数理解析研究所講究録1033 巻 1998 年 110-118

110

A localization lemma and its applications*Tomoaki Honda
Hokkaido University

Abstract
In this article, we give alternative proofs of two famous facts, thePoincare’-Hopf index theorem and the compatibility of two definitionsof the degree of a divisor on a compact Riemann surface, and define
a generalization of the tangential index [Br] and [Ho] and prove its
index theorem by the method of the localization of the Chern classof a virtual bundle.
tangential index and its index formula wasordinary defined and proved by M.Brunella [Br] for a curve and a singular foliation on a compact complex surface and the author reproved
it for a compact curve and a singular foliation on a complex surface[Ho].
$\mathrm{T}\mathrm{h}\dot{\mathrm{e}}$1Introduction

Let $X$ be amanifold of dimension $m$ and acomplex vector bundleof rank . We consider the Chern class $c(E)\in H^{*}(X;\mathrm{c})$ of $E$ . Note that
we use the complex number fieldas the coefficient of the cohomology
groups although in fact $c(E)$ itself is in, since we use the ChernWeil theory for the construction of Chern classes. If $E$ has a global section: $Xarrow E$ , which is not identically zero, we can make a frame, including, of the ristriction of $E$ to the complelnent of the zero set of . Therefore
the top Chern class can be localized to the neighborhood of the zero set of. This fact have many applications. In this article, we consider a simple
generalization of this fact.
Letbe an open covering of $X$ such that the vector bundle $E$
has a section
on an open set , which is not a zero section.
:
Assume that there exist non-vanishing functions
onsuch that$E$$C^{\infty}$$C^{\infty}$$n$$\mathrm{C}$$H^{*}(X;\mathrm{Z})$

$s$

$s$

$s$

$s$

$\mathcal{V}=\{V_{\alpha}\}$$s_{\alpha}$$V_{\alpha}arrow E$$V_{\alpha}$$f_{\alpha\beta}$$V_{\alpha}\cap V_{\beta}$*This article is based on the author’s talk “A vanishing lemma and some
RIMS. The title was changed.

$\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{i}_{\mathrm{C}\mathrm{e}}\mathrm{S}^{\text{ノ}j}$

at

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and the system
is a cocycle. We denote by $F$ the line bundle
defined by. Then we consider the Chern class of the virtual bundle$E-F$ . It is localized to the neighborhood of each connected component of
the union of the zero set of each . Then we can define the index of $E$ by$F$
and get its index formula.
In section 2, we consider a localization lemma and, as examples, thePoincar\’e-Hopf index formula and the compatibility of two definitions of thedegree of a divisor on a compact Riemann surface. Although the \v{C}ech-de
Rham cohomology theory and its integration theory play important roles inthis article, we refer to [BT], [Lehl], [Leh2], [LS] and [Su] for the details ofthese theories. In section 3, a generalization of the tangential index [Br] and[Ho] are defined and we prove its index formula. This index can be considered
to represent how a variety and a one dimentional singular foliation intersect,and it is a kind of indices relative not only to a singular foliation but also
to a variety. The tange.ntial index is defined by M.Brunella [Br] for a curveand a singular foliation on a compact complex surface. We generalize it fora variety and a dimension one singular foliation on $X$ .
The author would like to thank Tatsuo Suwa for many helpful comments
and suggestions.
$s_{\beta}=s_{\alpha}f\alpha\beta$$\{f_{\alpha\beta}\}$$\{f_{\alpha\beta}\}$$s_{\alpha}$2Localization lemma

Let $X$ be amanifold of dimension $m,$ $E$ a complex vector bundle of rank$n,$an open covering of $X$ and:asection,$E$$E$
of on each . We can assume thatis trivial on eachif necessary
taking a refinement of . Moreover we assume the following condition.
$C^{\infty}$$\mathcal{V}=\{V_{\alpha}\}_{\alpha\in A}$$V_{\alpha}arrow E$$s_{\alpha}$$V_{\alpha}$$C^{\infty}$$l_{\alpha}^{\gamma}$$\mathcal{V}$Assumption 2.1 For any
non-vanishing
functionand the system$C^{\infty}$$\alpha,$$f_{\alpha\beta}$$\{f_{\alpha\beta}\}$$V_{\alpha}\cap V_{\beta}$$\beta\in A$:

such that

there exsits asuch thaton$V_{\alpha}\cap V_{\beta}\neq\emptyset_{f}$$V_{\alpha}\cap V_{\beta}arrow \mathrm{C}^{*}$$s_{\beta}=s_{\alpha}f_{\alpha\beta}$

forms a cocycle.

We denote by $F$ the line bundle which is defined by this cocycle
Then there exists a bunble map : $Farrow E$ such that$\{f_{\alpha\beta}\}$

.

$f$

(1)

$f(F_{p})\subset E_{p}$for all $p\in E$(2) there exist a subset $S\subset X$ such that$f_{p}$:$F_{p}arrow E_{p}$is injective for$p\in X-^{s}$ .

We call the set of singularities of the line bundle $F$ . Let
be the decomposition to connected components. We assume that eachcompact. Take an open set
for each such that
and$S$

$S=\coprod_{\lambda\in\Lambda}\tau_{\lambda}$$U_{\lambda}$$\lambda$$U_{\lambda}\supset T_{\lambda}$$T_{\lambda}$is$U_{\lambda}\cap U_{\mu}=\emptyset$

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for
. Then, where $U_{0}=X-S$ , is an open coveringof $X$ .
We consider the \v{C}ech-de Rham cohomology groupassociatedwith this open covering . Note that this cohomology is isomorphic to the deRhamology (see [BT]). The n-th Chern class $c_{n}(E-F)$ of the virtual
bundle $E-F$ has a representativein the \v{C}ech-de Rham
cohomology groupof degree $2n$ , whereandare -closed
forms which are representatives of $c_{n}(E-F)$ on, respectively,
andin the de Rham cohomology group andis a $(2n-1)$ -form onsuch that. Note that we can construct
and
from
$F$
$E$
connections of and , using the Chern-Weil theory, and the \v{C}ech-de Rham
cohomology class represented by these forms is independent on the choice ofthe connections.
$\lambda\neq\mu$$\mathcal{U}=\{U_{0}, (U_{\lambda})_{\lambda\in\Lambda}\}$$H^{*}(A(\mathcal{U}))$$\mathcal{U}$$\mathrm{c}\mathrm{o}\mathrm{h}_{0}\dot{\mathrm{m}}$$(\sigma_{n}^{0}, (\sigma_{n}^{\lambda})_{\lambda},$$(\sigma_{n}^{0\lambda})_{\lambda})$$H^{2n}(A(\mathcal{U}))$$\sigma_{n}^{0}$$\sigma_{n}^{\lambda}$$U_{0}$$2\mathrm{n}$$U_{\lambda}$$\sigma_{n}^{0\lambda}$$U_{0}\cap U_{\lambda}$$d\sigma_{n}^{0\lambda}=\sigma_{n}^{\lambda}-\sigma_{n}^{0}$$\sigma_{n}^{0},$$\sigma_{n}^{\lambda}$$\sigma_{n}^{0\lambda}$

Lemma 2.2 (localization) Let: $H^{2n}(X, X-s;\mathrm{c})arrow H^{2n}(X;\mathrm{c})$ be
natural map. Then there exists $c\in H^{2n}(X, X-S;\mathrm{C})$ such that $j^{*}(c)=$$j^{*}$$c_{n}(E-F)$ .

Proof. Since

can be considered a subbundle of $E$ on , there exists
the decomposition $F\oplus E’$ of . The systemforms a frame of . Let
be a trivial connection of respect to the frame,a conncetion of$E’$ onand the curvature matrix of the connectionof$E$ on. Then $\sigma_{n}^{0}=\det k=0$ . We can construct
and
from
and
$E$
conncetions of and on .
Therefore the representative of $c_{n}(E-F)$ in the \v{C}ech-de Rham coholnology is. This is a 2 -cocyle in. the \v{C}ech-de Rhamrelative to $X-S$ . Let be a 2 -form on $X$ corresponding to .
Then $c=[\tau]\in H^{2n}(X, X-^{s};\mathrm{c})$ and $j^{*}(c)=c_{n}(E-F)$ .
$F$$U_{0}$$E$$F$$\{s_{\alpha}\}$$F$$\nabla_{0}^{F}$$U_{0}$$\nabla_{0}^{E’}$$\nabla_{0}=\nabla_{0}^{F}\oplus\nabla_{0}^{E’}$$\kappa$$U_{0}$$\sigma_{n}^{\lambda}$$F$$\sigma=(0, (\sigma_{n}^{\lambda})_{\lambda},$$\sigma_{n}^{0\lambda}$$\nabla_{0}$$U_{\lambda}$$(\sigma_{n}^{0\lambda})_{\lambda})$$\mathrm{C}\mathrm{O}\ln_{\mathrm{P}^{\mathrm{l}\mathrm{e}\mathrm{X}}}$$n$$\tau$$n$$\sigma$1

We denote $c\in H^{2n}$ ( $X,$ $X-S$ ; C) by $c_{n}(E;F)$ . This is a localization of$.c_{n}(E-F).$ .

If $X$ is compact, there exists following commutative diagram.
$H^{2n}(X, X-^{s};\mathrm{c})-^{A}H_{m-2n}(S;\mathrm{c})=\oplus\lambda\in\Lambda H-m2n(T\lambda;\mathrm{C})$$j^{*}\downarrow$$H^{2n}(X;\mathrm{c})$$\downarrow i_{*}$$arrow\wedge[X]$$H_{m-2n}(X;\mathrm{c})$,where is the Alexander duality, the natural inclusion and [X] the fundamental class of $X$ .
$A$$\dot{i}$113
Definition 2.3 We define an index $I(E.F;T_{\lambda})\in H_{m-2n}(T\lambda;\mathrm{C})$ of $E$ by
at
by
.

$F$

$T_{\lambda}$$A(c)=(I(E, F.\cdot \tau_{\lambda})’)_{\lambda\in\Lambda}$

Remark 2.4 We can define the index $I(E, F;T_{\lambda})$ if

is compact.

$S$

From the commutativity of the above diagram, we have following theorem.

Theorem 2.5 If $X$ is compact, we have$\sum_{\lambda\in\Lambda}\dot{i}_{*}I(E, F;\tau_{\lambda})=Cn(E-F)arrow[X]$.

In the rest of this section, we assume that $X$ is compact andconsists only of isolated points. Since eachconsists of a point
under thisassumption, we can take a sufficiently small open neighborhoodof .
Then we can assume eachis without loosing generalities. Hence thelocalized Chern class $c_{n}(E;F)$ has a representative. So it is important to write the $(2n-1)$ -form
explicitly. We have to mention the
Bochner-Martinelli kernel for the purpose of writing
clearly.
$S$$T_{\lambda}$$p_{\lambda}$$U_{\lambda}$$p_{\lambda}$$\sigma_{n}^{\lambda}$$0$$(0,0, (\sigma_{n}^{0\lambda})_{\lambda})$$\sigma_{n}^{0\lambda}$$\sigma_{n}^{0\lambda}$

Definition 2.6 We call following $(n, n-1)$ -form
Martinelli kemel;$\beta_{n}$on$\mathrm{C}^{n}$the Bochner-$\beta_{n}=c_{n^{\sum_{=1}^{n}(-1)^{i}}}i-1_{\frac{\overline{z}_{i}d\overline{z}_{1}\wedge\cdots\wedge d_{\sim i^{\wedge}}^{-}\wedge\wedge\ldots\wedge dz_{n}^{-}\wedge dZ_{1^{\wedge}}\cdots\wedge dzn}{||z||^{2n}}}$

,

where
$n(n-1) \frac{(n-1)!}{(2\pi\sqrt{-1})^{n}}$$C_{n}=(-1)\overline{2}$.

Remark 2.7 Let
be a $(2n-1)$ -sphere centered at the $or\dot{i}gin0$ .
Then the Bochner-Martinelli kemel
is real on $S^{2n-1}$ and a generator of
the cohomology group $H^{2n-1}(s^{2n-}1\mathrm{c};)$ :$S^{2n-1}\subset \mathrm{C}^{n}$$\beta_{n}$$\int_{S^{2n-1}}\beta n=1$

Theorem 2.8 Assume that$p_{\lambda}\in V_{\alpha}$

for some. . Then we have$\sigma_{n}^{0\lambda*}=-s_{\alpha}\beta_{n}$$\alpha$,To prove this theorem, the Chen-Weil theory and the integration along
the fiber are needed. Here the proof is omitted.

114
Hereafter we assume that $X$ is oriented. We introduce the integration on
the \v{C}ech-de Rham cohomology group.
for each .
be a closed neighborhood ofsuch thatLet.
andare oriented asandPutas the boundary ofsubmanifolds of $X$ for each anda system of honey-comb. We call a familycells adapted to the open covering .
Then we can define the integration on the \v{C}ech-de Rham cohomology
when $X$ is compact. For any agroup
associated with, we define the integration by$R_{\lambda}$$R_{0\lambda}=R_{0}\cap R_{\lambda}$$R_{0}=X- \bigcup_{\lambda\in\Lambda\lambda}\mathrm{i}\mathrm{n}\mathrm{t}R$$\lambda$$\lambda$$R_{\lambda}\subset U_{\lambda}$$p_{\lambda}$$R_{0}$$R_{\lambda}$$R_{0};R_{0\lambda}=\partial R_{0}=$$R_{0\lambda}$$\mathcal{R}=\{R_{0}, (R_{\lambda})_{\lambda}, (R_{0\lambda})_{\lambda}\}$$-\partial R_{\lambda}$$\mathcal{U}$$H^{m}(A(\mathcal{U}))$$=$$\mathcal{U}$$[(\sigma_{0}, (\sigma_{\lambda})_{\lambda}, (\sigma_{0\lambda})_{\lambda})]\in H^{m}(A(\mathcal{U}))$$\int_{X}\sigma=\int R\mathrm{o}\int\sigma 0+\sum_{\lambda\in\Lambda}R_{\lambda}\sigma\lambda+\sum_{\lambda\in\Lambda}\int_{R_{0\lambda}}\sigma 0\lambda$.

This definition is well-defined and compatible with the integration on thede Rham cohomology group;$\int_{X}\sigma=\int_{X}\tau$,where is a 2 -form on $X$ corresponding to a. See [Lehl], [Leh2], [LS] and[Su] for the details and more general definitions.
Then we describe examples.
$\tau$$n$Corollary 2.9 Let $C$ be a compact Riemann surface, $D=\{(U_{i}, f_{i})\}a$divisor on $C,$ $D’=\Sigma_{i=1}^{n}n_{ipi}$ the Weil divisor corresponding to $D$ .
Then$Cart_{\dot{i}}er$$\int_{c^{C_{1}([D}}])=\sum_{i=1}^{n}$ni,

where $[D]$ is the line bundle associated with $D$ .

Proof. We can assume that each point

andin $D’$ is contained inon eachsuch. There exists a coordinate
not contained in other, where $U_{0}=C-$. Thenthat $z_{i}(p_{i})=0$ and$\{p_{1},p2, \cdots,pn\}$ is an open covering of $C$ .
be a system of honeyLet
comb cell adapted to . Note that each is a section of $[D]$ on . From
theorem(2.5) and (2.8), we have$U_{i}$$p_{i}$$U_{j}$$z_{i}$$f_{i}(z_{i})=z_{i}^{n_{i}}$$U_{i}$$\mathcal{U}=\{U_{0)}(U_{i})_{i}\}$$\mathcal{R}$$\mathcal{U}$$\int_{C}c_{1}([D])$$f_{i}$$=$$\sum_{i=1}^{n}\int R_{0i}f-i^{*}\beta_{1}$$=$$\sum_{i=1}^{n}\frac{1}{2\pi\sqrt{-1}}\int_{S_{p^{1}}i}\frac{df_{i}}{f_{i}}=\sum_{=i1}^{n}\frac{1}{2\pi\sqrt{-1}}\int_{S^{1}}\frac{n_{i}dz}{z}p_{i}$$=$$\sum_{i=1}^{n}n_{i}$,$U_{i}$

115
where

is a 1-sphere in$S_{p_{i}}^{1}$

centerd at$\mathrm{C}$$p_{i}$and oriented naturally.
1

As the second example, we consider the Poincar\’e-Hopf index formula fora dimension one reduced singular foliation.

Definition 2.10 A dimension one singular foliation on a complex manisuch thatfold $X$ is determined by a triple$\mathcal{F}$$(\{V_{\alpha}\}, v_{\alpha’\alpha\beta}e)$is an open covering of $X$ and, for each
vector field on(1)$\{V_{\alpha}\}$$\alpha,$$v_{\alpha}$

is a holomorphic$V_{\alpha J}$(2) for each pairis a non-vanishing holomorphic function onwhich satisfies the cocyleon$(\alpha, \beta)_{f}e_{\alpha\beta}$$V_{\alpha^{\cap}}V_{\beta}$(3)$cond\dot{i}ti_{on,}e\alpha\gamma=e_{\alpha\beta}e_{\beta\gamma}$$v_{\beta}=v_{\alpha}e_{\alpha\beta}$on$V_{\alpha}\cap V_{\beta}$$V_{\alpha}\cap V_{\beta}\mathrm{n}V_{\gamma f}$.

The cocycle
defines a line bundle which is called the holomorphic tan$F$gent bundle of .
$\{e_{\alpha\beta}\}$Note that this definition is adapted to the assumption (2.1) if we regard
the holomorphic tangent bundle $TX$ as acomplex vector bundle .
The singular set of a foliation is defined similary. A dimension one singularfoliation is said to
reduceed if its singular set consists only of isolated$E$$C^{\infty}$$\mathrm{b}.\mathrm{e}$points.

Corollary 2.11 (Poincar\’e-Hopf) Let $X$ be a compact complex
ofcomplex dimensona reduced dimension one singular$F$foliation and a holomorphic tangent bundle of . Then we have$man\dot{i}f_{\mathit{0}}ld$$n_{f}\mathcal{F}=(\{V_{\alpha}\}, v_{\alpha’\alpha\beta}e)$$\mathcal{F}$-$\sum_{p\in S}PH(v,p)=\int Xc_{n}(\tau X-F)$

where
at .

$v$

$S$

is the singular set of$\mathcal{F}$,and $PH(v,p)$ is the Poincar\’e-Hopf index of$p$

Proof. Note that the Poincar\’e-Hopf index $PH(v,p)$ is written as$\int_{S_{p}^{2n-}}1v\beta_{n}*=\int_{R_{0\lambda}}\sigma_{n}^{0\lambda}$,

for some . Hence this fomula is an obvious corollary of the theorem(2.5)
and (2.8).
$\lambda$1116Remark 2.12 The-Hopf index formula is$or\dot{i}d_{\dot{i}}nalPo\dot{i}ncar\acute{e}$$\sum_{p\in S}PH(v,p)=x(X)$ ,where is a vector field on X. If there exists a global vector field with onlyisolated zero
, the tangent bundle $F$ isand we get the classical formula, using the fact $\int_{x^{C_{n}}}(X)=\chi(X)$ , from the theorem $(2.11)$ . This$v$

$po\dot{i}nts$$fo7mu\iota_{a}$$tr\dot{i}v\dot{i}al$

is a special case of the Baum-Bott residue theorem [BB].

Notesome other formulas, for example, the Riemann-Hurwitz formula, can be proved in this way.
$\dot{\mathrm{t}}\mathrm{h}\mathrm{a}\mathrm{t}$3Tangential index

Let $X$ be a complex manifold of dimension $n+k,$ $V\subset X$ a strong locallycomplete intersection (SLCI) of dimension (See [LS] for the definition of
SLCI),
an open vering of $X$ anda regular part of . Since is an SLCI, there exists acomplex vector
on a neighborhood of in $X$ such that the restriction
bundleisthe normal bundle
of $V’$ .
$n$$\mathcal{V}=\{V_{\alpha}\}$$\mathrm{c}\dot{\mathrm{o}}$$V$$V’=\mathrm{R}\mathrm{e}\mathrm{g}(V)=V-\mathrm{S}\mathrm{i}\mathrm{n}\mathrm{g}(V)$$V$$C^{\infty}$

$V$

$U$

$\tilde{N}$$\tilde{N}|_{V’}$$N_{V’}$

Assumption 3.1 There exists a
diagram$bun\dot{d}le$map$\tilde{\pi}$:$TX|_{U}arrow\tilde{N}$$0arrow TV’arrow TX|_{V^{J}}arrow\pi N_{V’}arrow 0$$i\downarrow$such that a(exact)$\downarrow i$$\tilde{N}$$TX|_{U}arrow\overline{\pi}$is commutative.

The above assumption(3.1) is satisfied, for example, when an SLCI
is defined by a holomorphic section of a holomorphic vector bundle$V=s^{-1}(0)$ . In this case, $E$ is isomorphic to.
be defining functions of on ;
Let
$s$

$V$

.

$E\text{ノ}$$\tilde{N}$$f_{1}^{\alpha},$$f_{2}^{\alpha},$$\cdots,$$V$$f_{k}^{\alpha}$$V_{\alpha}$$V\cap V\alpha=\{f_{1}\alpha=f_{2}^{\alpha}=\cdots=f_{k}^{\alpha}=0\}$We can take coordinates$\dot{i}=1,2,$, . Then$\cdots$$(x_{1}^{\alpha}, x_{2}^{\alpha}, \cdots , x_{n+k}^{\alpha})$on$V_{\alpha}$such that$k$$\pi\frac{\partial}{\partial x_{n+1}^{\alpha}},$$\pi\frac{\partial}{\partial x_{n+2}^{\alpha}},$$\cdots,$.

$\pi\frac{\partial}{\partial x_{n+k}^{\alpha}}$$x_{n+i}^{\alpha}=f_{i}^{\alpha}$

for

117
form a frame of $N_{V’}=(TX|_{V}J)/TV’$ . We can assume
is trivial on eachand there exists a frameof such that$\tilde{N}$$V_{\alpha}\cap U$$\tilde{N}$$\{e_{1}^{\alpha}, e_{2’ k}^{\alpha}\ldots, e^{\alpha}\}$$e_{i}^{\alpha}|_{V’}= \pi\frac{\partial}{\partial x_{n+i}^{\alpha}}$for each . This frame is said to be associated with.
Let
be a dimension one singular foliation and $F$ theholomorphic tangent bundle of .
$i$$\{f_{1}^{\alpha},$$f_{2’\cdot,f_{k}\}}\alpha..\alpha$$\mathcal{F}=\{(V_{\alpha}, v_{\alpha}, e_{\alpha\beta})\}$$\mathcal{F}$

Assumption 3.2 The SLCI is notby.
, whereis the ideal of holomorphic functions vanishing on
and generated by the defining functions of $V$ on .
$V$$\dot{i}nvar\cdot iant$$v_{\alpha}(f_{\alpha,i})\not\in I(V\cap$$\mathcal{F},\dot{i}.e$$I(V\cap V_{\alpha})$$V_{\alpha})$$V\cap V_{\alpha}$$V_{\alpha}$

Take a frame$\{e_{i}^{\alpha}\}$of$\tilde{N}$associated with$\{f_{i}^{\alpha}\}$. Then we get$\tilde{\pi}(v_{\alpha})|V=\sum_{=i1}kv\alpha(f\alpha,i)e_{i}^{\alpha}$.

Let$T_{\alpha}=\mathrm{S}\mathrm{i}\mathrm{n}\mathrm{g}V\cup\{p\in V’\cap V_{\alpha}|v_{\alpha}(p)\in T_{p}V’\}$.

Then we haveSo

$T= \bigcup_{\alpha}T_{\alpha}$Let$\tilde{\pi}(v_{\alpha})|_{V}$$=$$\tilde{\pi}(v_{\beta})|_{V\beta\alpha}e$$T_{\alpha}$$=$$\{p\in V\cap V_{\alpha}|\tilde{\pi}(v_{\alpha})(p)=0\}$.

is the set of tangential points of $F$ and $V$ .
is well-defined.
be the decomposition to connected components and weis compact. Then there exists generalized tangential index.
$T$$T=\coprod_{\lambda\in\Lambda}T_{\lambda}$

assume that

$T$

Theorem 3.3 (tangential index) The$7’ e$

exists index$I(N, F;T_{\lambda})\in H_{2(n-k})(\tau_{\lambda;}\mathrm{c})$

of

$N$

by $F$ at$T_{\lambda}$. Moreover if$V$is compact,$\sum_{\lambda\in\Lambda}\dot{i}_{*}I(N, F;T_{\lambda})=ck(N-F)-[V]$Theorem 3.4 If $n=k$ and$T$consists only of isolated points, then$I(N, F;p)= \int_{L}(\pi(v))^{*}\beta_{k}$where $p\in T$ and$L$is a link of $V$ at$p$,with a usual$or\dot{i}entat\dot{i}on_{f}\cdot L=\{f_{1}=$$f_{2}=\cdots=f_{k}=0,$ $|v(f_{1})|^{2}+|v(f_{2})|^{2}+\cdots+|v(f_{k})|^{2}=\epsilon\}$small$\epsilon>0$

for a sufficientry

and $d\arg v(f1)\wedge d\arg v(f_{2})\wedge\cdots\wedge d\arg v(f_{k})>0$.

118
These two theorems are corollaries of theorem(2.5) and (2.8), respectively.
Apply these theorems to a virtual bundle
and .
This index can be considered to represent how tangent a variety and on
dimensional singular foliation. If $n=k=1$ then we have$\tilde{N}-F$$I(N, F;p)= \frac{1}{2\pi\sqrt{-1}}\int_{L}\frac{dv(f)}{v(f)}$$V$.

This coincides with an intersection number $(v(f), f)_{p}$ at5) and the original tangential index [Br] and [Ho].

$p$(See [GH] Chapter

References[BB]

P. Baum and R. Bott, Singularities of holomorphic foliations, J. Differential Geom. 7(1972), 279-342[Br]

M. Brunella, Feuilletage holomo 7phes sur les surfaces complexes compactes, preprint.

[BT]

R. Bott and L. W. Tu, Differntial fo7mS in algebraic Topology, Graduate text in mathematics 82, Springer-Verlag, New York, Heidelberg,
Berlin, 1982.

[GH]

P. Griffiths and J. Harris,
Wiley&Sons, 1978.

$P_{\dot{\mathcal{H}}nCi}pleS$[Lehl] D. Lehmann, Integration sur lesParis 307, S\’er.I (1988), 603-606.

of Algebraic Geometw, John$var\dot{i}\acute{e}t\acute{e}sstrat_{ifi\acute{e}}es$, C.R.Acade.Sci.

[Leh2] D. Lehmann,de \v{C}ech-de Rhamet th\’eory de Chem- Wei, Geometry and Topology of Submanifolds II,
Proc. Conf. May 30-June 3, 1988, Avignon, France, World Scientific,
Singapore, 1990, 205-248.
$Var\dot{i}\acute{e}t\acute{e}SStrat_{\dot{i}fi\acute{e}e}sc\infty:Int\acute{e}grat\dot{i}on$[LS]

D. Lehmann and T.Suwa, Resdues of holomorphic vector fields relative
to singular invariant, J.Differntial Geom. 42(1995), 165-192.
$var\dot{i}et_{\dot{i}e}s$[Ho]

T. Honda, Tangential index of foliations with curves on surfaces,
preprint

T. Suwa, Residues of complex analytic foliations relative to singulainvawiant subvarieties, Studies in Advanced Mathematics, 5, .1997,230-245.
Department of Mathematics, Hokkaido University, Sapporo 060, JAPAN.
-mail address : t-honda@math.sci.hokudai.ac.jp[Su]$\mathrm{e}$