数理解析研究所講究録
第 845 巻 1993 年 1-7

1

A MICROLOCAL VERSION OF THE RIEMANN-HILBERT CORRESPONDANCE

Emmanuel Andronikof \dag er(univ. Paris 13)

1. –Introduction
Let $X$ be a complex n-dimensional manifold. Recall that the “Riemann-Hilbert correspondance” consistsof the two following commutative diagrams, together with the assertion that all the arrows are equivalencesof categories :

(1.1)

(1.2)

We make use of the following notations :is the derived category of bounded complexes of sheaves of C-vector spaces on $X$ withC-constructible cohomology.
$D_{C-c}^{b}(X)$Regho1 $(D_{X})$ is the abelian category of regular holonomic (left)$7to1(D_{X}^{\infty})$is the category of modules of the form

$D_{X}^{\infty} \bigotimes_{D_{X}}\mathcal{M}$$D_{X}$

where-modules.

$\Lambda 4$

is a holonomic$\mathcal{D}$-module.

-modules with regular holonomic
is the derived category of bounded complexes of-modules (in theis the derived catgeory of bounded complexes of admissible
cohomology.
.
sense of [S-K-K]) with cohomology in$D_{r-h}^{b}(D_{X})$$\mathcal{D}_{X}$$D_{h}^{b}(\prime D_{X}^{\infty})$$D_{X}^{\infty}$$?to1(D_{X}^{\infty})$

Perv(X) is the full abelian subcategory of “perverse sheaves” of $D_{C-c}^{b}(X)$ , where we adopt forour purpose a definition shifted by $n=\dim_{C}X$ from the usual one, i.e. given $F\in ObD_{C-c}^{b}(X)$ , we say$F$ is an object of Perv(X) iff $F[n]$ is perverse in the usual sense of [BBD] (e.g. if $Y\subset X$ is a purely
d-codimensional complex set then we say that $C_{Y}[-d]$ is perverse; see \S 4).

oraccordingly, and that the arrows
Recall that one sets Sol(M)bearing that name in (1.1) and (1.2) were constructed in $[K1]$ , that the equivalence under
was proven in [K-K] and that the construction of the temperate $R’ttom(\cdot, O)$-functor $RH$ and proof that
$RH$ is an equivalence was performed in $[K1,2]$ .
$=R\mathcal{H}om_{D}(\mathcal{M}, O)$$ffltom\prime p\infty(\mathcal{M}, O)$$D_{x\otimes v_{X}}^{\infty^{L}}(\cdot)$

D\’epartement de Math\’ematiques-Laboratoire C.N.R.S. URA 742-Universit\’e Paris XIII93430 Villetaneuse-France

2An independent proof that Sol is an equivalence is performed in [$M1$ and 2]. See [B] for a review of theseresults.
The point of interest here is to give a microlocal version of (1.2). Namely, if$\pi$: $T^{*}Xarrow X$ is the cotangent, one has the abelian category Regho1
bundle of $X$ , andof germs of regularholonomic modules over the ring of microdifferential operatorsof [S-K-K] which should be equivalent
to a category defined by a suitable microlocalization of Perv(X). The precise statement goes as follows.
$p\in\tau^{o_{*}}x_{d}=_{ef}T^{*}X\backslash T_{X}^{*}X$$(\mathcal{E}_{X,p})$$\mathcal{E}_{X,p}$

We set$C^{\cross}$$:=C\backslash \{0\}$and$\gamma$: $T^{*}Xarrow T^{*}X/C^{\cross}$ .

THEOREM 1. –One has the following commutative diagram (1.3) and all the honzontal arrows areequivalences of categories.

(1.3)

Here :$\mathcal{E}_{X}^{\infty}$$\mathcal{E}_{X}^{IR}$is the sheaf of infinite order microdifferential operators of [S-K-K],
is the sheaf of holomorphic microlocal operators of [S-K-K] and$\mathcal{E}_{X}^{IR,f}$

is its temperate analogueof [A].
An object ofdefinition for$Regho1\mathcal{E}_{X,p^{f}}^{1R}$$\mathcal{H}ol(\mathcal{E}_{X}^{\infty_{p}},)$

The categories Perv(X;$\mu hom(\cdot, \cdot)$

version ofis by definition of the form

and$C^{\cross}p$$\prime Hol(\mathcal{E}_{X}^{IR_{p}})$$\mathcal{E}_{X,p}^{IR,f}\bigotimes_{\mathcal{E}_{X}}\mathcal{M}$with$\mathcal{M}\in ObRegho1\mathcal{E}_{X,p}$, with a similar.

) and Perv(X; ) are defined below.
$p$is Kashiwara and Schapira) functor of [K-S 2], andof [A], while $\mu RH:=\gamma^{-1}R\gamma_{*}T-\mu hom(\cdot, O_{X})$ .
$s$$T-\mu hom(\cdot, \mathcal{O}_{X})$is the temperate$\mu hom(\cdot, O_{X})$

Assuming the definition of Perv(X; ), the construction of$p$$So1_{p}$is implicit in [K-S 1].

The various microlocalizations of Perv(X) are performed by essential use of the microlocal theory ofsheaves of Kashiwara and Schapira [K-S 2] and by using the microlocal characterisation of perverse
sheaves of loc. cit.

We stress the point that these microlocalizations rely on$nece\grave{s}sary$real (subanalytic) geometry.

The main tool in the proof is the invariance by canonical transformations which allows one to makeuse of the generic position theorem of [K-K] which reduces the situation to that of (regular holonomic)
D-modules.

32. –The category$D_{IR-c}^{b}(X;\Omega)$

Let $X$ be a real analytic manifold, $D^{b}(X)$ the derived category of the category of bounded complexesof sheaves on $X$ and $D_{R-c}^{b}(X)$ its full triangulated subcategory of complexes with IR-constructible
cohomology. The following is detailed in [ , Appendix].
$A$If$\Omega\subset T^{*}X$

is any subset of the cotangent bundle of $X$ the fundamental category occuring in [K-S 2] is$D^{b}(X,\cdot\Omega)=D^{b}(X)/\mathcal{N}_{\Omega}def$where$N_{\Omega}$is the null-system of objects$F$

whose micro-support $SS(F)$ does not meet$\Omega$(cf. loc.cit).

We set here.

$D_{IR-c}^{b}(X;\Omega)=D_{IR-c}^{b}(X)/\mathcal{N}_{\Omega}\cap ObD_{IR-c}^{b}(X)$Note that if$\Omega’\subset\Omega$

there is a canonical functor$D_{Et-c}^{b}(X;\Omega)arrow D_{IR-c}^{b}(X;\Omega’)$.

If $\Omega=\{p\}$ is a point we write $D^{b}(X;p)$ instead of $D^{b}(X;\{p\})$ and so forth.
By the results of [K-S 2] if is easy to see that

LEMMA 2.1. $-D_{IR-c}^{b}(X;p)$ is a full triangulated subcategory of $D^{b}(X;p)$ .
An adaptation of the microlocal kernel operations of [K-S 2] yields also the invariance under “extendedcanonical transformations” of loc.cit.
More precisely, let be another real manifold and denote by
the antipodal map of $T^{*}Y$ .
$Y$$q_{j}$the j-th projection of $X\cross Y$ and by$(\cdot)^{a}$

Let $p_{X}\in T^{*}X,$ $p_{Y}\in T^{*}Y$ and $K\in ObD_{IR-c}^{b}(X\cross Y)$ satisfying the following condition :(2.1)$SS(K)\cap(\{p_{X}\}\cross T^{*}Y)\subset\{(px,p_{Y}^{a})\}$

in the neighborhood of that point.

For $F\in ObD_{R-c}^{b}(Y)$ one defines a pro-object of $D_{Ft-c}^{b}(X;p_{X})$ by setting

(2.2)

‘

$\Phi_{K}^{\mu}(F)=\prime\prime\varliminf’’Rq_{1!}(K_{X\cross V}\otimes q_{2}^{-1}F)$whereruns over the set of relatively compact open subanalytic neighborhoods of $x=\pi(p)$ . Actually
one has the$V$

LEMMA 2.2. –For $K\in ObD_{1R-c}^{b}(X\cross Y)$ satisfying (2.1), this pro-object is an object of $D_{IR-c}^{b}(X;p_{X})$and the functor: $D_{IR-c}^{b}(Y;p_{Y})arrow D_{IR-c}^{b}(X;p_{X})$ is well defined.
$\Phi_{K}^{\mu}$Note that the functor $\Phi_{K}(\cdot)=Rq_{1!}(K\otimes q_{2}^{-1}(.))$ would not be defined here in general.
its: $(T^{*}Y)_{p_{Y}}arrow(T^{*}X)_{Px}$ be a germ of canonical transformation andassociated germ of Lagrangian manifold in $T^{*}(X\cross Y)$ . One may find $K\in ObD_{IR-c}^{b}(X\cross Y)$ with$SS(K)\subset\Lambda$ in the neighborhood, such that
: $D_{IR-c}^{b}(Y;p_{Y})arrow D_{IR-c}^{b}(X;px)$ is anofequivalence of categories.

PROPOSITION 2.3. –Let$\Lambda$$\varphi$$(p_{X},p_{Y}^{a})$$\Phi_{K}^{\mu}$4
3. –The category $D_{C-c}^{b}(X;\Omega)$the underlying real manifold. Recall that for
Let now $X$ be a complex n-dimensional manifold, and$F\in ObD_{IR-c}^{b}(X)$ one has the following characterisation (cf. [K-S 2]) :$X_{Et}$(3.1)$(F\in ObD_{\mathbb{C}-c}^{b}(X))\Leftrightarrow$thus we may define for any subset(3.2)( $SS(F)$ is$\Omega\subset T^{*}X$$C^{\cross}- conica1$)$\Leftrightarrow$( $SS(F)$ is C-Lagrangian),a full triangulated subcategory of$\{def$$D_{Et-c}^{b}(X;\Omega)$of

PROPOSITION 3.1 (See $[A$ , Appendix]). –Letof complex canonical tmnsformation andThen$Y$$\Omega$by settin$g$.

be another copy of $X,$ : $(T^{*}Y)_{p_{Y}}arrow(T^{*}X)_{px}$ be a germ
its associated complex Lagrangian submanifold.
$\varphi$$\Lambda\subset T^{*}(X\cross Y)$(i) there exists $K\in Ob(D_{\mathbb{C}-c}^{b}(X\cross Y;(px,p_{Y}^{a})))$ with $SS(K)\subset$ A in a neighborhood ofthat the functor of proposition 2.3 induces an equivalence of categones$\Phi_{K}^{\mu}$$(p_{X}, p_{Y}^{a})$such: $D_{C-c}^{b}(Y;p_{Y})arrow D_{C-c}^{b}(X;p_{X})$ ,then there is K E $Ob(D_{C-c}^{b}(X\cross Y;C^{\cross} (p_{X},p_{Y}^{a})))$ ,(ii) if moreover is globally defined on the orbit C’such that
C’A in a neighborhood ofinduces an equivalence ofwith $SS(K)\subset$ A
categories:( , C’ ) $arrow D_{C-c}^{b}(X, C^{\cross}p_{X})$ .
$p_{Y}$$\varphi$$C^{\cross}(p_{X},p_{Y}^{a})$$=$$\Phi_{K}^{\mu}$$D_{C-c}^{b}$$Y$$\Phi_{K}^{\mu}$$p_{Y}$

preserves local-conicity, then (ii)
Point (i) follows easily from proposition 2.3 by (3.1) because
is defined at any point in the fiber of over $\pi(p)$ .
stems from (i) and formula (2.2) that shows that
$C^{\cross}$$\Phi_{K}^{\mu}$$\Phi_{K}^{\mu}$$\pi$

For example one has $D_{C-c}^{b}(X;T^{*}X)=D_{C-c}^{b}(X)$ and if $x\in X\cong T_{X}^{*}X$ one has the equivalence$(F\in ObD_{\mathbb{C}-c}^{b}(X;x))\Leftrightarrow(F\in ObD_{IR-c}^{b}(X)$ andfor some open neighborhood$V$$F_{1_{V}}\in ObD_{\mathbb{C}-c}^{b}(V)$$ofx)$ .

do not have C-constructible cohomologies and the natural
Note that, in general the objects of$D_{C-c}^{b}(X)/\mathcal{N}_{p}\cap
D_{C-c}^{b}(X)arrow
D_{C-c}^{b}(X;p)$is not an equivalence.
functor$D_{\mathbb{C}-c}^{b}(X;p)$On the other hand, one has the following geometrical version of the generic position theorem. Recall (cf.
[K-K]) that a complex Lagrangian subset A C $T^{*}X$ is said to have a generic position at $p\in\tau^{o}*x$ iff(3.3)$\Lambda\cap\pi^{-1}\pi(p)=C^{\cross}p$

in a neighborhood of .
$p$

PROPOSITION 3.2. –Let FE $ObD_{C-c}^{b}(X;p)$ such that $SS(F)$ is in a generic position at . Then there.
exists $F’\in ObD_{C-c}^{b}(X;\pi(p))$ such that $F’\simeq F$ in$p$$D^{b}(X,\cdot p)$,The proof goes by showing that one may “cut-off” the non C-Lagrangian part of $SS(F)$ inand an open subanalytic neighborhood
inofi.e. one finds kernels $K,$
satisfy the conditions of proposition 3.1 (i),is a quasi-inverse of
and
in $X$ such that $K,$
$F’$is such that $SS(F’)$ is C’-invariant in $\pi^{-1}(U)$ . Thus $F’\in ObD_{C-c}^{b}(X;\pi(p))$ by(3.1) and $F’\simeq F$ in $D^{b}(X;p)$ by proposition 3.1.
$\pi^{-1}\pi(p)$$K^{*}$$K^{*}$$D_{\mathbb{C}-c}^{b}(X\cross X;(p,p^{a}))$$\Phi_{K}^{\mu}$.

$U$

$x$

$\Phi_{K}^{\mu}$$:=\Phi_{K}^{\mu}.((\Phi_{K}^{\mu}F)_{U})$One may get a quicker proof by using a refined version, obtained in [D’A-S], of a microlocal cut-off lemmaof [K-S 2] where one is allowed non-convex sets.

54. –Microlocalization of Perverse Sheaves
In [K-S 2] one finds the following microlocal characterisation of perverse sheaves :On object $F\in ObD_{\mathbb{C}-c}^{b}(X)$ is a perverse sheaf iff it satisfies the following condition(4.1)$\{\begin{array}{l}Foranynon- singularpointp\in SS(F)suchthat\pi\cdot.SS(F)arrow Xhasconstantrankinaneighborhoodofp,thereexistsacomplexd- codimensionalsubmanifoldY\subset XsuchthatF\simeq C_{Y}[-d]inD^{b}(X\cdot.p)(cf.[K- S2,(10.3.7)])\end{array}$Thus for any subset
manner.

$\Omega\subset T^{*}X$

we may define a full subcategory Perv(X; ) of$\Omega$$D_{\mathbb{C}-c}^{b}(X;\Omega)$in the following

DEFINITION 4.1. $-ObPerv(X;\Omega)=\{Fdef\in ObD_{\mathbb{C}-c}^{b}(X,\cdot\Omega);F$ satisfies condition (4.1) at any
neighborhood of$\Omega$$p$in a}.

Then the following results from \S 3 and the characterisation (4.1).
PROPOSITION 4.2. –Let $\Omega=\{p\}$ (resp.

).

$\Omega=C^{\cross}p$(i) Perv(X; ) is invariant by extended canonical transformation in the sense of proposition 3.1 (i)(resp. proposition 3.1 $(ii)$).
$\Omega$(ii) Let $F\in Perv(X;p)$ (resp. Perv(X; C’ )) such that $SS(F)$ is in a generic position at . Then thereis $F’\in Perv(X;\pi(p))$ such that $F\simeq F’$ in $D^{b}(X;p)$ .
$p$

$p$

(iii) Perv(X;$\Omega$) is a full abelian subcategory of

5. –The equivalence Perv

$D_{\mathbb{C}-c}^{b}(X;\Omega)$.

$(X; C^{\cross}p)^{o_{arrow}^{\mu RH}}Regho1\mathcal{E}_{X,p}$

Recall that Kashiwara’s functor $RH$ of cohomology with bounds of $[K2,3]$ is defined on IR-constructible
complexes, more precisely$RH$ : $D_{IR-c}^{b}(X)^{o}arrow D^{b}(D_{X})$ ,(where$D^{b}(\mathcal{D}_{X})$stands for$D^{b}(Mod\mathcal{D}_{X})$), and it is microlocalized in [A] as a functor: $D_{IR-c}^{b}(X)^{o}arrow D_{IR>0}^{b}(\pi^{-1}D_{X})$ ,$T-\mu hom(\cdot, O_{X})$where the latter category is the full subcategory subcategory of the complexes of $D^{b}(\pi^{-1}D_{X})$$D^{b}(Mod(\pi^{-1}\mathcal{D}_{X}))$ with $IR>0$ -homogenous cohomology.
Since one has$:=$$supp(T-\mu hom(F, O_{X}))\subset SS(F)$ ,then for any subset$\Omega\subset T^{*}X$, the functor of triangulated categories$T-\mu hom(\cdot, O_{X})$is well-defined, where$F\in ObD_{IR-c}^{b}(X)$ :(5.1)$\pi_{\Omega};=\pi_{1_{\Omega}}$:$D_{IR-c}^{b}(X;\Omega)^{0}arrow D_{IR>0}^{b}(\pi_{\Omega}^{-1}D_{X})$: $\Omegaarrow X$ . If moreover$\Omega=C^{\cross}\Omega$

is a C’-invariant subset we set for$\mu RH(F)=\gamma^{-1}R\gamma_{*}T-\mu hom(F, O_{X})def\in ObD_{IR_{>0}}^{b}(\pi_{\Omega}^{-1}D_{X})$.

6
Recall also the following facts
For any F E $ObD_{Ft-c}^{b}(X)$ and any $j\in Z,$ $H^{j}T-\mu hom(F, O_{X})$ is an$\mathcal{E}_{X}^{R,f}$-module,

and,is faithfully flat on
and we have invariance by canonical transformations, that is, with the hypotheses of proposition 3.1 (i),one may find a section,means the sheaf of maximum degree forms relative to $X\cross Yarrow X$ ) such that(where$\mathcal{E}_{X}^{IR,f}$$\gamma^{-1}R\gamma_{*}\mathcal{E}_{X}^{IR,j}\cong \mathcal{E}_{X}$$\mathcal{E}_{X}$$s\in H^{o}(T-\mu hom(K, \Omega_{X\cross Y/X}))_{(p_{X},p_{Y}^{\sigma})}$$\Omega_{X\cross Y/X}$

the correspondance$P\in \mathcal{E}_{X,p^{j_{X}}}^{R}rightarrow Q\in \mathcal{E}_{Y,p^{f_{Y}}}^{IR}$compatible with a natural isomorphism

such that $Ps=sQ$ is a ring isomorphism$T-\mu hom(F, \mathcal{O}_{Y})_{p_{Y}}arrow T-\mu hom(\Phi_{K[n]}^{\mu}F, \mathcal{O}_{X})_{px}\sim$.

Finally we have a basic formula :$T-\mu hom(F, O_{X})\simeq \mathcal{E}_{X}^{R,j}\otimes_{\pi^{-i}P_{X}}\pi^{-1}RH(F)$,for F E$ObD_{\mathbb{C}-c}^{b}(X)$,from which we get(5.2)$\mu RH(F)=\mathcal{E}_{X}\otimes_{\pi^{-1}D_{X}}\pi^{-1}RH(F)$for $F\in ObD_{\mathbb{C}-c}^{b}(X)$ .

The key point is then theLEMMA 5.1. –Formula (5.1) actually defines a functor$\mu RH:Perv(X;C^{x}p)^{o}arrow Regho1(\mathcal{E}_{X,p})$.

Proof: Let $F\in Ob$ Perv( $X$ ; C’ ). By the invariance by extended (resp. quantized) canonical transformations, we may assume that $SS(F)$ has generic position at , thus, by proposition 4.2 (iii) we may find$F’\in Perv(X;\pi(p))$ such that $F\simeq F’$ in $D^{b}(X;p)$ , thus
$p$

$p$

$\mu RH(F)_{p}\simeq\mu RH(F’)_{p}\simeq(\mathcal{E}x\otimes_{\pi^{-1}\mathcal{D}_{X}}\pi^{-1}RH(F’))_{p}$,by (5.2), and the latter is an object concentrated in degree zero, which coincides with the germ at-module.
regular holonomic

$p$

of a

$\mathcal{E}_{X}$$\wp$

That $\mu RH$ : Perv $(X; C^{\cross}p)^{o}arrow Reghol(\mathcal{E}_{X,p})$ is an equivalence is then readily deduced, by using again
invariance by canonical transformations, from Kashiwara and Kawai’s generic position theorem of [K-K].
Details will appear elsewhere.
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-Bernstein, J.

-Deligne, P. : Faisceaux pervers. Ast\’erisque 100

Bjork, J-E. : Analytic D-Modules. Kluwer (1993).

7[D’A-S]

D’Agnolo, A.

-Schapira, P. : in preparation.

[K1]

Kashiwara, M. : On the maximally overdetermined systems of linear differentialequations. Publ. R.I.M.S. Kyoto Univ. 10, 563-579 (1975).

[K2]

Kashiwara, M. : Faisceaux constructibles et syst\‘emes holon\^omes \‘a points singuliers
r\’eguliers. S\’em. Goulaouic-Schwartz. Ecole Polytechnique (1979-80).

[K3]

Kashiwara, M. : The Riemann-Hilbert problem for holonomic systems. Publ. R.I.M.S.,
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[K-K]

Kashiwara, M. Kawai, T. : On the holonomic systems of microdifferential equations,III. Systems with regular singularities. Publ. R.I.M.S., Kyoto, Univ., t. 17, 813-979 (1981).
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[K-S 2]

Kashiwara, M. Schapira, P. : Sheaves on manifolds. Grundlehren der math. Wiss.
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[M1 and 2]

Mebkhout, Z. : Une \’equivalence de cat\’egories Comp. Math. 51, 51-62 (1984), et : Uneautre \’equivalence de cat\’egories. Comp. Math. 51, 63-88 (1984).

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--Sato, M.
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