数理解析研究所講究録
第 845 巻 1993 年 49-52

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REGULARITY THEOREMS FOR HOLONOMIC MODULES(本多尚文)NAOFUMI HONDA(北海道大学)

University of Hokkaido

0. INTRODUCTION
The regularity theorem for the ordinary differential equations has a long history.
Malgrange [17] has shown the regular singularity of the system is equivalent to theconvergence of its formal power series solutions. Ramis [18] extend the results tothe irregular singular case, that is, the irregularity is characterized by the Gevrey
growth order of its formal power series solutions. In the real domain, Komatsu [15]also has a similar result comparing ultra-distribution and hyperfunction solutions.
One of the important problems is to extend these results to the higher dimension.
The deep study of holonomic systems due to $Kashiwara- Kawai^{\iota}[10]$ and Kashiwara[9] established the regularity theorems for holonomic modules in the regular singularcase. The purpose of this paper is to give several regularity theorem for the irregular
singular holonomic modules.

1. PRELIMINARY
Let $X$ be a complex manifold of dimension$n$and$\pi$: $T^{*}Xarrow X$ its cotangent.
bundle. Set
and denote by the restriction of to$T^{*}X$$X$$(x_{1},\cdotsx_{n})$asWe choose a local coordinate system ofasand that ofof micro-differential. $T^{*}X$ is endowed with the sheaf
operators of infinite order constructed by Sato-Kashiwara-Kawai [19].
(resp.
) the subsheaf ofWe denote byconsisting of micro-differential
operators of finite order (resp. micro-differential operators of order at most $m$ ). For, refer to [19] and Schapira [20]. Now we define the subsheaf
the theory ofof micro-differential operators of Gevrey growth order $(s)$ for any $s\in(1, \infty)$ .
$\mathring{T}^{*}X=T^{*}X\backslash T_{X}^{*}X$$T^{*}X\circ$$\pi\circ$$\pi$$(x_{1}, \cdots x_{n}; \xi_{1}, \cdots\xi_{n})$$\mathcal{E}_{X}^{\infty}$$\mathcal{E}_{X}(m)$$\mathcal{E}_{X}$$\mathcal{E}_{X}^{\infty}$$\mathcal{E}_{X}^{(s)}$$\mathcal{E}_{X}$Definition 1.1. For an open $su$ bsetof $T^{*}X$ , a umif and only if
belongs tosatisfies the following estimate (1.1); forany compact set $K$ of , there exists a positi $ve$ constantsuch that$U$$\mathcal{E}_{X}^{(s)}(U)$$\sum_{i\in \mathbb{Z}}P_{i}(x, \xi)\in \mathcal{E}_{X}^{\infty}(U)$$s$$\{P_{i}\}_{i\in N}$

$U$

(1.1)

$C_{K}$$\sup_{K}|P_{i}(x, \xi)|\leq\frac{C_{K}^{i}}{i!^{s}}$$(i\geq 0)$.

For convenience, we set.
andNext we review briefly the definition of the sheaf of holomorphic microfunctionsin Gevrey class. Let
be a complex submanifold of $X$ andits conormal$\mathcal{E}_{X}^{(1)}$$Y$$:=\mathcal{E}_{X}^{\infty}$$\mathcal{E}_{X}^{(\infty)}$$:=\mathcal{E}_{X}$$T_{Y}^{*}X$

Typeset by$\mathcal{A}_{\mathcal{M}}S-TkX$

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NAOFUMI HONDA

bundle. Then we define the subsheafas$C_{Y|X}^{\mathbb{R},(s)}$of the holomorphic microfunctions$C_{Y|X}^{\mathbb{R}}$$C_{Y|X}^{\mathbb{R},(s)}$$:=\mathcal{E}_{X}^{(s)}C_{Y|X}^{\mathbb{R},f}$where
is the sheaf of tempered holomorphic microfunctions (for the definition,refer to Andronikof $[1,2]$ ). Remark that these sheaves are also defined by the functor, which is a natural extension of tempered microlocalization functor
constructed by Andronikof $[1,2]$ ,$C_{Y|X}^{\mathbb{R},f}$$T-\mu^{(s)}(\mathcal{O})$$T-\mu.(\mathcal{O})$$\mathcal{E}_{X}^{(s)}$where$:=\tau^{-1}\tau_{*}T-\mu_{X}^{(s)}(\mathcal{O}_{X\cross X})\otimes\Omega_{X}[\dim X]$: $T^{*}X\circarrow P^{*}X$ is a canonical projection, and$\tau$$C^{\mathbb{R},(s)}$$:=T-\mu_{Y}^{(s)}(\mathcal{O}_{X})$$Y|X$[codim $Y$ ].

For the definition and properties of the Gevrey microlocalization functor, refer toHonda [5].
Let be a regular or maximally degenerate involutive submanifold of codimen$V$by. We define the subsheaf.
Here we denote the symbol map of degree 1 by. Now we define the sheaf ofsion $d\geq 1$ in$T^{*}X\circ$$:=T^{*}X\backslash T_{X}^{*}X$$I_{V}$$I_{V}\subset \mathcal{E}_{X}(1)$$:=\{P\in \mathcal{E}_{X}(1);\delta_{1}(P)|_{V}\equiv 0\}$$\delta_{1}(\cdot)$

rings

in

$\mathcal{E}_{V}^{(\sigma)}$$T^{*}X\circ$for a rational number).

$\sigma\in[1, \infty$$\mathcal{E}_{V}^{(\sigma)}$$;= \sum_{n\geq 0}\mathcal{E}_{X}(\frac{(1-\sigma)n}{\sigma})I_{V}^{n}$.

In case $\sigma=1$ , this sheaf coincides with the sheaf
defined in Kashiwara-Oshima[12] and [10].
.
We list up the main properties of the sheaf$\mathcal{E}_{V}$$\mathcal{E}_{V}^{(\sigma)}$.
is a subring of, andis a left and right
module.
is a sheaf of Noetherian ring, and any coherent
module is pseudo.
coherent over, then its formal ajoint operator(4) If.
belongs to(1)(2)(3)$\mathcal{E}_{V}^{(\sigma)}$$\mathcal{E}_{X}$$\mathcal{E}_{X}(0)\subset \mathcal{E}_{V}^{(\sigma)}$$\mathcal{E}_{V}^{(\sigma)}$$\mathcal{E}_{X}(0)$$\mathcal{E}_{V}^{(\sigma)}$$\mathcal{E}_{X}$$\mathcal{E}_{V}^{(\sigma)}$$P\in \mathcal{E}_{V}^{(\sigma)}$$P^{*}$$\mathcal{E}_{V}^{(\sigma)}$Let
be a holonomic
modules in a neighborhood of. We firstat a smooth point of its support. Given
define the weak irregularity of$\mathcal{M}$$p\in T^{*}X\circ$$\mathcal{E}_{X}$$\mathcal{M}$$supp(\mathcal{M})_{sing}\cup T_{X}^{*}X$$p\not\in$.

Deflnition 1.2.
has weak irregularity at $most\sigma$ at if and only ifsatisfies
the follo wing conditions.
There exist an open neighborhood of , maxim ally degenerate involu tive $su$ b, and an
manifold with its singular locus
modules
in which
generates
overand is finitely generated overat any point of a dense$su$ bset in $supp(\mathcal{M})\cap U$ .
$\mathcal{M}$

$U$

$V$

$\mathcal{M}$$p$$supp(\mathcal{M})$$\mathcal{E}_{X}$$\mathcal{M}$$p$$\mathcal{E}_{V}^{(\sigma)}$$\mathcal{E}_{X}(0)$Next we define weak irregularity in the general case.

$\mathcal{M}_{0}$

$U$

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REGULARITY THEOREMS FOR HOLONOMIC MODULES

Deflnition 1.3.
has wea irregularity at most at if and onlymodule(1) A holonomicof and a closed analytic $su$ bset
if there exist an open neighborhood
has weak irregularity at most atwith codim $Z\geq\dim X+1,$$\mathcal{M}$$\mathcal{E}_{X}$$p$$\sigma$$U$$Z\supset$$p$$supp(\mathcal{M})_{sing}$$\mathcal{M}$$\sigma$any point in.
(2) A holonomi $cD_{X}$ module has weak irregularity at most
$U\backslash Z\cap T^{*}X\circ$$\mathcal{N}$$\sigma$

if and only if$\mathcal{E}_{X}\otimes v_{X}$$0$.Af has irregularity at most$\sigma$

at any point in $T^{*}X$ .

2. STATEMENT OF MAIN THEOREM
a holonomic
Main Theorem. Let $U\subset T^{*}X$ be C’ conic open set,a $ra$ tional number. The follo wing conditions (1),(2) andmodules in and(3) are $eq$ uivalent.
(1) There exists a holonomi
modulewith regu lar singularities satisfying$\mathcal{M}$

$a$

$U$

$\mathcal{E}_{X}$$\sigma\geq 1$$c\mathcal{E}_{X}$$\mathcal{M}_{reg}$$\mathcal{E}_{X}^{(s)}\otimes_{\mathcal{E}_{X}}\mathcal{M}\simeq \mathcal{E}_{X}^{(s)}\otimes_{\mathcal{E}_{X}}\mathcal{M}_{reg}$.
in for all(2) For any submanifold $Y\subset X$ and any$U$$s \in[1, \frac{\sigma}{\sigma-1}]$$s \in[1, \frac{\sigma}{\sigma-1}]$, we have$\mathbb{R}\mathcal{H}om_{\mathcal{E}_{X}}(\mathcal{M},C_{Y|X}^{\mathbb{R},(s)})|_{U}\simeq \mathbb{R}\mathcal{H}om_{\mathcal{E}_{X}}(\mathcal{M},C_{Y|X}^{\mathbb{R}})|_{U}$(3)$\mathcal{M}$has weak irregularity at most$\sigma$

in

$U$.

.

Using the same technique of the proof, we deduce the following corollary. Let$M$ be a real analytic manifold with its complexification $X$ .
modules at $p\in T^{*}X$ with weak irregube a holonomic
Corollary 1. Let,larity at most . Then we have the isomorphisms for all$\mathcal{M}$$\mathcal{E}_{X}$$s \in[1, \frac{\sigma}{\sigma-1}]$$\sigma$$\mathbb{R}\mathcal{H}om_{\mathcal{E}_{X}}(\mathcal{M},C_{M}^{(s)})\simeq \mathbb{R}\mathcal{H}om_{\mathcal{E}_{X}}(\mathcal{M},C_{M})$and (resp.
whereGevrey clas $s(s))$ .
$C_{M}$$C_{M}^{(s)}$) is the she af of microfunctions (resp. microfunctions ofis regular singular and the solution sheaf is tempered miIn the case thatcrofunctions, this result is already obtained by Andronikof [3]. Finally we remark
to the result (2) of the main theorem, we can recoverthat applying the functor
the results of Laurent [16].
$\mathcal{M}$$\tau_{*}$

be a holonomic
modules at $p\in T^{*}X$ with weak
Corollary 2 [16]. Letand for
irregularity at most . Then. we have the isomorphisms for all$Y\subset
X$,
any submanifold$\mathcal{M}$$\mathcal{E}_{X}$$s \in[1, \frac{\sigma}{\sigma-1}]$$\sigma$$R\mathcal{H}om_{\mathcal{E}_{X}}(\mathcal{M},C_{Y|X}^{(s)})\simeq \mathbb{R}\mathcal{H}om_{\mathcal{E}_{X}}(\mathcal{M},C_{Y|X}^{\infty})$.

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$\Pi$$c\infty$$\Pi$