aspects of the fourier-laplace transform
TRANSCRIPT
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
Aspects of the Fourier-Laplace transform – p. 2/21
Introduction
DEFINITION (A. Schwarz & I. Shapiro):
Aspects of the Fourier-Laplace transform – p. 2/21
Introduction
DEFINITION (A. Schwarz & I. Shapiro):
Physics is a part of mathematics devoted to the
calculation of integrals of the form∫h(x)eg(x)dx.
Aspects of the Fourier-Laplace transform – p. 2/21
Introduction
DEFINITION (A. Schwarz & I. Shapiro):
Physics is a part of mathematics devoted to the
calculation of integrals of the form∫h(x)eg(x)dx.
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.
Aspects of the Fourier-Laplace transform – p. 2/21
Introduction
DEFINITION (A. Schwarz & I. Shapiro):
Physics is a part of mathematics devoted to the
calculation of integrals of the form∫h(x)eg(x)dx.
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.
Aspects of the Fourier-Laplace transform – p. 2/21
Exp. twisted D-modules
Aspects of the Fourier-Laplace transform – p. 3/21
Exp. twisted D-modules
Work of Céline Roucairol
Aspects of the Fourier-Laplace transform – p. 3/21
Exp. twisted D-modules
Work of Céline Roucairol
U smooth affine complex variety
Aspects of the Fourier-Laplace transform – p. 3/21
Exp. twisted D-modules
Work of Céline Roucairol
U smooth affine complex variety, f, g : U −→ A1
Aspects of the Fourier-Laplace transform – p. 3/21
Exp. twisted D-modules
Work of Céline Roucairol
U smooth affine complex variety, f, g : U −→ A1
Exp. twisted Gauss-Manin systems∫ k
fOUe
g
Aspects of the Fourier-Laplace transform – p. 3/21
Exp. twisted D-modules
Work of Céline Roucairol
U smooth affine complex variety, f, g : U −→ A1
Exp. twisted Gauss-Manin systems∫ k
fOUe
g
Solutions:∫
γegω,
Aspects of the Fourier-Laplace transform – p. 3/21
Exp. twisted D-modules
Work of Céline Roucairol
U smooth affine complex variety, f, g : U −→ A1
Exp. twisted Gauss-Manin systems∫ k
fOUe
g
Solutions:∫
γegω,
ω: f -relative algebraic k-form,
Aspects of the Fourier-Laplace transform – p. 3/21
Exp. twisted D-modules
Work of Céline Roucairol
U smooth affine complex variety, f, g : U −→ A1
Exp. twisted Gauss-Manin systems∫ k
fOUe
g
Solutions:∫
γegω,
ω: f -relative algebraic k-form,γ: closed k-cycle in the fibres of f ,
Aspects of the Fourier-Laplace transform – p. 3/21
Exp. twisted D-modules
Work of Céline Roucairol
U smooth affine complex variety, f, g : U −→ A1
Exp. twisted Gauss-Manin systems∫ k
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 ∼ ∞.
Aspects of the Fourier-Laplace transform – p. 3/21
Exp. twisted D-modules
Work of Céline Roucairol
U smooth affine complex variety, f, g : U −→ A1
Exp. twisted Gauss-Manin systems∫ k
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 ∼ ∞.
Question:
Aspects of the Fourier-Laplace transform – p. 3/21
Exp. twisted D-modules
Work of Céline Roucairol
U smooth affine complex variety, f, g : U −→ A1
Exp. twisted Gauss-Manin systems∫ k
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 ∼ ∞.
Question: Describe the irregular singular points of∫ k
fOUe
g in terms of the geometry of f and g.
Aspects of the Fourier-Laplace transform – p. 3/21
Exp. twisted D-modules
Work of Céline Roucairol
First step : reduction to 2 var’s.
Aspects of the Fourier-Laplace transform – p. 4/21
Exp. twisted D-modules
Work of Céline Roucairol
First step : reduction to 2 var’s.(f, g) : U −→ A2, f = t, g = x
Aspects of the Fourier-Laplace transform – p. 4/21
Exp. twisted D-modules
Work of Céline Roucairol
First step : reduction to 2 var’s.(f, g) : U −→ A2, f = t, g = x∫ j
tMex, M =
∫ `
(f,g)OU ,
Aspects of the Fourier-Laplace transform – p. 4/21
Exp. twisted D-modules
Work of Céline Roucairol
First step : reduction to 2 var’s.(f, g) : U −→ A2, f = t, g = x∫ j
tMex, M =
∫ `
(f,g)OU ,
M has regular singularities and singularsupport S ⊂ A2.
Aspects of the Fourier-Laplace transform – p. 4/21
Exp. twisted D-modules
Work of Céline Roucairol
First step : reduction to 2 var’s.(f, g) : U −→ A2, f = t, g = x∫ j
tMex, M =
∫ `
(f,g)OU ,
M has regular singularities and singularsupport S ⊂ A2.S = discriminant of (f, g).
Aspects of the Fourier-Laplace transform – p. 4/21
Exp. twisted D-modules
Work of Céline Roucairol
First step : reduction to 2 var’s.(f, g) : U −→ A2, f = t, g = x∫ j
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.
Aspects of the Fourier-Laplace transform – p. 4/21
Exp. twisted D-modules
Work of Céline Roucairol
Second step : G :=
∫ j
tMex,
Aspects of the Fourier-Laplace transform – p. 5/21
Exp. twisted D-modules
Work of Céline Roucairol
Second step : G :=
∫ j
tMex, M reg. sing. on A2
(coord. (t, x)),
Aspects of the Fourier-Laplace transform – p. 5/21
Exp. twisted D-modules
Work of Céline Roucairol
Second step : G :=
∫ j
tMex, M reg. sing. on A2
(coord. (t, x)), S = sing. supp. of M .
Aspects of the Fourier-Laplace transform – p. 5/21
Exp. twisted D-modules
Work of Céline Roucairol
Second step : G :=
∫ j
tMex, M reg. sing. on A2
(coord. (t, x)), S = sing. supp. of M .
THEOREM:
Aspects of the Fourier-Laplace transform – p. 5/21
Exp. twisted D-modules
Work of Céline Roucairol
Second step : G :=
∫ j
tMex, M reg. sing. on A2
(coord. (t, x)), S = sing. supp. of M .
THEOREM:
to irreg. sing. of∫ j
tMex =⇒
t = to is an asymptotic dir. of S.
Aspects of the Fourier-Laplace transform – p. 5/21
Exp. twisted D-modules
Work of Céline Roucairol
∞
S
to
Aspects of the Fourier-Laplace transform – p. 6/21
Exp. twisted D-modules
Work of Céline Roucairol
6
-t
Si
xx =∞
tos
Aspects of the Fourier-Laplace transform – p. 6/21
Exp. twisted D-modules
Work of Céline Roucairol
Second step : G :=
∫ j
tMex, M reg. sing. on A2
(coord. (t, x)), S = sing. supp. of M .
THEOREM:
to irreg. sing. of∫ j
tMex =⇒
t = to is an asymptotic dir. of S.
Aspects of the Fourier-Laplace transform – p. 7/21
Exp. twisted D-modules
Work of Céline Roucairol
Second step : G :=
∫ j
tMex, M reg. sing. on A2
(coord. (t, x)), S = sing. supp. of M .
THEOREM:
to irreg. sing. of∫ j
tMex =⇒
t = to is an asymptotic dir. of S.Si = fi(t, x) = 0
Aspects of the Fourier-Laplace transform – p. 7/21
Exp. twisted D-modules
Work of Céline Roucairol
Second step : G :=
∫ j
tMex, M reg. sing. on A2
(coord. (t, x)), S = sing. supp. of M .
THEOREM:
to irreg. sing. of∫ j
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),
Aspects of the Fourier-Laplace transform – p. 7/21
Exp. twisted D-modules
Work of Céline Roucairol
Second step : G :=
∫ j
tMex, M reg. sing. on A2
(coord. (t, x)), S = sing. supp. of M .
THEOREM:
to irreg. sing. of∫ j
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,
Aspects of the Fourier-Laplace transform – p. 7/21
Exp. twisted D-modules
Work of Céline Roucairol
Second step : G :=
∫ j
tMex, M reg. sing. on A2
(coord. (t, x)), S = sing. supp. of M .
THEOREM:
to irreg. sing. of∫ j
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 .
Aspects of the Fourier-Laplace transform – p. 7/21
Exp. twisted D-modules
Work of Céline Roucairol
Second step : G :=
∫ j
tMex, M reg. sing. on A2
(coord. (t, x)), S = sing. supp. of M .
THEOREM:
to irreg. sing. of∫ j
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
Aspects of the Fourier-Laplace transform – p. 7/21
Exp. twisted D-modules
Work of Céline Roucairol
Second step : G :=
∫ j
tMex, M reg. sing. on A2
(coord. (t, x)), S = sing. supp. of M .
THEOREM:
to irreg. sing. of∫ j
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→ info on the corresp. reg. part
of G
Aspects of the Fourier-Laplace transform – p. 7/21
Exp. twisted D-modules
Work of Céline Roucairol
Second step : G :=
∫ j
tMex, M reg. sing. on A2
(coord. (t, x)), S = sing. supp. of M .
THEOREM:
to irreg. sing. of∫ j
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→ info on the corresp. reg. part
of G (e.g. char. pol. of formal monodromy ).Aspects of the Fourier-Laplace transform – p. 7/21
Exp. twisted D-modules
Sketch of proof.
Aspects of the Fourier-Laplace transform – p. 8/21
Exp. twisted D-modules
Sketch of proof.
First step: ramification.
Aspects of the Fourier-Laplace transform – p. 8/21
Exp. twisted D-modules
Sketch of proof.
First step: ramification.
6
-t
Si
xx =∞
tos
Aspects of the Fourier-Laplace transform – p. 8/21
Exp. twisted D-modules
Sketch of proof.
First step: ramification.
6
-t1/p
Sij
xx =∞
tos
→
6
-t
Si
xx =∞
tos
Aspects of the Fourier-Laplace transform – p. 8/21
Exp. twisted D-modules
Sketch of proof.
Second step: twist by e−ϕ
Aspects of the Fourier-Laplace transform – p. 9/21
Exp. twisted D-modules
Sketch of proof.
Second step: twist by e−ϕ
Take any ϕ ∈ t−1C[t−1] and consider G ⊗ e−ϕ.
Aspects of the Fourier-Laplace transform – p. 9/21
Exp. twisted D-modules
Sketch of proof.
Second step: twist by e−ϕ
Take any ϕ ∈ t−1C[t−1] and consider G ⊗ e−ϕ.
point: Reg. part of G rel. eϕ ←→ grV (G ⊗ e−ϕ)
Aspects of the Fourier-Laplace transform – p. 9/21
Exp. twisted D-modules
Sketch of proof.
Second step: twist by e−ϕ
Take any ϕ ∈ t−1C[t−1] and consider G ⊗ e−ϕ.
point: Reg. part of G rel. eϕ ←→ grV (G ⊗ e−ϕ)V = V-filtration of Kashiwara and Malgrange.
Aspects of the Fourier-Laplace transform – p. 9/21
Exp. twisted D-modules
Sketch of proof.
Second step: twist by e−ϕ
Take any ϕ ∈ t−1C[t−1] and consider G ⊗ e−ϕ.
point: Reg. part of G rel. eϕ ←→ grV (G ⊗ e−ϕ)V = V-filtration of Kashiwara and Malgrange.
(Laurent & Malgrange) grV = ψmodt compatible
with∫
t.
Aspects of the Fourier-Laplace transform – p. 9/21
Exp. twisted D-modules
Sketch of proof.
Second step: twist by e−ϕ
Take any ϕ ∈ t−1C[t−1] and consider G ⊗ e−ϕ.
point: Reg. part of G rel. eϕ ←→ grV (G ⊗ e−ϕ)V = V-filtration of Kashiwara and Malgrange.
(Laurent & Malgrange) grV = ψmodt compatible
with∫
t. → compute ψmod
t (M ⊗ e−ϕ).
Aspects of the Fourier-Laplace transform – p. 9/21
Exp. twisted D-modules
Sketch of proof.
Second step: twist by e−ϕ
Take any ϕ ∈ t−1C[t−1] and consider G ⊗ e−ϕ.
point: Reg. part of G rel. eϕ ←→ grV (G ⊗ e−ϕ)V = V-filtration of Kashiwara and Malgrange.
(Laurent & Malgrange) grV = ψmodt compatible
with∫
t. → compute ψmod
t (M ⊗ e−ϕ).
Simplify the computation by blowing-ups→ S is aN.C.D.
Aspects of the Fourier-Laplace transform – p. 9/21
Exp. twisted D-modules
Sketch of proof.
Third step: local computations in dim. 2.
Aspects of the Fourier-Laplace transform – p. 10/21
Exp. twisted D-modules
Sketch of proof.
Third step: local computations in dim. 2.f(x1, x2) = xm1
1 xm2
2 ,
Aspects of the Fourier-Laplace transform – p. 10/21
Exp. twisted D-modules
Sketch of proof.
Third step: local computations in dim. 2.f(x1, x2) = xm1
1 xm2
2 ,R reg. sing. with poles ⊂ x1x2 = 0,
Aspects of the Fourier-Laplace transform – p. 10/21
Exp. twisted D-modules
Sketch of proof.
Third step: local computations in dim. 2.f(x1, x2) = xm1
1 xm2
2 ,R reg. sing. with poles ⊂ x1x2 = 0,fix n = (n1, n2)
Aspects of the Fourier-Laplace transform – p. 10/21
Exp. twisted D-modules
Sketch of proof.
Third step: local computations in dim. 2.f(x1, x2) = xm1
1 xm2
2 ,R reg. sing. with poles ⊂ x1x2 = 0,fix n = (n1, n2)
Compute ψmodf (R ⊗ e1/x
n
).
Aspects of the Fourier-Laplace transform – p. 10/21
Application to Fourier-Laplace
Aspects of the Fourier-Laplace transform – p. 11/21
Application to Fourier-LaplaceData.
R a finite dim. C((u)) vect. space
Aspects of the Fourier-Laplace transform – p. 11/21
Application to Fourier-LaplaceData.
R a finite dim. C((u)) vect. space with reg. sing.connection∇,
Aspects of the Fourier-Laplace transform – p. 11/21
Application to Fourier-LaplaceData.
R a finite dim. C((u)) vect. space with reg. sing.connection∇,
ϕ ∈ C((u))/C[[u]], ϕ(u) = u−q · unit,
Aspects of the Fourier-Laplace transform – p. 11/21
Application to Fourier-LaplaceData.
R a finite dim. C((u)) vect. space with reg. sing.connection∇,
ϕ ∈ C((u))/C[[u]], ϕ(u) = u−q · unit,
ρ ∈ uC[[u]], ρ(u) = up · unit,
Aspects of the Fourier-Laplace transform – p. 11/21
Application to Fourier-LaplaceData.
R a finite dim. C((u)) vect. space with reg. sing.connection∇,
ϕ ∈ C((u))/C[[u]], ϕ(u) = u−q · unit,
ρ ∈ uC[[u]], ρ(u) = up · unit,
M =
∫ 0
ρeϕ ⊗R: C((ρ)) vect. space with
connection.
Aspects of the Fourier-Laplace transform – p. 11/21
Application to Fourier-LaplaceData.
R a finite dim. C((u)) vect. space with reg. sing.connection∇,
ϕ ∈ C((u))/C[[u]], ϕ(u) = u−q · unit,
ρ ∈ uC[[u]], ρ(u) = up · unit,
M =
∫ 0
ρeϕ ⊗R: C((ρ)) vect. space with
connection.Assume ρ = up
Aspects of the Fourier-Laplace transform – p. 11/21
Application to Fourier-LaplaceData.
R a finite dim. C((u)) vect. space with reg. sing.connection∇,
ϕ ∈ C((u))/C[[u]], ϕ(u) = u−q · unit,
ρ ∈ uC[[u]], ρ(u) = up · unit,
M =
∫ 0
ρeϕ ⊗R: C((ρ)) vect. space with
connection.Assume ρ = up→M is a free C[ρ, ρ−1]-modulewith connection.
Aspects of the Fourier-Laplace transform – p. 11/21
Application to Fourier-LaplaceData.
R a finite dim. C((u)) vect. space with reg. sing.connection∇,
ϕ ∈ C((u))/C[[u]], ϕ(u) = u−q · unit,
ρ ∈ uC[[u]], ρ(u) = up · unit,
M =
∫ 0
ρeϕ ⊗R: C((ρ)) vect. space with
connection.Assume ρ = up→M is a free C[ρ, ρ−1]-modulewith connection.
Local Fourier-Laplace transform (t = new var.):
F (0,∞)M = M =
∫ 1
te−ρ/t ⊗M((t))
Aspects of the Fourier-Laplace transform – p. 11/21
Application to Fourier-LaplaceData.
R a finite dim. C((u)) vect. space with reg. sing.connection∇,
ϕ ∈ C((u))/C[[u]], ϕ(u) = u−q · unit,
ρ ∈ uC[[u]], ρ(u) = up · unit,
M =
∫ 0
ρeϕ ⊗R: C((ρ)) vect. space with
connection.Assume ρ = up→M is a free C[ρ, ρ−1]-modulewith connection.
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
Application to Fourier-Laplace
THEOREM:
Aspects of the Fourier-Laplace transform – p. 12/21
Application to Fourier-Laplace
THEOREM: M =
∫ 0
bρe bϕ ⊗ R,
Aspects of the Fourier-Laplace transform – p. 12/21
Application to Fourier-Laplace
THEOREM: M =
∫ 0
bρe bϕ ⊗ R,
ρ(u) = ρ′(u)/ϕ′(u),
Aspects of the Fourier-Laplace transform – p. 12/21
Application to Fourier-Laplace
THEOREM: M =
∫ 0
bρe bϕ ⊗ R,
ρ(u) = ρ′(u)/ϕ′(u), ϕ(u) = ϕ(u)−ρ(u)
ρ′(u)ϕ′(u),
Aspects of the Fourier-Laplace transform – p. 12/21
Application to Fourier-Laplace
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).
Aspects of the Fourier-Laplace transform – p. 12/21
Application to Fourier-Laplace
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,
Aspects of the Fourier-Laplace transform – p. 12/21
Application to Fourier-Laplace
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,
Aspects of the Fourier-Laplace transform – p. 12/21
Application to Fourier-Laplace
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),
Aspects of the Fourier-Laplace transform – p. 12/21
Application to Fourier-Laplace
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).
Aspects of the Fourier-Laplace transform – p. 12/21
Application to Fourier-Laplace
Sketch of proof.
Aspects of the Fourier-Laplace transform – p. 13/21
Application to Fourier-Laplace
Sketch of proof.
M =
∫ 1
te−ρ/t⊗M((t))
Aspects of the Fourier-Laplace transform – p. 13/21
Application to Fourier-Laplace
Sketch of proof.
M =
∫ 1
te−ρ/t⊗M((t)) =
∫ 1
teϕ(u)−ρ(u)/t⊗R((t)).
Aspects of the Fourier-Laplace transform – p. 13/21
Application to Fourier-Laplace
Sketch of proof.
M =
∫ 1
te−ρ/t⊗M((t)) =
∫ 1
teϕ(u)−ρ(u)/t⊗R((t)).
g(t, u) = ϕ(u)− ρ(u)/t,
Aspects of the Fourier-Laplace transform – p. 13/21
Application to Fourier-Laplace
Sketch of proof.
M =
∫ 1
te−ρ/t⊗M((t)) =
∫ 1
teϕ(u)−ρ(u)/t⊗R((t)).
g(t, u) = ϕ(u)− ρ(u)/t,
Aspects of the Fourier-Laplace transform – p. 13/21
Application to Fourier-Laplace
Sketch of proof.
M =
∫ 1
te−ρ/t⊗M((t)) =
∫ 1
teϕ(u)−ρ(u)/t⊗R((t)).
g(t, u) = ϕ(u)− ρ(u)/t,
π : (t, u) 7−→ (t, ϕ(u)− ρ(u)/t),
Aspects of the Fourier-Laplace transform – p. 13/21
Application to Fourier-Laplace
Sketch of proof.
M =
∫ 1
te−ρ/t⊗M((t)) =
∫ 1
teϕ(u)−ρ(u)/t⊗R((t)).
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
Aspects of the Fourier-Laplace transform – p. 13/21
Application to Fourier-Laplace
Sketch of proof.
M =
∫ 1
te−ρ/t⊗M((t)) =
∫ 1
teϕ(u)−ρ(u)/t⊗R((t)).
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
projection formula: M =
∫ 1
tex ⊗M ,
Aspects of the Fourier-Laplace transform – p. 13/21
Application to Fourier-Laplace
Sketch of proof.
M =
∫ 1
te−ρ/t⊗M((t)) =
∫ 1
teϕ(u)−ρ(u)/t⊗R((t)).
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
projection formula: M =
∫ 1
tex ⊗M ,
M =
∫ 0
πR((t)).
Aspects of the Fourier-Laplace transform – p. 13/21
Application to Fourier-Laplace
Sketch of proof.
S = Sing. supp. M = discriminant of π,
Aspects of the Fourier-Laplace transform – p. 14/21
Application to Fourier-Laplace
Sketch of proof.
S = Sing. supp. M = discriminant of π,
Crit. locus of π : (t, u) 7−→ (t, ϕ(u)− ρ(u)/t):
Aspects of the Fourier-Laplace transform – p. 14/21
Application to Fourier-Laplace
Sketch of proof.
S = Sing. supp. M = discriminant of π,
Crit. locus of π : (t, u) 7−→ (t, ϕ(u)− ρ(u)/t):
det
(1 ρ(u)/t2
0 ϕ′(u)− ρ′(u)/t
)= ϕ′(u)− ρ′(u)/t = 0,
Aspects of the Fourier-Laplace transform – p. 14/21
Application to Fourier-Laplace
Sketch of proof.
S = Sing. supp. M = discriminant of π,
Crit. locus of π : (t, u) 7−→ (t, ϕ(u)− ρ(u)/t):
det
(1 ρ(u)/t2
0 ϕ′(u)− ρ′(u)/t
)= ϕ′(u)− ρ′(u)/t = 0,
i.e. t = ρ(u),
Aspects of the Fourier-Laplace transform – p. 14/21
Application to Fourier-Laplace
Sketch of proof.
S = Sing. supp. M = discriminant of π,
Crit. locus of π : (t, u) 7−→ (t, ϕ(u)− ρ(u)/t):
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:
Aspects of the Fourier-Laplace transform – p. 14/21
Application to Fourier-Laplace
Sketch of proof.
S = Sing. supp. M = discriminant of π,
Crit. locus of π : (t, u) 7−→ (t, ϕ(u)− ρ(u)/t):
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)),
Aspects of the Fourier-Laplace transform – p. 14/21
Application to Fourier-Laplace
Sketch of proof.
S = Sing. supp. M = discriminant of π,
Crit. locus of π : (t, u) 7−→ (t, ϕ(u)− ρ(u)/t):
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)).
Aspects of the Fourier-Laplace transform – p. 14/21
Fourier-Laplace and Hodge
Aspects of the Fourier-Laplace transform – p. 15/21
Fourier-Laplace and Hodge
Deligne (1984).
Aspects of the Fourier-Laplace transform – p. 15/21
Fourier-Laplace and Hodge
Deligne (1984).
X compact Riemann surf.,
Aspects of the Fourier-Laplace transform – p. 15/21
Fourier-Laplace and Hodge
Deligne (1984).
X compact Riemann surf., g : X −→ P1,
Aspects of the Fourier-Laplace transform – p. 15/21
Fourier-Laplace and Hodge
Deligne (1984).
X compact Riemann surf., g : X −→ P1,S ⊃ g−1(∞), U = X r S
Aspects of the Fourier-Laplace transform – p. 15/21
Fourier-Laplace and Hodge
Deligne (1984).
X compact Riemann surf., g : X −→ P1,S ⊃ g−1(∞), U = X r S
(V,∇) hol. vect. bdle with connection on U ,
Aspects of the Fourier-Laplace transform – p. 15/21
Fourier-Laplace and Hodge
Deligne (1984).
X compact Riemann surf., g : X −→ P1,S ⊃ g−1(∞), U = X r S
(V,∇) hol. vect. bdle with connection on U ,
(M ,∇) Deligne merom. extension of (V,∇)
Aspects of the Fourier-Laplace transform – p. 15/21
Fourier-Laplace and Hodge
Deligne (1984).
X compact Riemann surf., g : X −→ P1,S ⊃ g−1(∞), U = X r S
(V,∇) hol. vect. bdle with connection on U ,
(M ,∇) Deligne merom. extension of (V,∇) :merom. bdle on X with poles on S, ∇ reg. sing.
Aspects of the Fourier-Laplace transform – p. 15/21
Fourier-Laplace and Hodge
Deligne (1984).
X compact Riemann surf., g : X −→ P1,S ⊃ g−1(∞), U = X r S
(V,∇) hol. vect. bdle with connection on U ,
(M ,∇) Deligne merom. extension of (V,∇) :merom. bdle on X with poles on S, ∇ reg. sing.
H∗DR(X,M ⊗ eg): de Rham cohomology of∇+ dg
Aspects of the Fourier-Laplace transform – p. 15/21
Fourier-Laplace and Hodge
Deligne (1984).
X compact Riemann surf., g : X −→ P1,S ⊃ g−1(∞), U = X r S
(V,∇) hol. vect. bdle with connection on U ,
(M ,∇) Deligne merom. extension of (V,∇) :merom. bdle on X with poles on S, ∇ reg. sing.
H∗DR(X,M ⊗ eg): de Rham cohomology of∇+ dg
=
∫ ∗
X→ptMeg.
Aspects of the Fourier-Laplace transform – p. 15/21
Fourier-Laplace and Hodge
Deligne (1984).
X compact Riemann surf., g : X −→ P1,S ⊃ g−1(∞), U = X r S
(V,∇) hol. vect. bdle with connection on U ,
(M ,∇) Deligne merom. extension of (V,∇) :merom. bdle on X with poles on S, ∇ reg. sing.
H∗DR(X,M ⊗ eg): de Rham cohomology of∇+ dg
=
∫ ∗
X→ptMeg.
THEOREM (Deligne):
Aspects of the Fourier-Laplace transform – p. 15/21
Fourier-Laplace and Hodge
Deligne (1984).
X compact Riemann surf., g : X −→ P1,S ⊃ g−1(∞), U = X r S
(V,∇) hol. vect. bdle with connection on U ,
(M ,∇) Deligne merom. extension of (V,∇) :merom. bdle on X with poles on S, ∇ reg. sing.
H∗DR(X,M ⊗ eg): de Rham cohomology of∇+ dg
=
∫ ∗
X→ptMeg.
THEOREM (Deligne): Assume mono.(V,∇) unitary .
Aspects of the Fourier-Laplace transform – p. 15/21
Fourier-Laplace and Hodge
Deligne (1984).
X compact Riemann surf., g : X −→ P1,S ⊃ g−1(∞), U = X r S
(V,∇) hol. vect. bdle with connection on U ,
(M ,∇) Deligne merom. extension of (V,∇) :merom. bdle on X with poles on S, ∇ reg. sing.
H∗DR(X,M ⊗ eg): de Rham cohomology of∇+ dg
=
∫ ∗
X→ptMeg.
THEOREM (Deligne): Assume mono.(V,∇) unitary .Can define canonically a “Hodge ” filtration F •
Del onH∗DR(X,M ⊗ eg),
Aspects of the Fourier-Laplace transform – p. 15/21
Fourier-Laplace and Hodge
Deligne (1984).
X compact Riemann surf., g : X −→ P1,S ⊃ g−1(∞), U = X r S
(V,∇) hol. vect. bdle with connection on U ,
(M ,∇) Deligne merom. extension of (V,∇) :merom. bdle on X with poles on S, ∇ reg. sing.
H∗DR(X,M ⊗ eg): de Rham cohomology of∇+ dg
=
∫ ∗
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.
Aspects of the Fourier-Laplace transform – p. 15/21
Fourier-Laplace and Hodge
Deligne (1984).
REMARKS:
Aspects of the Fourier-Laplace transform – p. 16/21
Fourier-Laplace and Hodge
Deligne (1984).
REMARKS:Without eg:mixed Hodge structure on H∗DR(X,M ).
Aspects of the Fourier-Laplace transform – p. 16/21
Fourier-Laplace and Hodge
Deligne (1984).
REMARKS:Without eg:mixed Hodge structure on H∗DR(X,M ).F •
Del indexed by (possibly) real numbers α
Aspects of the Fourier-Laplace transform – p. 16/21
Fourier-Laplace and Hodge
Deligne (1984).
REMARKS:Without eg:mixed Hodge structure on H∗DR(X,M ).F •
Del indexed by (possibly) real numbers α,exp 2πiα: eigenvalue of mono. of (V,∇) aroundg =∞.
Aspects of the Fourier-Laplace transform – p. 16/21
Fourier-Laplace and Hodge
Deligne (1984).
REMARKS:Without eg:mixed Hodge structure on H∗DR(X,M ).F •
Del indexed by (possibly) real numbers α,exp 2πiα: eigenvalue of mono. of (V,∇) aroundg =∞.
MOTIVATION/EXAMPLES:
Aspects of the Fourier-Laplace transform – p. 16/21
Fourier-Laplace and Hodge
Deligne (1984).
REMARKS:Without eg:mixed Hodge structure on H∗DR(X,M ).F •
Del indexed by (possibly) real numbers α,exp 2πiα: eigenvalue of mono. of (V,∇) aroundg =∞.
MOTIVATION/EXAMPLES:∫
R
e−x2
dx = π1/2 → F •
Del jumps at 1/2 only
for∫
A1→ptOA1e−x
2
.
Aspects of the Fourier-Laplace transform – p. 16/21
Fourier-Laplace and Hodge
Deligne (1984).
REMARKS:Without eg:mixed Hodge structure on H∗DR(X,M ).F •
Del indexed by (possibly) real numbers α,exp 2πiα: eigenvalue of mono. of (V,∇) aroundg =∞.
MOTIVATION/EXAMPLES:∫ ∞
0e−xxαdx/x = Γ(α) → F •
Del jumps at
1− α only for∫
A1→ptOA1xαe−x.
Aspects of the Fourier-Laplace transform – p. 16/21
Fourier-Laplace and Hodge
Deligne (1984).
REMARKS:Without eg:mixed Hodge structure on H∗DR(X,M ).F •
Del indexed by (possibly) real numbers α,exp 2πiα: eigenvalue of mono. of (V,∇) aroundg =∞.
MOTIVATION/EXAMPLES:∫ ∞
0e−xxαdx/x = Γ(α) → F •
Del jumps at
1− α only for∫
A1→ptOA1xαe−x.
Lamentation:Aspects of the Fourier-Laplace transform – p. 16/21
Fourier-Laplace and Hodge
Deligne (1984).
REMARKS:Without eg:mixed Hodge structure on H∗DR(X,M ).F •
Del indexed by (possibly) real numbers α,exp 2πiα: eigenvalue of mono. of (V,∇) aroundg =∞.
MOTIVATION/EXAMPLES:∫ ∞
0e−xxαdx/x = Γ(α) → F •
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
Aspects of the Fourier-Laplace transform – p. 17/21
Wild Hodge theory
Various origins/motivations:
Aspects of the Fourier-Laplace transform – p. 17/21
Wild Hodge theory
Various origins/motivations:Analogy with pure perverse `-adic sheaves,
Aspects of the Fourier-Laplace transform – p. 17/21
Wild Hodge theory
Various origins/motivations:Analogy with pure perverse `-adic sheaves,Hard Lefschetz theorem with irregular singularities(Kashiwara’s conjecture),
Aspects of the Fourier-Laplace transform – p. 17/21
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,
Aspects of the Fourier-Laplace transform – p. 17/21
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.
Aspects of the Fourier-Laplace transform – p. 17/21
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.
Starting point for the new point of view: work ofSimpson,
Aspects of the Fourier-Laplace transform – p. 17/21
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.
Starting point for the new point of view: work ofSimpson,
Results by Biquard-Boalch, Szabo,Hertling-Sevenheck, C.S.,
Aspects of the Fourier-Laplace transform – p. 17/21
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.
Starting point for the new point of view: work ofSimpson,
Results by Biquard-Boalch, Szabo,Hertling-Sevenheck, C.S.,
Main complete results obtained by Mochizuki.
Aspects of the Fourier-Laplace transform – p. 17/21
Fourier-Laplace and Hodge
Aspects of the Fourier-Laplace transform – p. 18/21
Fourier-Laplace and Hodge
X = P1, g = x, S 3 ∞, U = P1 r S.
Aspects of the Fourier-Laplace transform – p. 18/21
Fourier-Laplace and Hodge
X = P1, g = x, S 3 ∞, U = P1 r S.
(V,∇) underlies a polarized var. of Hodgestructure on U :
Aspects of the Fourier-Laplace transform – p. 18/21
Fourier-Laplace and Hodge
X = P1, g = x, S 3 ∞, U = P1 r S.
(V,∇) underlies a polarized var. of Hodgestructure on U :
e.g.f : Y → U ,
Aspects of the Fourier-Laplace transform – p. 18/21
Fourier-Laplace and Hodge
X = P1, g = x, S 3 ∞, U = P1 r S.
(V,∇) underlies a polarized var. of Hodgestructure on U :
e.g.f : Y → U , f projective and smooth over U ,
Aspects of the Fourier-Laplace transform – p. 18/21
Fourier-Laplace and Hodge
X = P1, g = x, S 3 ∞, U = P1 r S.
(V,∇) underlies a polarized var. of Hodgestructure on U :
e.g.f : Y → U , f projective and smooth over U ,
(V,∇) = Gauss-Manin connection of f
Aspects of the Fourier-Laplace transform – p. 18/21
Fourier-Laplace and Hodge
X = P1, g = x, S 3 ∞, U = P1 r S.
(V,∇) underlies a polarized var. of Hodgestructure on U :
e.g.f : Y → U , f projective and smooth over U ,
(V,∇) = Gauss-Manin connection of f :∫ ∗
fOU .
Aspects of the Fourier-Laplace transform – p. 18/21
Fourier-Laplace and Hodge
X = P1, g = x, S 3 ∞, U = P1 r S.
(V,∇) underlies a polarized var. of Hodgestructure on U :
e.g.f : Y → U , f projective and smooth over U ,
(V,∇) = Gauss-Manin connection of f :∫ ∗
fOU .
C∞ bdle C∞U ⊗OUV =
⊕p+q=wH
p,q,
Aspects of the Fourier-Laplace transform – p. 18/21
Fourier-Laplace and Hodge
X = P1, g = x, S 3 ∞, U = P1 r S.
(V,∇) underlies a polarized var. of Hodgestructure on U :
e.g.f : Y → U , f projective and smooth over U ,
(V,∇) = Gauss-Manin connection of f :∫ ∗
fOU .
C∞ bdle C∞U ⊗OUV =
⊕p+q=wH
p,q,
Hodge holomorphic sub-bdlesF p =
⊕p′>pH
p′,w−p′
,
Aspects of the Fourier-Laplace transform – p. 18/21
Fourier-Laplace and Hodge
X = P1, g = x, S 3 ∞, U = P1 r S.
(V,∇) underlies a polarized var. of Hodgestructure on U :
e.g.f : Y → U , f projective and smooth over U ,
(V,∇) = Gauss-Manin connection of f :∫ ∗
fOU .
C∞ bdle C∞U ⊗OUV =
⊕p+q=wH
p,q,
Hodge holomorphic sub-bdlesF p =
⊕p′>pH
p′,w−p′
,
∇ : F p −→ F p−1 ⊗ Ω1U ,
Aspects of the Fourier-Laplace transform – p. 18/21
Fourier-Laplace and Hodge
X = P1, g = x, S 3 ∞, U = P1 r S.
(V,∇) underlies a polarized var. of Hodgestructure on U :
e.g.f : Y → U , f projective and smooth over U ,
(V,∇) = Gauss-Manin connection of f :∫ ∗
fOU .
C∞ bdle C∞U ⊗OUV =
⊕p+q=wH
p,q,
Hodge holomorphic sub-bdlesF p =
⊕p′>pH
p′,w−p′
,
∇ : F p −→ F p−1 ⊗ Ω1U ,
polarization .
Aspects of the Fourier-Laplace transform – p. 18/21
Fourier-Laplace and Hodge
Definition of F •
Del(M ⊗ ex).
Aspects of the Fourier-Laplace transform – p. 19/21
Fourier-Laplace and Hodge
Definition of F •
Del(M ⊗ ex).
Schmid (1973):
Aspects of the Fourier-Laplace transform – p. 19/21
Fourier-Laplace and Hodge
Definition of F •
Del(M ⊗ ex).
Schmid (1973): → F •(M ) (on P1): holomorphicsub-bdles.
Aspects of the Fourier-Laplace transform – p. 19/21
Fourier-Laplace and Hodge
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:
Aspects of the Fourier-Laplace transform – p. 19/21
Fourier-Laplace and Hodge
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:Res∞∇ on V γ
y (M ) has eigenval. in [γ, γ + 1).
Aspects of the Fourier-Laplace transform – p. 19/21
Fourier-Laplace and Hodge
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:Res∞∇ on V γ
y (M ) has eigenval. in [γ, γ + 1).
Def. at finite distance :
Aspects of the Fourier-Laplace transform – p. 19/21
Fourier-Laplace and Hodge
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:Res∞∇ on V γ
y (M ) has eigenval. in [γ, γ + 1).
Def. at finite distance : F •
Del(M ⊗ ex) :=F •(M ),
Aspects of the Fourier-Laplace transform – p. 19/21
Fourier-Laplace and Hodge
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:Res∞∇ on V γ
y (M ) has eigenval. in [γ, γ + 1).
Def. at finite distance : F •
Del(M ⊗ ex) :=F •(M ),
Def. at infinity :
Aspects of the Fourier-Laplace transform – p. 19/21
Fourier-Laplace and Hodge
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:Res∞∇ on V γ
y (M ) has eigenval. in [γ, γ + 1).
Def. at finite distance : F •
Del(M ⊗ ex) :=F •(M ),
Def. at infinity :
FαDel(M⊗ex) :=
Aspects of the Fourier-Laplace transform – p. 19/21
Fourier-Laplace and Hodge
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:Res∞∇ on V γ
y (M ) has eigenval. in [γ, γ + 1).
Def. at finite distance : F •
Del(M ⊗ ex) :=F •(M ),
Def. at infinity :
FαDel(M⊗ex) := F dαe+k
Aspects of the Fourier-Laplace transform – p. 19/21
Fourier-Laplace and Hodge
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:Res∞∇ on V γ
y (M ) has eigenval. in [γ, γ + 1).
Def. at finite distance : F •
Del(M ⊗ ex) :=F •(M ),
Def. at infinity :
FαDel(M⊗ex) := F dαe+k∩V α−dαe
y
Aspects of the Fourier-Laplace transform – p. 19/21
Fourier-Laplace and Hodge
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:Res∞∇ on V γ
y (M ) has eigenval. in [γ, γ + 1).
Def. at finite distance : F •
Del(M ⊗ ex) :=F •(M ),
Def. at infinity :
FαDel(M⊗ex) :=
(F dαe+k∩V α−dαe
y ⊗e1/y)
Aspects of the Fourier-Laplace transform – p. 19/21
Fourier-Laplace and Hodge
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:Res∞∇ on V γ
y (M ) has eigenval. in [γ, γ + 1).
Def. at finite distance : F •
Del(M ⊗ ex) :=F •(M ),
Def. at infinity :
FαDel(M⊗ex) := ∂kyy
−1(F dαe+k∩V α−dαe
y ⊗e1/y)
Aspects of the Fourier-Laplace transform – p. 19/21
Fourier-Laplace and Hodge
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:Res∞∇ on V γ
y (M ) has eigenval. in [γ, γ + 1).
Def. at finite distance : F •
Del(M ⊗ ex) :=F •(M ),
Def. at infinity :
FαDel(M⊗ex) :=
∑
k∈N
∂kyy−1(F dαe+k∩V α−dαe
y ⊗e1/y)
Aspects of the Fourier-Laplace transform – p. 19/21
Fourier-Laplace and Hodge
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:Res∞∇ on V γ
y (M ) has eigenval. in [γ, γ + 1).
Def. at finite distance : F •
Del(M ⊗ ex) :=F •(M ),
Def. at infinity :
FαDel(M⊗ex) :=
∑
k∈N
∂kyy−1(F dαe+k∩V α−dαe
y ⊗e1/y)
Filtered de Rham complex FαDel DR(M⊗ex):
Aspects of the Fourier-Laplace transform – p. 19/21
Fourier-Laplace and Hodge
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:Res∞∇ on V γ
y (M ) has eigenval. in [γ, γ + 1).
Def. at finite distance : F •
Del(M ⊗ ex) :=F •(M ),
Def. at infinity :
FαDel(M⊗ex) :=
∑
k∈N
∂kyy−1(F dαe+k∩V α−dαe
y ⊗e1/y)
Filtered de Rham complex FαDel DR(M⊗ex):
0→ FαDel(M⊗ex)
∇−→ Ω1
P1 ⊗ Fα−1Del (M⊗ex)→ 0
Aspects of the Fourier-Laplace transform – p. 19/21
Fourier-Laplace and Hodge
Taking hypercohomology→ FαDelH∗DR(X,M ⊗ ex).
Aspects of the Fourier-Laplace transform – p. 20/21
Fourier-Laplace and Hodge
Taking hypercohomology→ FαDelH∗DR(X,M ⊗ ex).
THEOREM:
Aspects of the Fourier-Laplace transform – p. 20/21
Fourier-Laplace and Hodge
Taking hypercohomology→ FαDelH∗DR(X,M ⊗ ex).
THEOREM: The Hodge⇒ de Rham spectral seq.degenerates at E1.
Aspects of the Fourier-Laplace transform – p. 20/21
Fourier-Laplace and Hodge
Taking hypercohomology→ FαDelH∗DR(X,M ⊗ ex).
THEOREM: The Hodge⇒ de Rham spectral seq.degenerates at E1.
By rescaling x 7−→ −x/t, t ∈ C∗,
Aspects of the Fourier-Laplace transform – p. 20/21
Fourier-Laplace and Hodge
Taking hypercohomology→ FαDelH∗DR(X,M ⊗ ex).
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
Aspects of the Fourier-Laplace transform – p. 20/21
Fourier-Laplace and Hodge
Taking hypercohomology→ FαDelH∗DR(X,M ⊗ ex).
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
→FαDelM ,
Aspects of the Fourier-Laplace transform – p. 20/21
Fourier-Laplace and Hodge
Taking hypercohomology→ FαDelH∗DR(X,M ⊗ ex).
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
→FαDelM , ∇ : FαDelM −→ Ω1 ⊗ Fα−1Del M ,
Aspects of the Fourier-Laplace transform – p. 20/21
Fourier-Laplace and Hodge
Taking hypercohomology→ FαDelH∗DR(X,M ⊗ ex).
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
→FαDelM , ∇ : FαDelM −→ Ω1 ⊗ Fα−1Del M ,
But no corresp. Hodge decomp. (lamentation).
Aspects of the Fourier-Laplace transform – p. 20/21
Fourier-Laplace and Hodge
Taking hypercohomology→ FαDelH∗DR(X,M ⊗ ex).
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
→FαDelM , ∇ : FαDelM −→ Ω1 ⊗ Fα−1Del M ,
But no corresp. Hodge decomp. (lamentation).
On the other hand, wild Hodge Theory →
Aspects of the Fourier-Laplace transform – p. 20/21
Fourier-Laplace and Hodge
Taking hypercohomology→ FαDelH∗DR(X,M ⊗ ex).
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
→FαDelM , ∇ : FαDelM −→ Ω1 ⊗ Fα−1Del M ,
But no corresp. Hodge decomp. (lamentation).
On the other hand, wild Hodge Theory → Hodgedecomp. for each H∗DR(X,M ⊗ e−x/t),
Aspects of the Fourier-Laplace transform – p. 20/21
Fourier-Laplace and Hodge
Taking hypercohomology→ FαDelH∗DR(X,M ⊗ ex).
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
→FαDelM , ∇ : FαDelM −→ Ω1 ⊗ Fα−1Del M ,
But no corresp. Hodge decomp. (lamentation).
On the other hand, wild Hodge Theory → Hodgedecomp. for each H∗DR(X,M ⊗ e−x/t), but noHodge bundles F •.
Aspects of the Fourier-Laplace transform – p. 20/21
Fourier-Laplace and Hodge
Taking hypercohomology→ FαDelH∗DR(X,M ⊗ ex).
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
→FαDelM , ∇ : FαDelM −→ Ω1 ⊗ Fα−1Del M ,
But no corresp. Hodge decomp. (lamentation).
On the other hand, wild Hodge Theory → Hodgedecomp. for each H∗DR(X,M ⊗ e−x/t), but noHodge bundles F •.Why?
Aspects of the Fourier-Laplace transform – p. 20/21
Fourier-Laplace and Hodge
Taking hypercohomology→ FαDelH∗DR(X,M ⊗ ex).
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
→FαDelM , ∇ : FαDelM −→ Ω1 ⊗ Fα−1Del M ,
But no corresp. Hodge decomp. (lamentation).
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.
Aspects of the Fourier-Laplace transform – p. 20/21
Fourier-Laplace and Hodge
THEOREM:
Aspects of the Fourier-Laplace transform – p. 21/21
Fourier-Laplace and Hodge
THEOREM: Asymptotic vanishing of the lamentationwhen t −→∞.
Aspects of the Fourier-Laplace transform – p. 21/21
Fourier-Laplace and Hodge
THEOREM: Asymptotic vanishing of the lamentationwhen t −→∞.
0 ∞ t
Aspects of the Fourier-Laplace transform – p. 21/21
Fourier-Laplace and Hodge
THEOREM: Asymptotic vanishing of the lamentationwhen t −→∞.
0 ∞
Jump of FDel
t
Aspects of the Fourier-Laplace transform – p. 21/21
Fourier-Laplace and Hodge
THEOREM: Asymptotic vanishing of the lamentationwhen t −→∞.
0 ∞
Jump of FDel
Jump of Hodge decomp.(“new supersymm.
index” Cecotti-Vafa)
t
Aspects of the Fourier-Laplace transform – p. 21/21