Introduction to twistor D-modules (II)ramadas/NS@50/Sabbah2.pdf · Introduction to twistor D-modules ... Deﬁne a pure twistor D-moduleas an object ... (M. Saito, mixed Hodge module

Introduction to twistor D-modules(II)

Claude Sabbah

Centre de Mathematiques Laurent Schwartz

UMR 7640 du CNRS

Ecole polytechnique, Palaiseau, France

Introduction to twistor D-modules (II) – p. 1/20

Conjecture of Kashiwara (weak form)

X projective, c = c1 of ample line bundle,

V semi-simple local system on Xo ⊂ X smoothquasi-projective,

then HLT holds for IH∗(X, V ):

∀k > 1, Lkc : IHn−k(X, V )∼

−→ IHn+k(X, V )

Moreover, the Decomposition Theorem holds forany projective morphism f : X −→ Y , i.e.,

Rf∗ ICX(V ) ≃⊕

i,j

ICYi(Vi,j)[j]

with Yi ⊂ Y closed irred., Vi,j semi-simple on Y oi

smooth Zariski open in Yi.



Example of the decomposition theorem (Simpson):

X,Y smooth projective, f : X −→ Y smooth.

V semi-simple local system on X,

then

each local system Rjf∗V is semi-simple,

the relative HLT holds:

∀k > 1, Lkc : Rn−kf∗V

∼−→ Rn+kf∗V

Deligne (1968) =⇒ the Decomposition Theoremholds, i.e.,

Rf∗V ≃⊕

j

Rjf∗V [−j]



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




Semi-simple perverse

sheaf on X





sheaf on Y


sheaf on X

Conjecture of Kashiwara

regular case



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


sheaf on Y


sheaf on X





sheaf on Y

Polarized regular twistor

D-module on X


sheaf on X





D-module on Y


sheaf on Y

decomposition

theorem (C.S.)


D-module on X


sheaf on X





D-module on Y


sheaf on Y

decomposition

theorem (C.S.)

Simpson

+ Hamm-Le D.T.


D-module on X


sheaf on X





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 on X

Semi-simple local

system on X




Po la ri ze d regular twisto r

D-modul e o n Y

Semi-simple per ve rs e

sheaf on Y

decompositio n

theorem (C .S .)

Simpso n

+ Hamm- L e D .T .


D-modul e o n X T. Mochizuk i


sheaf on X





D-modul e o n Y


sheaf on Y

decompositio n

theorem (C .S .)

Simpso n

+ Hamm- L e D .T .




sheaf on X

Conjecture of Kashiw ar a

regular case


λ-Differential operators

DX , loc. ≃ OX〈∂x1, . . . , ∂xn〉,

ր filtr. FpDX : ∂-ord. 6 p, grFDX : gr. comm. ring.

Rees ring (graded ring)

RFDX :=⊕

p λpFpDX

loc.≃ OX [λ]〈ðx1, . . . , ðxn

〉

(ðxi:= λ∂xi

)

DX =RFDX/(λ− 1)RFDX ,

grFDX =RFDX/λRFDX



DX , loc. ≃ OX〈∂x1, . . . , ∂xn〉,

ր filtr. FpDX : ∂-ord. 6 p, grFDX : gr. comm. ring.

Rees ring (graded ring)

RFDX :=⊕

p λpFpDX

loc.≃ OX [λ]〈ðx1, . . . , ðxn

〉

(ðxi:= λ∂xi

)

DX =RFDX/(λ− 1)RFDX ,

grFDX =RFDX/λRFDX

X = X × Cλ, RX = OX ⊗OX [λ] RFDX (gr. lost)

DX = RX /(λ− 1)RX ,

OT ∗X = RX /λRX (grading lost)



Standard objects

Flat holom. bdle ⇒ OX -coh. left DX -mod.

Holom. Higgs bdle ⇒ OX -coh. OT ∗X -mod.

Hol. bdle with flat λ-conn. ⇒ OX -coh. left RX -mod.



Standard objects

Flat holom. bdle ⇒ OX -coh. left DX -mod.

Holom. Higgs bdle ⇒ OX -coh. OT ∗X -mod.

Hol. bdle with flat λ-conn. ⇒ OX -coh. left RX -mod.

Singular objects

Holonomic DX -module M : DX -coh. + its char. var.is Lagrangean (and conic) in T ∗X.

Holonomic OT ∗X -module: OT ∗X -coh. + its supp. isLagrangean (maybe not conic) in T ∗X.

Holonomic RX -module: RX -coh. + its char. var. is⊂ Λ × C in (T ∗X) × C, Λ conic Lagrangean in T ∗X.

Strict RX -mod.: OC-flat, i.e., no OC-torsion.



THEOREM (Gabber): M = strict holonomic RX -mod.Then

1. ∀λo 6= 0, Mλo:= M/(λ− λo)M is DX -holonomic.

2. M0 := M/λM is a holonomic Higgs sheaf.

Singular objects

Holonomic DX -module M : DX -coh. + its char. var.is Lagrangean (and conic) in T ∗X.

Holonomic OT ∗X -module: OT ∗X -coh. + its supp. isLagrangean (maybe not conic) in T ∗X.

Holonomic RX -module: RX -coh. + its char. var. is⊂ Λ × C in (T ∗X) × C, Λ conic Lagrangean in T ∗X.

Strict RX -mod.: OC-flat, i.e., no OC-torsion.


Variation of twistor structure(C. Simpson)

X: complex manifold, X := X × P1.

Twistor conjugation: ordinary conjugation on X andtwistor conjugation on P1.

H : C∞ vect. bdle on X, holom. w.r.t. P1,

Relative connections D′,D′′:

D′ : H −→ Ω1

X/P1(1 · λ=0) ⊗ H ,

D′′ : H −→ σ∗Ω1

X/P1(1 · λ=∞) ⊗ H ,

Flatness: D2 = (D′ + D

′′)2 = 0.

On X = X × C, (H ′ := ker D′′, λD

′): holonomicRX -mod.


Variation of twistor structure

Two issues to extend the notion to singular objects:

Should not work with C∞ sheaves (pbs withcoherence)

Should allow more complicated objects than bundleson X × P1.

Solution:

Express H as the result of the gluing of H ′ (on X )and σ∗H ′′ with H ′′ holom. bdle on X .

To make the gluing degenerate, consider it as anondegenerate pairing

H′

|X×C∗ ⊗ σ∗H ′′∨

|X×C∗ −→ C∞,anX×C∗


R-triples T

DEFINITION: Object of R- Triples(X) are triplesT = (M ′,M ′′, C) s.t.

M ′,M ′′ holonomic RX -mod.

C : M ′|X×C∗ ⊗ σ∗M ′′

|X×C∗ −→ DbX×C∗/C∗

is R ⊗ σ∗R-linear: for P loc. sect. of R,C(Pm′, σ∗m′′) = PC(m′, σ∗m′′)

C(m′, σ∗Pm′′) = σ∗PC(m′, σ∗m′′)


R-triples T

DEFINITION: Object of R- Triples(X) are triplesT = (M ′,M ′′, C) s.t.

M ′,M ′′ holonomic RX -mod.

C : M ′|X×C∗ ⊗ σ∗M ′′

|X×C∗ −→ DbX×C∗/C∗

is R ⊗ σ∗R-linear: for P loc. sect. of R,C(Pm′, σ∗m′′) = PC(m′, σ∗m′′)

C(m′, σ∗Pm′′) = σ∗PC(m′, σ∗m′′)

Morphism ϕ : T1 → T2:

ϕ′ : M ′2 → M ′

1

ϕ′′ : M ′′1 → M ′′

2

are R-linear

andC1(ϕ

′(m′2), σ

∗m′′1 ) = C2(m

′2σ

∗ϕ′′(m′′1 ))


R-triples T

Tate twist: For k ∈ 12Z,

T (k) := (M ′,M ′′, (iλ)−2kC).

Adjunction: T ∗ := (M ′′,M ′, C∗) with

C∗(m′′, σ∗m′) := σ∗C(m′, σ∗m′′)

w-Hermitian duality: S : T∼

−→ T ∗(−w) s.t.S ∗ = (−1)wS .

Proper push-forward: f : X → Y proper. Can define

f j† T := Hjf†T (j ∈ Z).

e.g., f : X → pt, fj† M ′ = Hn+j(X,DR M ′).

Lefschetz morph.: c ∈ H2(X,C) real (1, 1)-class.Can define Lc : f

j† T −→ f

j+2† T .


Specialization ofR-triples T

Idea: Define a pure twistor D-module as an objectof R- Triples(X) which gives rise to a pure twistorstructure after restriction to each x ∈ X.

Problem: How to define the restriction to a point?

Solution (M. Saito, mixed Hodge module theory):Iterate the nearby cycle functor of Deligne.

Need to adapt the nearby cycle functor ofKashiwara-Malgrange for holonomic D-modules.

analogue of a parabolic filtration along any germof holom. fnct. on X.



Fix a local coordinate syst. (t, x2, . . . , xn).X0 := t = 0, M : hol. RX -mod.

DEFINITION: M is specializable along t = 0 if each loc.sect. m satisfies an eqn.

bm(tðt)m = tP (t, x, tðt, ðx)m

with bm(s) ∈ C[λ, s]:

bm(s) =∏

u=(a,α)∈R×C

(

s+ (α− aλ− αλ2))νu .

∀λo ∈ C, ր Filtr. V (λo)•

by the parab. order at λo:par-ordλo

(m) = maxνu 6=0(a+ 2 Re(αλo)).



DEFINITION: M is strictly specializable along t = 0 it isspecializable and

∀λo ∈ C, ∀b ∈ R, V(λo)b M/V

(λo)<b M is strict , i.e. OC-flat.

∀λo ∈ C, b ∈ R, action of tðt on grV(λo)

b M withminimal polyn. in C[λ, s].

ψ(λo)t,u M := gener. eigen submod. of

⊕

b grV(λo)

b M

independent of λo.

ψt,uM with nilp . operator N induced by(tðt + (α− aλ− αλ2)), u = (a, α).

“Monodromy filtr.” M•ψt,u and, for each ℓ ∈ N,primitive part P grMℓ ψt,uM : coh. RX0

-mod.


Pure twistor D-module (reg. case)INDUCTIVE DEFINITION W.R.T. SUPPORT:T = (M ′,M ′′, C) is a pure twistor D-mod. of wght wand support of dim. 6 d if

M ′,M ′′ strictly specializable along any holom. germf : X → C, and have support of dim. 6 d,

∀f , ∀u ∈ R × C, ∀ℓ ∈ N, P grMℓ ψt,uT is a puretwistor D-mod. of weight w + ℓ and dim. supp.6 d− 1,

If d = 0, pure twistor D-mod. of weight w ⇐⇒ puretwistor structure of weight w.


Pure twistor D-module (reg. case)INDUCTIVE DEFINITION W.R.T. SUPPORT:T = (M ′,M ′′, C) is a pure twistor D-mod. of wght wand support of dim. 6 d if

M ′,M ′′ strictly specializable along any holom. germf : X → C, and have support of dim. 6 d,

∀f , ∀u ∈ R × C, ∀ℓ ∈ N, P grMℓ ψt,uT is a puretwistor D-mod. of weight w + ℓ and dim. supp.6 d− 1,

If d = 0, pure twistor D-mod. of weight w ⇐⇒ puretwistor structure of weight w.

S : T∼

−→ T ∗(−w) is a polarization if

∀f , ∀u ∈ R × C, ∀ℓ ∈ N, P grMℓ ψt,uS is a pol. ofP grMℓ ψt,uT ,

If d = 0, S is a pol. of the pure twistor structure.Introduction to twistor D-modules (II) – p. 15/20




sheaf on Y


D-module on X


sheaf on X





D-module on Y


sheaf on Y

decomposition

theorem (C.S.)


D-module on X


sheaf on X





D-module on Y


sheaf on Y

decomposition

theorem (C.S.)

Simpson

+ Hamm-Le D.T.


D-module on X


sheaf on X





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 on X

Semi-simple local

system on X





D-modul e o n Y


sheaf on Y

decompositio n

theorem (C .S .)

Simpso n

+ Hamm- L e D .T .




sheaf on X


Tame harmonic flat bundles

Z: smooth cplx mfld, X ⊂ Z irred. closed analyticsubset, Xo smooth Zariski open subset of X.

(H,D) C∞ flat bdle on Xo, D = D′ +D′′,(V = kerD′′,∇ = D′|V ) holom. flat bdle,




(H,D) C∞ flat bdle on Xo, D = D′ +D′′,(V = kerD′′,∇ = D′|V ) holom. flat bdle,h = a metric: ∃! (Dh,Θ) s.t.Dh comp. with h,

Θ = θ + θ∗, θ := Θ(1,0), θ∗ = h-adjoint of θ (and= Θ(0,1)),D′ +D′′ = Dh + Θ.

DEFINITION: (H,D, h) harmonic if (D′′h + θ)2 = 0

( ⇐⇒ (D′′h)

2 = 0, D′′h(θ) = 0, θ ∧ θ = 0).

⇒ E = kerD′′h , θ = holom. Higgs field.




(H,D) C∞ flat bdle on Xo, D = D′ +D′′,(V = kerD′′,∇ = D′|V ) holom. flat bdle,h = (pluri)harmonic metric.

(V,∇, h) is tame if ∀xo ∈ X rXo, ∃π = proj.

modif. ˜nb(xo, X) −→ nb(xo, X) s.t.π−1(X rXo) = ncd and in loc. coord. w.r.t. the ncd,

θ =m∑

i=1

θidxi

xi+

n∑

j=m+1

θjdxj ,

with char. pol. θi, θj in O ˜nb(xo,X)[T ]


Comparison theorem 1

THEOREM (MOCHIZUKI 2007): (T ,S ) pure polarizedreg. twistor D-mod. of weight 0 supp. on X closedanalytic subset in Z7→ (T ,S )|Xo smooth object (= var. pol. twistor struct.on Xo, = harmonic flat bdle on Xo)induces an equivalence with tame harmonic flat bdleson Xo.


Comparison theorem 1“Proof” of ⇐:

Choose π : Y −→ X projective modif. s.t.D := π−1(X rX0) is a ncd.

tame harmonic bdle on (Y,D).

Asymptotic analysis of the metric ⇒ the extension byfixing the growth of sections gives a coherentmulti-parabolic filtr.

Analyze the nearby cycles of the extended objectwith respect to any function and prove that it satisfiesthe requirements for a pure pol. twistor D-module.→ For this, reduce to the case where the function isa monomial by resol. sing. of the fnct. and then usethe push-forward (decomp. thm already proved).

Use the decom. thm for π to get the corresp. twistorD-module.



THEOREM (SIMPSON 1990, BIQUARD 1997, JOST-ZUO1998, MOCHIZUKI 2007):Z cplx proj. mfld., X closed analytic subset in Z, thenthe correspondence

(V,∇, h) purely imaginary tame harm. bdle on smoothZar. open Xo ⊂ X

7−→ V ∇|Xo local syst. on Xo

induces an equiv. with semi-simple loc. syst. on Xo

( ⇐⇒ semi-simple perverse sheaf supported on Z,smooth on Xo).



THEOREM (SIMPSON 1990, BIQUARD 1997, JOST-ZUO1998, MOCHIZUKI 2007):Z cplx proj. mfld., X closed analytic subset in Z, thenthe correspondence

(V,∇, h) purely imaginary tame harm. bdle on smoothZar. open Xo ⊂ X

7−→ V ∇|Xo local syst. on Xo

induces an equiv. with semi-simple loc. syst. on Xo

( ⇐⇒ semi-simple perverse sheaf supported on Z,smooth on Xo).

Not known: How to characterize Higgs sheavesassociated to pure twistor D-modules?


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Introduction to twistor D-modules (II)ramadas/NS@50/Sabbah2.pdf · Introduction to twistor D-modules ... Deﬁne a pure twistor D-moduleas an object ... (M. Saito, mixed Hodge module