aspects of the fourier-laplace transform...aspects of the fourier-laplace transform claude sabbah...

Aspects ofthe Fourier-Laplace transform

Claude Sabbah

Centre de Mathematiques Laurent Schwartz

UMR 7640 du CNRS

Ecole polytechnique, Palaiseau, France

Aspects of the Fourier-Laplace transform – p. 1/21

Introduction


Introduction

DEFINITION (A. Schwarz & I. Shapiro):


Introduction


Physics is a part of mathematics devoted to the

calculation of integrals of the form∫h(x)eg(x)dx.


Introduction




Different branches of physics are distinguished bythe range of the variable x and by the names usedfor h(x), g(x) and for the integral.


Introduction




Different branches of physics are distinguished bythe range of the variable x and by the names usedfor h(x), g(x) and for the integral.

Of course this is a joke, physics is not a part ofmathematics. However, it is true that the mainmathematical problem of physics is the calculation of

integrals of the form∫h(x)eg(x)dx.


Exp. twisted D-modules



Work of Céline Roucairol




U smooth affine complex variety




U smooth affine complex variety, f, g : U −→ A1





Exp. twisted Gauss-Manin systems∫ k

fOUe

g






fOUe

g

Solutions:∫

γegω,






fOUe

g

Solutions:∫

γegω,

ω: f -relative algebraic k-form,






fOUe

g

Solutions:∫

γegω,

ω: f -relative algebraic k-form,γ: closed k-cycle in the fibres of f ,






fOUe

g

Solutions:∫

γegω,

ω: f -relative algebraic k-form,γ: closed k-cycle in the fibres of f ,rapid decay: e−g −→ 0 when γ ∼ ∞.






fOUe

g

Solutions:∫

γegω,


Question:






fOUe

g

Solutions:∫

γegω,


Question: Describe the irregular singular points of∫ k

fOUe

g in terms of the geometry of f and g.




First step : reduction to 2 var’s.




First step : reduction to 2 var’s.(f, g) : U −→ A2, f = t, g = x




First step : reduction to 2 var’s.(f, g) : U −→ A2, f = t, g = x∫ j

tMex, M =

∫ `

(f,g)OU ,





tMex, M =

∫ `

(f,g)OU ,

M has regular singularities and singularsupport S ⊂ A2.





tMex, M =

∫ `

(f,g)OU ,

M has regular singularities and singularsupport S ⊂ A2.S = discriminant of (f, g).





tMex, M =

∫ `

(f,g)OU ,

M has regular singularities and singularsupport S ⊂ A2.S = discriminant of (f, g).

e.g. M is a bundle with flat connection on A2 r S.




Second step : G :=

∫ j

tMex,




Second step : G :=

∫ j

tMex, M reg. sing. on A2

(coord. (t, x)),




Second step : G :=

∫ j


(coord. (t, x)), S = sing. supp. of M .




Second step : G :=

∫ j



THEOREM:




Second step : G :=

∫ j



THEOREM:

to irreg. sing. of∫ j

tMex =⇒

t = to is an asymptotic dir. of S.




∞

S

to




6

-t

Si

xx =∞

tos




Second step : G :=

∫ j



THEOREM:


tMex =⇒

t = to is an asymptotic dir. of S.




Second step : G :=

∫ j



THEOREM:


tMex =⇒

t = to is an asymptotic dir. of S.Si = fi(t, x) = 0




Second step : G :=

∫ j



THEOREM:


tMex =⇒

t = to is an asymptotic dir. of S.Si = fi(t, x) = 0, Puiseux expansion (to = 0):ϕi(t) = t−qi/piv(t1/pi),




Second step : G :=

∫ j



THEOREM:


tMex =⇒

t = to is an asymptotic dir. of S.Si = fi(t, x) = 0, Puiseux expansion (to = 0):ϕi(t) = t−qi/piv(t1/pi), v(t1/pi) invertible,




Second step : G :=

∫ j



THEOREM:


tMex =⇒

t = to is an asymptotic dir. of S.Si = fi(t, x) = 0, Puiseux expansion (to = 0):ϕi(t) = t−qi/piv(t1/pi), v(t1/pi) invertible,then possible exp. factor are eϕi .




Second step : G :=

∫ j



THEOREM:


tMex =⇒

t = to is an asymptotic dir. of S.Si = fi(t, x) = 0, Puiseux expansion (to = 0):ϕi(t) = t−qi/piv(t1/pi), v(t1/pi) invertible,then possible exp. factor are eϕi .φfi

(DR(M ))|S∗

i




Second step : G :=

∫ j



THEOREM:


tMex =⇒


(DR(M ))|S∗

i→ info on the corresp. reg. part

of G




Second step : G :=

∫ j



THEOREM:


tMex =⇒


(DR(M ))|S∗

i→ info on the corresp. reg. part

of G (e.g. char. pol. of formal monodromy ).Aspects of the Fourier-Laplace transform – p. 7/21


Sketch of proof.



Sketch of proof.

First step: ramification.



Sketch of proof.


6

-t

Si

xx =∞

tos



Sketch of proof.


6

-t1/p

Sij

xx =∞

tos

→

6

-t

Si

xx =∞

tos



Sketch of proof.

Second step: twist by e−ϕ



Sketch of proof.


Take any ϕ ∈ t−1C[t−1] and consider G ⊗ e−ϕ.



Sketch of proof.



point: Reg. part of G rel. eϕ ←→ grV (G ⊗ e−ϕ)



Sketch of proof.



point: Reg. part of G rel. eϕ ←→ grV (G ⊗ e−ϕ)V = V-filtration of Kashiwara and Malgrange.



Sketch of proof.




(Laurent & Malgrange) grV = ψmodt compatible

with∫

t.



Sketch of proof.





with∫

t. → compute ψmod

t (M ⊗ e−ϕ).



Sketch of proof.





with∫

t. → compute ψmod

t (M ⊗ e−ϕ).

Simplify the computation by blowing-ups→ S is aN.C.D.



Sketch of proof.

Third step: local computations in dim. 2.



Sketch of proof.

Third step: local computations in dim. 2.f(x1, x2) = xm1

1 xm2

2 ,



Sketch of proof.


1 xm2

2 ,R reg. sing. with poles ⊂ x1x2 = 0,



Sketch of proof.


1 xm2

2 ,R reg. sing. with poles ⊂ x1x2 = 0,fix n = (n1, n2)



Sketch of proof.


1 xm2

2 ,R reg. sing. with poles ⊂ x1x2 = 0,fix n = (n1, n2)

Compute ψmodf (R ⊗ e1/x

n

).


Application to Fourier-Laplace


Application to Fourier-LaplaceData.

R a finite dim. C((u)) vect. space



R a finite dim. C((u)) vect. space with reg. sing.connection∇,




ϕ ∈ C((u))/C[[u]], ϕ(u) = u−q · unit,




ϕ ∈ C((u))/C[[u]], ϕ(u) = u−q · unit,

ρ ∈ uC[[u]], ρ(u) = up · unit,




ϕ ∈ C((u))/C[[u]], ϕ(u) = u−q · unit,


M =

∫ 0

ρeϕ ⊗R: C((ρ)) vect. space with

connection.




ϕ ∈ C((u))/C[[u]], ϕ(u) = u−q · unit,


M =

∫ 0


connection.Assume ρ = up




ϕ ∈ C((u))/C[[u]], ϕ(u) = u−q · unit,


M =

∫ 0


connection.Assume ρ = up→M is a free C[ρ, ρ−1]-modulewith connection.




ϕ ∈ C((u))/C[[u]], ϕ(u) = u−q · unit,


M =

∫ 0



Local Fourier-Laplace transform (t = new var.):

F (0,∞)M = M =

∫ 1

te−ρ/t ⊗M((t))




ϕ ∈ C((u))/C[[u]], ϕ(u) = u−q · unit,


M =

∫ 0



Local Fourier-Laplace transform (t = new var.):

F (0,∞)M = M =

∫ 1

te−ρ/t ⊗M((t))

finite dim. C((t)) vect. space with connection.Aspects of the Fourier-Laplace transform – p. 11/21


THEOREM:



THEOREM: M =

∫ 0

bρe bϕ ⊗ R,



THEOREM: M =

∫ 0

bρe bϕ ⊗ R,

ρ(u) = ρ′(u)/ϕ′(u),



THEOREM: M =

∫ 0

bρe bϕ ⊗ R,

ρ(u) = ρ′(u)/ϕ′(u), ϕ(u) = ϕ(u)−ρ(u)

ρ′(u)ϕ′(u),



THEOREM: M =

∫ 0

bρe bϕ ⊗ R,

ρ(u) = ρ′(u)/ϕ′(u), ϕ(u) = ϕ(u)−ρ(u)

ρ′(u)ϕ′(u),

R = R⊗ (uq/2).



THEOREM: M =

∫ 0

bρe bϕ ⊗ R,

ρ(u) = ρ′(u)/ϕ′(u), ϕ(u) = ϕ(u)−ρ(u)

ρ′(u)ϕ′(u),

R = R⊗ (uq/2).

± classical formula,



THEOREM: M =

∫ 0

bρe bϕ ⊗ R,

ρ(u) = ρ′(u)/ϕ′(u), ϕ(u) = ϕ(u)−ρ(u)

ρ′(u)ϕ′(u),

R = R⊗ (uq/2).

± classical formula, formulated by Laumon (1987) inthe `-adic case,



THEOREM: M =

∫ 0

bρe bϕ ⊗ R,

ρ(u) = ρ′(u)/ϕ′(u), ϕ(u) = ϕ(u)−ρ(u)

ρ′(u)ϕ′(u),

R = R⊗ (uq/2).

± classical formula, formulated by Laumon (1987) inthe `-adic case, proved by Fu in the `-adic case(2007),



THEOREM: M =

∫ 0

bρe bϕ ⊗ R,

ρ(u) = ρ′(u)/ϕ′(u), ϕ(u) = ϕ(u)−ρ(u)

ρ′(u)ϕ′(u),

R = R⊗ (uq/2).

± classical formula, formulated by Laumon (1987) inthe `-adic case, proved by Fu in the `-adic case(2007), proved by C.S./Fang (2007).



Sketch of proof.



Sketch of proof.

M =

∫ 1

te−ρ/t⊗M((t))



Sketch of proof.

M =

∫ 1

te−ρ/t⊗M((t)) =

∫ 1

teϕ(u)−ρ(u)/t⊗R((t)).



Sketch of proof.

M =

∫ 1


∫ 1


g(t, u) = ϕ(u)− ρ(u)/t,



Sketch of proof.

M =

∫ 1


∫ 1


g(t, u) = ϕ(u)− ρ(u)/t,

π : (t, u) 7−→ (t, ϕ(u)− ρ(u)/t),



Sketch of proof.

M =

∫ 1


∫ 1


g(t, u) = ϕ(u)− ρ(u)/t,

π : (t, u) 7−→ (t, ϕ(u)− ρ(u)/t),C∗t × C∗u −→ C∗t × A1

x



Sketch of proof.

M =

∫ 1


∫ 1


g(t, u) = ϕ(u)− ρ(u)/t,


x

projection formula: M =

∫ 1

tex ⊗M ,



Sketch of proof.

M =

∫ 1


∫ 1


g(t, u) = ϕ(u)− ρ(u)/t,


x

projection formula: M =

∫ 1

tex ⊗M ,

M =

∫ 0

πR((t)).



Sketch of proof.

S = Sing. supp. M = discriminant of π,



Sketch of proof.


Crit. locus of π : (t, u) 7−→ (t, ϕ(u)− ρ(u)/t):



Sketch of proof.



det

(1 ρ(u)/t2

0 ϕ′(u)− ρ′(u)/t

)= ϕ′(u)− ρ′(u)/t = 0,



Sketch of proof.



det

(1 ρ(u)/t2

0 ϕ′(u)− ρ′(u)/t

)= ϕ′(u)− ρ′(u)/t = 0,

i.e. t = ρ(u),



Sketch of proof.



det

(1 ρ(u)/t2

0 ϕ′(u)− ρ′(u)/t

)= ϕ′(u)− ρ′(u)/t = 0,

i.e. t = ρ(u),

Discriminant of π:



Sketch of proof.



det

(1 ρ(u)/t2

0 ϕ′(u)− ρ′(u)/t

)= ϕ′(u)− ρ′(u)/t = 0,

i.e. t = ρ(u),

Discriminant of π: image ofu 7−→ (t = ρ(u), ϕ(u)− ρ(u)ϕ′(u)/ρ′(u)),



Sketch of proof.



det

(1 ρ(u)/t2

0 ϕ′(u)− ρ′(u)/t

)= ϕ′(u)− ρ′(u)/t = 0,

i.e. t = ρ(u),

Discriminant of π: image ofu 7−→ (t = ρ(u), ϕ(u)− ρ(u)ϕ′(u)/ρ′(u)),

Puiseux param.: u −→ (ρ(u), ϕ(u)).


Fourier-Laplace and Hodge



Deligne (1984).



Deligne (1984).

X compact Riemann surf.,



Deligne (1984).

X compact Riemann surf., g : X −→ P1,



Deligne (1984).

X compact Riemann surf., g : X −→ P1,S ⊃ g−1(∞), U = X r S



Deligne (1984).


(V,∇) hol. vect. bdle with connection on U ,



Deligne (1984).



(M ,∇) Deligne merom. extension of (V,∇)



Deligne (1984).



(M ,∇) Deligne merom. extension of (V,∇) :merom. bdle on X with poles on S, ∇ reg. sing.



Deligne (1984).




H∗DR(X,M ⊗ eg): de Rham cohomology of∇+ dg



Deligne (1984).





=

∫ ∗

X→ptMeg.



Deligne (1984).





=

∫ ∗

X→ptMeg.

THEOREM (Deligne):



Deligne (1984).





=

∫ ∗

X→ptMeg.

THEOREM (Deligne): Assume mono.(V,∇) unitary .



Deligne (1984).





=

∫ ∗

X→ptMeg.

THEOREM (Deligne): Assume mono.(V,∇) unitary .Can define canonically a “Hodge ” filtration F •

Del onH∗DR(X,M ⊗ eg),



Deligne (1984).





=

∫ ∗

X→ptMeg.

THEOREM (Deligne): Assume mono.(V,∇) unitary .Can define canonically a “Hodge ” filtration F •

Del onH∗DR(X,M ⊗ eg), and the Hodge⇒ de Rhamspectral seq. degenerates at E1.



Deligne (1984).

REMARKS:



Deligne (1984).

REMARKS:Without eg:mixed Hodge structure on H∗DR(X,M ).



Deligne (1984).

REMARKS:Without eg:mixed Hodge structure on H∗DR(X,M ).F •

Del indexed by (possibly) real numbers α



Deligne (1984).


Del indexed by (possibly) real numbers α,exp 2πiα: eigenvalue of mono. of (V,∇) aroundg =∞.



Deligne (1984).



MOTIVATION/EXAMPLES:



Deligne (1984).



MOTIVATION/EXAMPLES:∫

R

e−x2

dx = π1/2 → F •

Del jumps at 1/2 only

for∫

A1→ptOA1e−x

2

.



Deligne (1984).



MOTIVATION/EXAMPLES:∫ ∞

0e−xxαdx/x = Γ(α) → F •

Del jumps at

1− α only for∫

A1→ptOA1xαe−x.



Deligne (1984).





Del jumps at

1− α only for∫

A1→ptOA1xαe−x.

Lamentation:Aspects of the Fourier-Laplace transform – p. 16/21


Deligne (1984).





Del jumps at

1− α only for∫

A1→ptOA1xαe−x.

Lamentation: No “Hodge decomposition” expected.Aspects of the Fourier-Laplace transform – p. 16/21

Wild Hodge theory


Wild Hodge theory

Various origins/motivations:


Wild Hodge theory

Various origins/motivations:Analogy with pure perverse `-adic sheaves,


Wild Hodge theory

Various origins/motivations:Analogy with pure perverse `-adic sheaves,Hard Lefschetz theorem with irregular singularities(Kashiwara’s conjecture),


Wild Hodge theory

Various origins/motivations:Analogy with pure perverse `-adic sheaves,Hard Lefschetz theorem with irregular singularities(Kashiwara’s conjecture),Rational structures in mirror symmetry,


Wild Hodge theory

Various origins/motivations:Analogy with pure perverse `-adic sheaves,Hard Lefschetz theorem with irregular singularities(Kashiwara’s conjecture),Rational structures in mirror symmetry,Nahm transform in differential geometry.


Wild Hodge theory


Starting point for the new point of view: work ofSimpson,


Wild Hodge theory



Results by Biquard-Boalch, Szabo,Hertling-Sevenheck, C.S.,


Wild Hodge theory



Results by Biquard-Boalch, Szabo,Hertling-Sevenheck, C.S.,

Main complete results obtained by Mochizuki.



X = P1, g = x, S 3 ∞, U = P1 r S.



X = P1, g = x, S 3 ∞, U = P1 r S.

(V,∇) underlies a polarized var. of Hodgestructure on U :



X = P1, g = x, S 3 ∞, U = P1 r S.


e.g.f : Y → U ,



X = P1, g = x, S 3 ∞, U = P1 r S.


e.g.f : Y → U , f projective and smooth over U ,



X = P1, g = x, S 3 ∞, U = P1 r S.



(V,∇) = Gauss-Manin connection of f



X = P1, g = x, S 3 ∞, U = P1 r S.



(V,∇) = Gauss-Manin connection of f :∫ ∗

fOU .



X = P1, g = x, S 3 ∞, U = P1 r S.




fOU .

C∞ bdle C∞U ⊗OUV =

⊕p+q=wH

p,q,



X = P1, g = x, S 3 ∞, U = P1 r S.




fOU .


⊕p+q=wH

p,q,

Hodge holomorphic sub-bdlesF p =

⊕p′>pH

p′,w−p′

,



X = P1, g = x, S 3 ∞, U = P1 r S.




fOU .


⊕p+q=wH

p,q,


⊕p′>pH

p′,w−p′

,

∇ : F p −→ F p−1 ⊗ Ω1U ,



X = P1, g = x, S 3 ∞, U = P1 r S.




fOU .


⊕p+q=wH

p,q,


⊕p′>pH

p′,w−p′

,

∇ : F p −→ F p−1 ⊗ Ω1U ,

polarization .



Definition of F •

Del(M ⊗ ex).



Definition of F •

Del(M ⊗ ex).

Schmid (1973):



Definition of F •

Del(M ⊗ ex).

Schmid (1973): → F •(M ) (on P1): holomorphicsub-bdles.



Definition of F •

Del(M ⊗ ex).

Schmid (1973): → F •(M ) (on P1): holomorphicsub-bdles.At x =∞, y = 1/x, V γ

y (M ), γ ∈ R:



Definition of F •

Del(M ⊗ ex).


y (M ), γ ∈ R:Res∞∇ on V γ

y (M ) has eigenval. in [γ, γ + 1).



Definition of F •

Del(M ⊗ ex).




Def. at finite distance :



Definition of F •

Del(M ⊗ ex).




Def. at finite distance : F •

Del(M ⊗ ex) :=F •(M ),



Definition of F •

Del(M ⊗ ex).





Del(M ⊗ ex) :=F •(M ),

Def. at infinity :



Definition of F •

Del(M ⊗ ex).





Del(M ⊗ ex) :=F •(M ),

Def. at infinity :

FαDel(M⊗ex) :=



Definition of F •

Del(M ⊗ ex).





Del(M ⊗ ex) :=F •(M ),

Def. at infinity :

FαDel(M⊗ex) := F dαe+k



Definition of F •

Del(M ⊗ ex).





Del(M ⊗ ex) :=F •(M ),

Def. at infinity :

FαDel(M⊗ex) := F dαe+k∩V α−dαe

y



Definition of F •

Del(M ⊗ ex).





Del(M ⊗ ex) :=F •(M ),

Def. at infinity :

FαDel(M⊗ex) :=

(F dαe+k∩V α−dαe

y ⊗e1/y)



Definition of F •

Del(M ⊗ ex).





Del(M ⊗ ex) :=F •(M ),

Def. at infinity :

FαDel(M⊗ex) := ∂kyy

−1(F dαe+k∩V α−dαe

y ⊗e1/y)



Definition of F •

Del(M ⊗ ex).





Del(M ⊗ ex) :=F •(M ),

Def. at infinity :

FαDel(M⊗ex) :=

∑

k∈N

∂kyy−1(F dαe+k∩V α−dαe

y ⊗e1/y)



Definition of F •

Del(M ⊗ ex).





Del(M ⊗ ex) :=F •(M ),

Def. at infinity :

FαDel(M⊗ex) :=

∑

k∈N


y ⊗e1/y)

Filtered de Rham complex FαDel DR(M⊗ex):



Definition of F •

Del(M ⊗ ex).





Del(M ⊗ ex) :=F •(M ),

Def. at infinity :

FαDel(M⊗ex) :=

∑

k∈N


y ⊗e1/y)

Filtered de Rham complex FαDel DR(M⊗ex):

0→ FαDel(M⊗ex)

∇−→ Ω1

P1 ⊗ Fα−1Del (M⊗ex)→ 0



Taking hypercohomology→ FαDelH∗DR(X,M ⊗ ex).




THEOREM:




THEOREM: The Hodge⇒ de Rham spectral seq.degenerates at E1.





By rescaling x 7−→ −x/t, t ∈ C∗,






H∗DR(X,M ⊗ ex)→

∫

tM [t, t−1]⊗ e−x/t = M







∫

tM [t, t−1]⊗ e−x/t = M

→FαDelM ,







∫

tM [t, t−1]⊗ e−x/t = M

→FαDelM , ∇ : FαDelM −→ Ω1 ⊗ Fα−1Del M ,







∫

tM [t, t−1]⊗ e−x/t = M


But no corresp. Hodge decomp. (lamentation).







∫

tM [t, t−1]⊗ e−x/t = M



On the other hand, wild Hodge Theory →







∫

tM [t, t−1]⊗ e−x/t = M



On the other hand, wild Hodge Theory → Hodgedecomp. for each H∗DR(X,M ⊗ e−x/t),







∫

tM [t, t−1]⊗ e−x/t = M



On the other hand, wild Hodge Theory → Hodgedecomp. for each H∗DR(X,M ⊗ e−x/t), but noHodge bundles F •.







∫

tM [t, t−1]⊗ e−x/t = M



On the other hand, wild Hodge Theory → Hodgedecomp. for each H∗DR(X,M ⊗ e−x/t), but noHodge bundles F •.Why?







∫

tM [t, t−1]⊗ e−x/t = M



On the other hand, wild Hodge Theory → Hodgedecomp. for each H∗DR(X,M ⊗ e−x/t), but noHodge bundles F •.Why? The jumps of the Hodge filtration (t fixed)depend on t.



THEOREM:



THEOREM: Asymptotic vanishing of the lamentationwhen t −→∞.




0 ∞ t




0 ∞

Jump of FDel

t




0 ∞

Jump of FDel

Jump of Hodge decomp.(“new supersymm.

index” Cecotti-Vafa)

t


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