数理解析研究所講究録1033 巻 1998 年 71-80

71

The local analytical trivialityof a complex analytic singular foliation
Hokkaido University

Junya Yoshizaki*(北海道大学

吉崎純也)

Abstract

A singular foliation on a complex manifold $M$ is defined as an integrable coherent subsheaf $E$ of the tangent sheaf of $M$ . In this talk we discuss the existenceof the “leaf (integral submanifold)” of $E$ at each point of $M$ (Theorem (3.3)).
The dimensions of the leaves are not constant on $M$ in general, so the singular
set $S(E)$ , which is in fact an analytic subset of $M$ , is given as the set of points
where the dimension of the leaf of $E$ is not maximal. As an application of the existence of the leaves, we can show that the structure of the foliation $E$ is locallyanalytically trivial along its each leaf (Theorem (3.4)). This kind of triviality
was studied by P.Baumfor the point such that is a non-singular point
of $S(E)$ and. D.Cerveau also took up asimilar problem from another viewpoint in [C] (for the real case, see [N], [Ss]and [St] . We generalize and arrange their theory, and add some new results.
This is a joint work with Yoshiki Mitera. In the process of this work, we
received many useful suggestions and advices ffom Tatsuo Suwa. We would liketo thank him for helpful conversations and comments.
$([\mathrm{B}])$

$p$

$p$

$\mathrm{d}i\mathrm{m}_{p}S(E)=\dim E(p)=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}E-1$$)$

1

Complex analytic singular foliations

At first, we recall some generalities about complex analytic singular foliations oncomplex manifolds. The notation in the following is originary due to T.Suwa. For
further details, see [B], [BB] and [Sw].
Let $M$ be a (connected) complex manifold of (complex) dimension , and let$_{M}$ anddenote, respectively, the heaf of holomorphic functions on $M$ , thetangent sheaf and the cotangent sheaf of $M$ .
$n$$\mathcal{O}_{M},$$\Omega_{M}$$.\mathrm{s}$*Research Fellow of the Japan Society for the Promotion of Science. This research is partially
supported by The Ministry of Education, Science and Culture, Japan, Grant-in-Aid for Scientific
Research.

72. Note that, in this case, $E$ is coherent if and
Let $E$ be a coherent subsheaf of. We set
only if $E$ is locally finitely generated, since $_{M}$ is locally$\mathrm{O}-_{M}$$\mathrm{h}\mathrm{e}\mathrm{e}$$S(E)=$ {$p\in M|(\mathrm{O}-_{M}/E)_{p}$ is not$(\mathcal{O}_{M})_{p}$-hee},and call it the singular set of $E$ . Each point of $S(E)$ is called a singular point of. If we restrict $E$ to a sufficiently small coordinate neighborhood with coordinates, we can express $E$ on explicitly as follows:
$p$

$U$

$E$

$U$

$(z_{1}, z_{2}, \ldots , z_{n})$(1.1)$E=(v_{1}, v_{2}, \ldots,v_{s})$,$v_{i}= \sum_{1j=}^{n}f_{i}j(z)\frac{\partial}{\partial z_{j}}$are holomorphic functions defined on
where
Then the singular set $S(E)$ is given on by$f_{ij}(z)$

$U$

,

, and

$1\leq i\leq s$$s$,is a non-negative integer.

$U$$S(E)\cap U=$

A coherent subsheaf $E$ ofof $M$ ,(1.2){$p\in U|\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}(f_{ij}(p))$$\mathrm{O}-_{M}$is not maximal}.

is said to be integrable (or involutive) if for any point$p$$[E_{p}, E_{p}]\subset E_{p}$holds (where [ , ] denotes the Lie bracket of smooth vector fields). Moreover, we
define the rank (we sometimes call it dimension) of to be the rank of locally free
sheaf $E|_{M-s}(E)$ , and denote it rankE. Using the notation in (1.1), we can rewrite itasrank $(f_{ij}(p))$ .
rankE
$E$

$= \max_{p\in M}$(complex analytic) singular foliation on $M$ is an integrable coDefinition 1.3
herent subsheaf $E$ of $_{M}$ .
$A$

It is clear that a singular foliation $E$ induces a non-singular foliation on $M-S(E)$ .

Definition 1.4

Let $E$ be a coherent subsheaf of$v\in\Gamma(U, \mathrm{O}-M),$holds for every open set$\mathrm{O}-_{M}$$v|_{U-S(E})(\in\Gamma U-S(E), E)$$U$. We say that$\Rightarrow$$E$

is reduced if$v\in\Gamma(U, E)$

in $M$ .

By the preceding two definitions, we can consider
foliations” in naturalsense, i.e., a reduced foliation on $M$ is a coherent subsheaf of $_{M}$ which is integrable
and reduced.
$‘ {}^{t}reduced$

Remark 1.5

We can check the following facts about reduced foliations:

73(i) If a singular foliation $E$ is locally free,$E$

is reduced.

$\mathrm{c}\mathrm{o}\dim S(E)\geq 2$$\Leftrightarrow$(ii) Let $E$ be a reduced coherent subsheaf of $_{M}$ . Then $E$ is integrable if (1.2) holds
for every point $p\in M-S(E)$ .
Next, let us represent singular foliations in terms of holomorphic 1-forms. It is not
the viewpoint of its “dual”, but there are several points
so difficult to rewrite it
which require a little care.
$\mathrm{h}\mathrm{o}\mathrm{m}$Let

Definition 1.6

be a coherent subsheaf of$F$$\Omega_{M}$$S(F)=$ {$p\in M|(\Omega_{M}/F)_{p}$ is not. Then we set$(\mathcal{O}_{M})_{p}$-free},and call it the singular set of F. Each point in $S(F)$ is oflen calledof $F$ .

A coherent subsheaf

Definition 1.7

$F$

of

$\sim dF_{p}\subset\Omega_{p}$

$F$

$A$

(

of

$\Omega_{M}$LetDefinition 1.9
reduced if$\mathrm{C}\mathrm{o}\mathrm{m}_{\mathrm{P}^{\mathrm{l}\mathrm{e}}}\mathrm{x}$$F$

is defined to be the rank of theanalytic) singular foliation on $M$ is an integrable co-.

$F(\subset\Omega_{M})$be a coherent subsheaf of$\omega|_{U-s(F})\mathrm{r}\in(U-s(F), F)$$\omega\in\Gamma(U, \Omega M),$holds for every open set/$F_{p}$

holds for every point $p\in M.$ Moreover, the rank oflocally free sheaf $F|_{M-s}(F)$ , and denoted rankF.

Definition 1.8
herent subsheaf

singular pointis said to be integrable when$\Omega_{M}$

A

$a$

$U$

$\Rightarrow$$\Omega_{M}$. We say that

$F$

is

$\omega\in\Gamma(U, F)$

in $M$ .

In the following we describe the relation between the two definitions, (1.3) and (1.8).

Definition 1.10

For singular foliations$E\subset \mathrm{O}-_{M}$and$F\subset\Omega_{M}$$E^{a}=${$\omega\in\Omega_{M}|\langle v,$ $\omega\rangle=0$

for all $v\in E$ },$F^{a}=${$v\in \mathrm{O}-_{M}|\langle v,\omega\rangle=0$

for all$\omega\in F$, we set},

denotes the natural pairing between a vector field and 1-form. Thenanddefine reduced singular foliations on M. We call (resp.
) is called the(resp.
the annihilator of $E$ (resp. $F$). Furthermore,
reduction of $E$ (resp. $F$ ).

whe-re$\langle$,$E^{a}(\subset\Omega_{M})$$F^{a})$$a$$\rangle$$E^{a}$$F^{a}(\subset \mathrm{O}-_{M})$$(E^{a})^{a}$$(F^{a})^{a}$74

Note that $S(E^{a})\subset S(E)$ and $S(F^{a})\subset S(F)$ hold.

Remark 1.11)
If we use the notation in (1.10), a singular foliation $E\subset_{M}$ (resp.
is reduced if and only if $(E^{a})^{a}=E$ (resp. $(F^{a})^{a}=F$ ). In this way we can makeany singular foliation reduced by taking its reduction. If we consider only reduced
foliations, then the two definitions of singular foliation stated above are equivalent,and in this occasion, moreover, there is no difference between the singular set in termsof vecter fields and that in terms of l-forms.
$F\subset\Omega_{M}$2

Singular set of a singular foliation

Next, let us summarize the basic properties of the singular set of a singular foliation.
Hereafter, we assume $E(\subset_{M})$ to be a singular foliation on a complex manifold $M$.
and set$r=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}E$For each point

Definition 2.1

$p$

in

$M_{f}$

we set$E(p)=\{v(p)|v\in E_{p}\}$ ,where $v(p)$ denotes theof the vector field
sub-vector space of the tangent space $T_{p}M$ .
$evaluati_{\mathit{0}}n$

For an integer

Definition 2.2$k$$ge7\mathrm{v}nv$

at . Note that $E(p)$ is a$p$with $0\leq k\leq r$ , we set$L^{(k)}=\{p\in M|\dim_{\mathrm{C}}E(p)=k\}$,$S^{(k)}=\{p\in M|\dim_{\mathrm{C}}E(p)\leq k\}$,and set $L^{(-1)}=S^{(-1)}=\emptyset$ for convenience. Clearly we have$L^{(k)}=s(k)-s(k-1)$ ,for $k=0,1,2,$
Remark 2.3$\ldots,$$r$$S^{(k)}= \bigcup_{i=0}^{k}L(i)$.

$L^{(k)}$and$S^{(k)}$are analytic sets for every integer$k$with $0\leq k\leq r$ .

By the remark stated above, we get the natural filtration which consists of analytic
sets:
.
(2.4)
$S(E)||$$S^{(r)}\supset s^{(r-1})\supset s^{(r-2})\supset\cdots\cdots\supset S^{(1)}\supset S^{(0)}\supset S^{(-1)}$$M||$$\emptyset||$This filtration seems to give us information only about the “dimension” of the space$E(p)$ atappearing in (2.4) also controlls. However, the local structure of each$p\inS^{(k)}$.
the “direction” of $E(p)$ at each point$p$$S^{(k)}$75

Example 2.5
defined byLet$v_{1},$$v_{2},$$v_{1}=$

(2.6)

$\{$

$v_{3}$

be holomorphic vector fields on $M=\mathrm{C}^{3}=\{(x, y, z)\}$$3y^{2} \frac{\partial}{\partial x}$$v_{2}=$$+2x \frac{\partial}{\partial y}$$(x^{2}-y^{3}) \frac{\partial}{\partial y}+3y^{2_{Z}}\frac{\partial}{\partial z}$$v_{3}=(x^{2}-y^{3}) \frac{\partial}{\partial x}$$-2xz \frac{\partial}{\partial z}$

Let $E(\subset_{M})$ be the coherent subsheaf generated by. We can easily check
$E$
$E$
that
is integrable, so defines a singular foliation on. Since the rank of $E$ is
two, allappearing in (2.4) are given by $S(E)=S^{(1)}=\{xz=yz=X^{2}-y^{3}=0\}=$$\{x=y=0\}\cup\{z=x^{2}-y^{3}=0\}$ and $S^{(0)}=\{x=y=0\}$ .
$v_{1},$$v_{2},$$v_{3}$$\mathrm{C}^{3}$$S^{(k)}$$S^{(1)}=L^{(0)}\cup L^{(1)}$$S^{(0)}=L^{(0)}$in the preceding example. For any point
Let us observe the analytic set, the dimension of the space $E(p)$ should be one or zero by the
belonging to. However we can obtain more information about $E(p)$ from justdefinition of. In fact, the direction of $E(p)$ is always “tangential
looking at the local structure of”at .
, in other words, $E(p)$ is always contained in the tangent cone oftoThis property can be stated precisely as follows.
$S^{(1)}$$p$$S^{(1)}$$S^{(1)}$$S^{(1)}$$S^{(1)}$$S^{(1)}$

Theorem 2.7 (TAN.GENCY LEMMA). Then we havepoint inLet$k$be an integer with $0\leq k\leq r$ and$S^{(k)}$$E(p)\subset\acute{C}_{p}s^{(k})$

where$C_{p}S^{(k}$)

denotes the tangent cone of$S^{(k)}$

,

at .
$p$

$p$

$p$

a

76

Remark 2.8 Theorem (2.7) was proved by P.Baum under the hypotheses that(see [B]). For the case of real singular
is reduced, and is a non-singular point offoliations, see [N], [Ss] and [St].
$E$$S^{(k)}$$p$, but weThis theorem is drawn as a corollary of a theorem by D.Cerveau
directly obtain a stronger result than (2.7) when is reduced. For the precise proofof the following proposition, which is originally due to T.Suwa, we refer to [Y].
$([\mathrm{C}])$$E$

Suppose $E(\subset_{M})$ is reduced
Proposition 2.9 ((STRONG) TANGENCY LEMMA)
and let $\{\varphi_{t}=\exp tv\}$ be the local
and is a point of M. Let be a germ in
1-parameter group of transformations induced by . For all sufficiently close to ,we have$E_{p}$

$v$

$p$

$t$

$v$

$(\varphi_{t})_{*}E_{p}=E\varphi t(p)$where$(\varphi_{t})_{*}$

denotes the differential map of$\varphi_{t}$$\mathit{0}$,.

We can check that proposition (2.9) is stronger than theorem (2.7) as follows. Takefor some . Then we have
. Supposea germ $v\in E_{p}$ and set$\varphi_{t}(p)\not\in S^{(k)}$$\varphi_{t}=\exp tv$$\dim E(p)\leq k<\dim E(\varphi_{t}(p))$

which contradicts proposition (2.9). So we haveto . Hence$t$,$\varphi_{t}(p)\in S^{(k)}$for all$t$

sufficiently close$0$$v(p)= \lim_{tarrow 0}\frac{\varphi_{t}(p)-p}{t}$is in the tangent cone3

$C_{p}S^{(k}$)

of$S^{(k)}$

at .
$p$

Main Results

Let $E$ be a singular foliation of rank on $M$ . We have already recalled thatinduces a non-singular foliation on $M-S(E)$ , so if a point $p\in M$ does not belong
to $S(E)$ , it is clear that there exists an integral submanifold (of dimension ) passing
through . As an application of theorem (2.7), we can show that there also exist
integral subm.anifolds on the singular set $S(E)$ , whose dimensions are lower than .
In order to prove the existence of the integral submanifolds on $S(E)$ , we have totake a stratification since the singular set $S(E)$ is not a smooth submanifold of $M$ ingeneral. However we must be careful in the choice of the stratification, because if wetake a stratification too much fine, then the space $E(p)$ is not always contained in thetangent space of the stratum at . As a “good” stratification of $S(E)$ , we adopt here
the famous method of the natural Whitney stratification which is due to H.Whitney.
For the generalities of the Whitney stratification, see [W].
$E$

$r$

$r$

$p$

$r$

$p$

77

Lemma 3.1Let
be a singular foliation on a complex manifold $M$ andan analytic subset of M. Suppose thatfor every point $p\in S(C_{p}S$denotes the tangent cone of at ). Let $S$ be the natural Whitney stratification of .
Then we have $E(p)\subset T_{p}X$ for every point $p\in S$ where $X(\in S)$ is the stratum passing
through .
$S$$E(\subset \mathrm{O}-_{M})$$E(p)\subset C_{p}Shold_{\mathit{8}}$

$S$

$S$

$p$

$p$

We can prove this lemma using theorem (2.7) and the way of construction of thenatural Whitney stratification. For the precise proof, we refer to [MY].
The following corollary is an immediate consequenceand (3.1).
$\mathrm{h}\mathrm{o}\mathrm{m}(2.7)$

Corollary 3.2
Let $E(\subset_{M})$ be a singular foliation of rank on a complex manithe natural Whitney stratification
fold M. Let be an integer with $0\leq k\leq r$ and$X\inS^{(k)}$and each point $p\in X$ we have $E(p)\subset T_{p}X$ .
of . Then for any stratum$r$$S^{(k)}$$k$$S^{(k)}$If we use this corollary, it is not so difficult to show the existence of the integral
manifolds on the singular set of a singular foliation $E$ .
Theorem 3.3 (EXISTENCE OF INTEGRAL SUBMANIFOLDS)
There exist integral
submanifolds (whose dimensions are lower than ) also on $S(E)$ . To be more precise,
there is a family of submanifolds of $M$ such that $M= \bigcup_{L\in c}L$ is a disjoint union
and that any
and $p\in L$ , we have $E(p)=T_{p}L$ .
$r$$\mathcal{L}$$L\in \mathcal{L}$

Proof. For each point $p\in M$ , take the unique integer such that $p\in L^{(k)}(=$$s^{(k)}-S(k-1))$ . Let
be the natural Whitney stratification of
and $X\in S^{(k)}$ the)unique stratum through . Since
is closed in $M,$ $X-S^{(k1}-$ ) has the structure of
a complex manifold. Corollary (3.2) implies that $E$ induces a non-singular foliation on$X-S^{(k}-1)$ (whose rank must be ). Therefore there exists a family
which consists))$X-S^{(k1}-$$X-S^{(k1}-$such thatof -dimensional complex submanifolds ofand $q\in L$ , we have $E(q)=T_{q}L$ . Then it isis a disjoint union and that any
obvious that$k$$S^{(k)}$$S^{(k)}$$s^{(k}-1$

$p$

$k$

$\mathcal{L}_{X}$$k$$= \bigcup_{L\in \mathcal{L}_{X}}L$$L\in \mathcal{L}_{X}$$\mathcal{L}=\bigcup_{k=0}^{r}X-s(k1)_{=\emptyset}X\in S^{(}k)\bigcup_{-}\mathcal{L}_{X}$is the family of submanifolds of $M$ which satisfies the conditions in the theorem.
Q.E.D.

Each element of is called a leaf ofThus, it turns out that $M$ is the disjoint union of the leaves of $E$ . Furthermore,
we can show that the structure of a singular foliation $E$ is locally analytically trivial
along the leaf at each point in $M$ . This claim can be expressed precisely as follows
in the case that $E$ is reduced.
$L$$\mathcal{L}$$E_{:}$$p$

78

Let $E(\subset_{M})$ be a reduced foTheorem 3.4 (LOCAL ANALYTICAL TRIVIALITY)
liation of rank on a complex manifold M. Let be an integer with $0\leq k\leq r$ and,a point in $L^{(k)}(=s^{(k)}-S(k-1))$ . Then there exist a neighborhood $D$ of ina singular foliation $E’$ on $D$ with $E’(0)=\{0\}$ , a neighborhoodof in $M$ and asubmersion : $U_{p}arrow D$ with $\pi(p)=0$ such that
$k$

$r$

$\mathrm{C}^{7\mathrm{P}k}$

$0$

$p$

$U_{p}$

$p$

$\pi$

$E|_{U_{\mathrm{p}}}=(\pi^{*}(E^{\prime^{a}}))a$This theorem is proved by taking a sufficient small neighborhoodofand. To be more concrete, if we take a small
constructing a good coordinates on, thenof and a good coordinatesoncoordinate neighborhood
is generated by the following $k+s$ vector fields:$U_{p}$$p$$U_{p}$$U_{p}$$(z_{1}, \ldots, z_{n})$$p$$U_{p}$$E|_{U_{p}}$$\frac{\partial}{\partial z_{1}},$$\frac{\partial}{\partial z_{2}}$, ...,$\frac{\partial}{\partial z_{k}}$,$v_{1}= \sum_{i=k+1}^{n}a^{1}i(z_{k1}+, \ldots, z_{n})\frac{\partial}{\partial z_{i}}$

(3.5)

,

. ..
$\cdot$$v_{s}= \sum_{i=k+1}^{n}a^{s_{i}}(Z_{k1}+, \ldots, z_{n})\frac{\partial}{\partial z_{i}}$,is a holomorphic function of $(n-k)$ -variables. We refer to [MY] forwhere eachthe precise proof of theorem (3.4).
$a_{i}^{j}$is generated by the $k+s$ vector fields of theRemark 3.6 The fact that
. From [MY] and [Y], we have anform (3.5) holds without assuming $E$ is reduced
the comments right
independent proof of this in the reduced case (see prop
before it).
$E|_{U_{p}}$$([\mathrm{C}])$$(2.9).\mathrm{a}\mathrm{n}\mathrm{d}$given in example (2.5).
Remark 3.7 Let us recall the singular foliation $E$ on
itself. Theorem (3.4), the leaf of passing through is
For any point of$E$ is locally analytically trivial at. On the other hand, if
along
tells us that$E$
$L^{(0)}-\{0\}$, the leaf ofpassing through consists of one
we consider a point ofcannot obtain any information from theorem (3.4) about the structure
point , soof singular foliation $E$ near . For the problem of the triviality along this type of
singular set, see [Y].
$\mathrm{C}^{3}$$p$$L^{(1)}$

$E$

$p$

$p$

$L^{(1)}$$L^{(1)}$

$q$

$q$

$\mathrm{w}\dot{\mathrm{e}}$

$q$

$q$

As an application of theorem (3.4), we can show the following proposition.
Proposition 3.8

If a singular foliation$E(\in \mathrm{O}-_{M})$is reduced, then$\mathrm{c}\mathrm{o}\dim S(E)\geq 2$.

79Remark 3.9 For the converse of this proposition, we have counterexamples. However, under the assumption that $E$ is locally free, the converse is also true (cf. remark (1.5) .
$)$. Set dimc $M=n$
Proof of (3.8). Suppose that is reduced and
$p\in
S(E)$
and
such thatand rankE $=r$ . First we choose a pointof and coordinates$\dim_{p}S(E)=n-1$ . Take a sufficiently small neighborhood$p=(0,\ldots, 0)$ . We setonsuch that $U\cap S(E)=\{z_{n}=0\}$ and, then clearly $0\leq k\leq r-1$ .
Next, choose a point in $U\cap S(E)$ such that $\dim_{\mathrm{C}}E(q)=k$ . For simplicity, we)
is a. Sinceon so that‘shift’ the coordinates. Thenof so thatclosed set, we can take a neighborhood
we have $U_{q}\cap S(E)=U_{q}\cap L^{(k)}$ . Applying theorem (3.4) (or (3.5)), we can retakeis generated by the following $k+S$ holomorphic vector
so that, ...,
andfields:$E$$\mathrm{c}\mathrm{o}\dim S(E)=1$$p\not\in \mathrm{S}\mathrm{i}\mathrm{n}\mathrm{g}(S(E))$

$U$

$p$

$U$

$(z_{1}, \ldots , z_{n})$$k= \max\{\mathrm{d}\mathrm{i}\mathrm{m}\mathrm{C}E(q)|q\in U\cap S(E)\}$

$q$

$U$

$(z_{1}, \ldots, z_{n})$$s^{(k}-1$$q=(\mathrm{O}, \ldots, 0)$$U_{q}(\subset U)$$U_{q}\cap S^{()}k-1=\emptyset$$q$$U_{q}$$z_{n})$$(z_{1}$$E|_{U_{q}}$$\partial$$\partial$$\partial z_{1}’\partial z_{2}’\ldots,$$\partial z_{k}$$\partial$$\underline{n}$1’$\backslash \partial$,$v_{1}= \sum_{i=k+1}ai(1Z_{k1}+, \ldots, z_{n})_{\overline{\partial z_{i}}}$(3.10),..
$\cdot$$v_{s}.= \sum_{i=k+1}^{n}a_{i}^{s}(zk+1, ..., z_{n})\frac{\partial}{\partial z_{i}}$.

If $s=0$ then $E$ gives a non-singular foliation on . This contradicts $q\in S(E)$ , so wehave $s\geq 1$ . On the other hand, $U_{q}\cap S(E)=U_{q}\cap L^{(k)}$ implies that $\dim_{\mathrm{C}}E(x)=k$
appearing in (3.10)
holds for every point $x\in U_{q}\cap S(E)$ , therefore allas$a_{i}^{j}(z_{k}+1, \ldots, z_{n-1},0)\equiv 0$ . For $i=k+1,$, we represent$U_{q}$$a_{i}^{j}$$\ldots,$$\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{S}}\mathrm{f}_{Y}$$a_{i}^{1}$$n$.

z_{n})$$a_{i}^{1}(Z_{k1}+, \ldots, z_{n})=Z_{ni}\alpha $b(Z_{k1}+’\ldots,i$are holomorphic functions such that
and
where
. We set
are uniquely determined andandNote thatbyondefine a holomorphic vector field$\tilde{v}_{1}$$U_{q}$$\tilde{v}_{1}=.\sum_{--,-}z_{n}-i\alpha b_{i}\alpha(n\downarrow \mathrm{t}Z_{k1}+’\ldots, z_{n})\frac{\partial}{\partial z_{i}}$$\sum_{i=k+1}^{n}z_{n}\alpha_{i}-\alpha bi(Z_{k1}+’\ldots, z_{n})\frac{\partial}{\partial z_{i}}(=\frac{1}{z_{n}^{\alpha}}v_{1)}$Then we haveis reduced.

$\alpha=\min\{\alpha_{i}\}$$\alpha_{i}\geq 1$$b_{i}$$\alpha_{i}$

.
, and

$b_{i}(z_{k+1}, \ldots , z_{n-1},0)\not\equiv 0$$b_{i}$$\alpha_{i}\in \mathrm{Z}$$\tilde{v}_{1}|_{U_{q}-}S(E)\in E|_{U_{q}-s}(E),$but$\cdot\tilde{v}_{1}\not\in E|_{U_{q}}$

since

.

$v_{1}^{-}\not\equiv 0$. This contradicts$\mathrm{t}\mathrm{h}\mathrm{a}\check{\mathrm{t}}E$

Q.E.D.

80

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[BB] P.Baum and R.Bott, Singula r.ities of holomorphic foliations, J. of Diff. Geom.
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[MY] Y.Mitera and J.Yoshizaki, The local analytical triviality of a complex analytic
singular foliation, preprint.
,

4[N] T.Nagano, Linear differential systems with singularities and application to transitive Lie algebras, J. Math. Soc. Japan 18, 1966.

[Ss] H.J.Sussmann, Orbits of families of vector fields and integrability of dist
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Math. Soc. 29, pp699-713, 1974.

$singula\dot{\mathcal{H}}ties$$r\cdot ibu-$, Proc. London[Sw] T.Suwa, Unfoldings of complex analytic foliations with singularities, JapaneseJ. of Math. 9, pp181-206, 1983.

[W] H.Whitney, Tangents to an analytic variety, Ann. of Math. 81, pp496-549,

1965.
[Y] J.Yoshizaki, On the structure of the singular set of a $c\dot{o}mpleX$ analytic foliation,
to appear in J. Math. Soc. Japan.

Junya Yoshizaki
Department of Mathematics, Hokkaido University, Sapporo 060, Japan-mail: j-yoshiz@math.sci.hokudai.ac.jp$\mathrm{e}$