Riemann-Hilbert correspondence for irregular holonomic D -modules - - PowerPoint PPT Presentation

Riemann-Hilbert correspondence for irregular holonomic D-modules

(joint work with Masaki KASHIWARA)

Andrea D’AGNOLO

Universit` a di Padova – Italy

Winter School on Higher Structures in Algebraic Analysis Padova, 18 February 2014

• A. D’Agnolo (Padova)

Riemann-Hilbert correspondence Padova, 18 February 2014 1 / 12

The classical RH problem

Hilbert’s 21st problem (1900) “A problem that Riemann himself may have in mind” “To show that there always exists a linear differential equation

• f the Fuchsian class, with given singular points and

monodromic group”

• A. D’Agnolo (Padova)

Riemann-Hilbert correspondence Padova, 18 February 2014 2 / 12

Fuchsian ODEs

P(z, ∂z) = am(z)∂m

z + · · · + a1(z)∂z + a0(z),

aj ∈ OC

Definition

z0 ∈ C Fuchsian singularity: am(z0) = 0,

• rdz=z0 am − m ≤ ordz=z0 aj − j

∀j Basis of m local solutions at z0 of the form: u(z) = (z − z0)λv(z) + (log terms), λ ∈ C, v ∈ OC,z0 λ monodromy

Corollary

{u ∈ OC ; Pu = 0} is a local system outside of the singular points

• A. D’Agnolo (Padova)

Riemann-Hilbert correspondence Padova, 18 February 2014 3 / 12

D-modules

X: complex manifold DX: sheaf of linear differential operators

Definition

M a DX-module Sol(M) = RHom DX (M, OX)

Example

P ∈ DX M = DX/DXP H0Sol(M) = {u ∈ OX ; Pu = 0} holonomic DX-module ODE regular holonomic DX-module Fuchsian ODE C-constructible sheaf local system

• A. D’Agnolo (Padova)

Riemann-Hilbert correspondence Padova, 18 February 2014 4 / 12

Regular RH correspondence

Db(DX): bounded derived category of DX-modules Db(CX): bounded derived category of sheaves of C-vector spaces

Theorem (Kashiwara 1984)

Db(DX)op

Sol

Db(CX)

Db

hol(DX)op

• Db

rh(DX)op ∼

• Db

C-c(CX)

• analysis

topology There is also an explicit reconstruction functor: Db

rh(DX) ∋ M F = Sol(M) ∈ Db C-c(CX)

RHom (F, O t

X) ≃ M

• A. D’Agnolo (Padova)

Riemann-Hilbert correspondence Padova, 18 February 2014 5 / 12

Subanalytic sheaves [Kashiwara-Schapira 2001]

Xsa the subanalytic site:

◮ open subanalytic subsets of X ◮ locally finite covers

Mod(CXsa) subanalytic sheaves ( ind-sheaves) Tempered distributions: Dbt(U) = image

• DbX(X) −

→ DbX(U)

• Ot

X = Dolbeault complex with coefficients in Dbt X

Sol t(M) = RHom DX (M, Ot

X)

Example

E1/z

C

= DCe1/z = DC/DCP, P(z, ∂z) = z2∂z − 1 not Fuchsian H0Sol t(E1/z

C

) = “lim − →”

c−

→+∞ C{Re(1/z)<c} Caveat: Sol t(E1/z

C

) ≃ Sol t(E2/z

C

)

• A. D’Agnolo (Padova)

Riemann-Hilbert correspondence Padova, 18 February 2014 6 / 12

Irregular ODEs

P(z, ∂z) = am(z)∂m

z + · · · + a0(z),

z0 ∈ C not Fuchsian For w = (z − z0)1/r, basis of m formal solutions:

• u(w) = eϕ(w)wλ

v(w)+(log terms), ϕ ∈ C[w−1], λ ∈ C, v ∈ OC,z0 ∀ direction θ, ∃ analytic solution u with u ∼ u on a sector S Caveat: u + u1 ∼ u if Re ϕ1 < Re ϕ at θ Stokes phenomenon

Theorem ([Deligne] and [Malgrange] in the 80s)

Irregular RH in dimension one, for fixed singular locus Idea: order as above the exponents, so that u is well defined in the graded part Caveat: difficult to extend in higher dimensions, cf [Sabbah 2013]

• A. D’Agnolo (Padova)

Riemann-Hilbert correspondence Padova, 18 February 2014 7 / 12

Enhanced sheaves

X × R∞ = bordered subanalytic site

◮ open subanalytic subsets of X × P1(R) included in X × R ◮ locally finite covers

Definition (influenced by [Tamarkin 2008])

Eb(CX) = Db(CX×R∞)/{K : K ≃ π−1Rπ∗K} π: X × R∞ − → Xsa It is a commutative tensor category K1

+

⊗ K2 = Rµ!!(q−1

1 K1 ⊗ q−1 2 K2)

convolution unit: C{t=0} = CX×{0} Six operations:

+

⊗, Ihom+, Rf∗, Rf!!, f −1, f !

Lemma

Db(CX) ֒ → Eb(CX) F → CE

X ⊗ π−1F,

CE

X = “lim

− →”

c−

→+∞ C{t≥c}

• A. D’Agnolo (Padova)

Riemann-Hilbert correspondence Padova, 18 February 2014 8 / 12

Reconstruction of exponential modules

DbE

X =

• Dbt

X×R∞ ∂t−1

− − − → Dbt

X×R∞

• = Hom DR∞(Et

R∞, Dbt X×R∞)[1]

OE

X = (Dolbeault complex with coefficients in DbE X)[1]

Sol E(M) = RHom DX (M, OE

X) ∼ Sol t(M ⊠ Et R∞)[1]

Theorem

Sol E(Eϕ

X ) ≃ “lim

− →”

c−

→+∞ C{t+Re ϕ(z)≥c} Eϕ

X = DXeϕ(∗D), ϕ ∈ OX(∗D)

Generalizes the example of E1/z

C

Theorem

Eϕ

X ≃ RHom ( “lim

− →”

c−

→+∞ C{t+Re ϕ(z)≥c}, OE

X),

RHom ∼ Rπ∗Ihom+ Related to [D’A 2013]

• A. D’Agnolo (Padova)

Riemann-Hilbert correspondence Padova, 18 February 2014 9 / 12

Structure of holonomic modules

Key result from [Mochizuki 2011] and [Kedlaya 2011]

Lemma

A statement QX(M) is true for any M ∈ Db

hol(DX) and any X if:

QX(M) ⇐ ⇒ QUi(M|Ui) ∀i ∈ I, for X =

i∈I Ui an open cover.

QX(M) = ⇒ QX(M[n]) ∀n ∈ Z. QX(M′)&QX(M′′) = ⇒ QX(M), for M′ − → M − → M′′ +1 − − → a d.t. QX(M ⊕ M′) = ⇒ QX(M). QX(M) = ⇒ QY(Df ∗M), for f : X − → Y projective. QX(M) holds for M with a normal form along a n.c. divisor. M has normal form if it is a direct sum of exponential D-modules

• n polysectors along the divisor.
• A. D’Agnolo (Padova)

Riemann-Hilbert correspondence Padova, 18 February 2014 10 / 12

RH correspondence

Theorem

Sol E : Db

hol(DX)op −

→ Eb(CX) is fully faithful Sol E(M) is R-constructible Reconstruction holds: Db

hol(DX) ∋ M K = Sol E(M) ∈ Eb R-c(CX)

RHom (K, OE

X) ≃ M

Compatibility with the regular case: Db

hol(DX)op Sol E

Eb

R-c(CX) RHom (∗,OE

X )

Db(DX)op

Db

rh(DX)op Sol t ∼

• Db

C-c(CX)

• RHom (∗,O t

X )

∼

Db

rh(DX)op

• A. D’Agnolo (Padova)

Riemann-Hilbert correspondence Padova, 18 February 2014 11 / 12

Stokes phenomenon

ϕ1, ϕ2 ∈ OC(∗0) Kj = Sol E(E

ϕj C ) ≃ “lim

− →”

c−

→+∞ C{t+Re ϕj(z)≥c} M flat meromorphic connection ∀θ: M ∼θ Eϕ1

C ⊕ Eϕ2 C K = Sol E(M) ∼θ K1 ⊕ K2

{Re(ϕ1 − ϕ2) = 0} = ⊔ℓn Ln Stokes lines

Lemma

S an open sector EndEb(CC)(π−1CS ⊗ (K1 ⊕ K2)) ≃

• b±,

S ⊂ {± Re(ϕ1 − ϕ2) > 0} t, S ⊃ Ln0, S ∩ Ln = ∅, n = n0 (b± upper/lower triangular in M2(C), t = b+ ∩ b−)

• A. D’Agnolo (Padova)

Riemann-Hilbert correspondence Padova, 18 February 2014 12 / 12