtration for M ′,M ′′ → ψfM .Barlet Hermitian form on nearby cycles→ ψfC.


Polarized pure twistor D-modules



Hodge Modules(M. Saito)

Twistor D-modules




Twistor D-modules

Filtered D-mod. (M,F•M) Triple T = (M ′,M ′′, C)




Twistor D-modules

Filtered D-mod. (M,F•M) Triple T = (M ′,M ′′, C)Q-structure: perverse FQ




Twistor D-modules


C⊗Q FQ ≃ DRM Hermitian pairing T ≃ T ∗




Twistor D-modules



Category MH6d(X,w) Category MT6d(X,w)




Twistor D-modules




∀ holom. germ f : (X,xo) −→ C, constraint onψf (M,F•M,FQ) ψfT




Twistor D-modules





∀ℓ ∈ Z, grMℓ ψf (M,F•M,FQ) ∈ MH6d−1(X,w)




Twistor D-modules





∀ℓ ∈ Z, grMℓ ψfT ∈MT6d−1(X,w)




Twistor D-modules






Polarization condition by “restriction” to any xo




Twistor D-modules






Polarization condition by “restriction” to any xo

Category MH6d(X,w)(p) Category MT6d(X,w)(p)


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





This is an advantage of this generalized frameworkcf. the original problemGeneral problem: Hodge theory in presence ofirregular singularities.





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.


The decomposition theorem



f : X −→ Y : a morphism between smooth complexprojective varieties,T : a pol. regular twistor D-module on X, weight w.





D-module on Y


sheaf on Y

decomposition

theorem (C.S.)


D-module onX


sheaf onX




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]






f+T ≃⊕k f

k+T [−k] (⇐= rel. Hard Lefschetz).


Conjecture of Kashiwara (general case)?






f : X −→ Y : a morphism between smooth complexprojective varieties.Semi-simple holonomic

D-module onX


general case ??

Semi-simple holonomic

D-module on Y



f : X −→ Y : a morphism between smooth complexprojective varieties.Semi-simple holonomic

D-module onX

Polarized wild twistor

D-module onX

decomposition

theorem ??


general case ??


D-module on Y??


D-module on Y

??


Holonomic D-modules on curves



D: disc with coordinate t




M : holonomic D-module on D




M : holonomic D-module on DM = C[[t]]⊗CtM the formalized module.





Turrittin-Levelt: M = Mreg ⊕ Mirr





Turrittin-Levelt: M = Mreg ⊕ Mirr and

∃ ρ : D′t′→7→

Dt=t′q

, ρ+Mirr ≃⊕

ϕ∈t′−1C[t′−1]

(E ϕ ⊗Rϕ),






∃ 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).






∃ ρ : D′t′→7→

Dt=t′q

, ρ+Mirr ≃⊕

ϕ∈t′−1C[t′−1]

(E ϕ ⊗Rϕ),


ψtM = ψtM = ψtMreg←→ Mreg,






∃ ρ : D′t′→7→

Dt=t′q

, ρ+Mirr ≃⊕

ϕ∈t′−1C[t′−1]

(E ϕ ⊗Rϕ),


ψtM = ψtM = ψtMreg←→ Mreg, and

ψtMirr = 0.






∃ ρ : D′t′→7→

Dt=t′q

, ρ+Mirr ≃⊕

ϕ∈t′−1C[t′−1]

(E ϕ ⊗Rϕ),



ψtMirr = 0.

M ←→ ψt′(E−η ⊗ ρ+M), q ≫ 0,

∀ η ∈ t′−1C[t′−1].






∃ ρ : D′t′→7→

Dt=t′q

, ρ+Mirr ≃⊕

ϕ∈t′−1C[t′−1]

(E ϕ ⊗Rϕ),



ψtMirr = 0.

M ←→ ψt′(E−η ⊗ ρ+M), q ≫ 0,

∀ η ∈ t′−1C[t′−1]. = ψt′Rη






∃ ρ : D′t′→7→

Dt=t′q

, ρ+Mirr ≃⊕

ϕ∈t′−1C[t′−1]

(E ϕ ⊗Rϕ),



ψtMirr = 0.

M ←→ ψt′(E−η ⊗ ρ+M), q ≫ 0,

∀ η ∈ t′−1C[t′−1]. = ψt′Rη






∃ ρ : D′t′→7→

Dt=t′q

, ρ+Mirr ≃⊕

ϕ∈t′−1C[t′−1]

(E ϕ ⊗Rϕ),



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




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′),








ϕ∈t′−1C[t′−1]

(E ϕ ⊗Rϕ)

holds over C(t′),Rϕ regular (ΘRϕ

is holomorphic),








ϕ∈t′−1C[t′−1]

(E ϕ ⊗Rϕ)


is holomorphic),E ϕ = (C(t′), dϕ).








ϕ∈t′−1C[t′−1]

(E ϕ ⊗Rϕ)


is holomorphic),E ϕ = (C(t′), dϕ).

No Stokes phenomenon .


Wild twistor D-modules on curves



DEFINITION:



DEFINITION: T = (M ′,M ′′, C) on X = D is a wildtwistor D-module of weight w at t = 0 if



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



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.





−ϕ/z ⊗ ρ+T )

is a pure twistor structure of weight w + ℓ.Problem:





−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 ?





−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:





−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).





−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?





−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).





−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



THEOREM 1:



THEOREM 1: Assume T is a polarized wild twistorD-module on D at t = 0.



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.



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).







T : a polarized regular twistor D-module of weight won P1.



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:





REMARK:

can define the Fourier-Laplace transformT = f0

+(E tτ /z ⊗ T ).





REMARK:


+(E tτ /z ⊗ T ).

T is a pure twistor D-module on Cτ which is regularat τ = 0.





REMARK:


+(E tτ /z ⊗ T ).


At τ =∞, one expects that T is a wild twistorD-module





REMARK:


+(E tτ /z ⊗ T ).


At τ =∞, one expects that T is a wild twistorD-module (cf. the work of S. Szabo, 2005).


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

),






ψDelf M :=

⊕

N

ψf(f+N ⊗M

),

N such that N is an irred. C((t))-module withconnection.



Problem:



Problem:

X = A2, coordinates x, y,



Problem:


f : X −→ A1, (x, y) 7−→ y,



Problem:


f : X −→ A1, (x, y) 7−→ y,

M = OA2[1/y],∇ = d+ d(x/y) = d+dx

y−xdy

y2.



Problem:


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).



Problem:


f : X −→ A1, (x, y) 7−→ y,

M = OA2[1/y],∇ = d+ d(x/y) = d+dx

y−xdy

y2.


Solution: Consider ψDelg M for various g in order to

recover information on M .



Problem:


f : X −→ A1, (x, y) 7−→ y,

M = OA2[1/y],∇ = d+ d(x/y) = d+dx

y−xdy

y2.



recover information on M .→MT

(s)6d(X,w) (‘s’ is for ‘sauvage’, i.e., ‘wild’).


Wild twistor D-modules


D-module onX


D-module onX

decomposition

theorem ??


general case ??


D-module on Y??


D-module on Y

??




D-module onX


D-module onX

decomposition

theorem ??


general case ??


D-module on Y??


D-module on Y

??

Work of O. Biquard and Ph. Boalch on curves.




D-module onX


D-module onX

decomposition

theorem ??


general case ??


D-module on Y??


D-module on Y

??


Recent work of T. Mochizuki.



T : a polarized regular twistor D-module of weight won P1.






















ψDelf M :=

⊕

N

ψf(f+N ⊗M

),






ψDelf M :=

⊕

N

ψf(f+N ⊗M

),

N such that N is an irred. C((t))-module withconnection.



Problem:



Problem:




Problem:


f : X −→ A1, (x, y) 7−→ y,



Problem:


f : X −→ A1, (x, y) 7−→ y,

M = OA2[1/y],∇ = d+ d(x/y) = d+dx

y−xdy

y2.



Problem:


f : X −→ A1, (x, y) 7−→ y,

M = OA2[1/y],∇ = d+ d(x/y) = d+dx

y−xdy

y2.




Problem:


f : X −→ A1, (x, y) 7−→ y,

M = OA2[1/y],∇ = d+ d(x/y) = d+dx

y−xdy

y2.



recover information on M .



Problem:


f : X −→ A1, (x, y) 7−→ y,

M = OA2[1/y],∇ = d+ d(x/y) = d+dx

y−xdy

y2.



recover information on M .→MT

(s)6d(X,w) (‘s’ is for ‘sauvage’, i.e., ‘wild’).




D-module onX


D-module onX

decomposition

theorem ??


general case ??


D-module on Y??


D-module on Y

??




D-module onX


D-module onX

decomposition

theorem ??


general case ??


D-module on Y??


D-module on Y

??





D-module onX


D-module onX

decomposition

theorem ??


general case ??


D-module on Y??


D-module on Y

??


Recent work of T. Mochizuki.


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Wild twistor D-modules