数理解析研究所講究録
第 1764 巻 2011 年 71-79

71

Hypersurfaces and Polar curves
by L\^e Dung Tr\’ang

Introduction
Let :be a non-constant germ of complex analytic function. From the book of
J. Milnor [11] it is known that:$(\mathbb{C}^{n+1},0)arrow(\mathbb{C}, 0)$$f$

Theorem 0.1 There issuch that, for any$\varphi_{\epsilon}(z)=f(z)/|f(z)|$ on the sphere centeredbyatdefined$\epsilon_{0}>0_{f}$$\epsilon,$, the map:with radius , is a locally trivial$0<\epsilon\leq\epsilon_{0}$$0$fibration.

In this theorem, J. Milnor does not assume that the function$\varphi_{\epsilon}$$S_{\epsilon}\backslash \{f=0\}arrow S^{1}$$\epsilon$$f$$C^{\infty}$

has isolated singularities.

The following theorem is consequence of [5] (Theorem 1.2.1):
Theorem 0.2 There is, such that, for any, there is $\eta_{0}>0$ , such that,:on theinduced byintersection of the open ball centered at with radius with the inverse image of the punctured
open disc, is a locally trivial
fibration.

for any$\eta,$$0<\eta\leq\eta_{0}$$\epsilon_{0}>0$, the map$\varphi_{\epsilon,\eta}$$(B_{\epsilon}\backslash \{f=0\})\cap f^{-1}(D_{\eta}^{*})arrow D_{\eta}^{*}$$0$$D_{\eta}^{*}$$0<\epsilon\leq\epsilon_{0}$$\epsilon,$$f$$\epsilon$$C^{\infty}$$:=D_{\eta}\backslash \{0\}$The proof of 0.2 uses that, locally, the special fiber $f^{-1}(0)$ can be stratified with Thom condition.
Namely, for $1\gg\epsilon>0$ , there is a stratification of , such thatis a union of strata andfor any sequencewhich converges tosuch that the sequence of$tangspT_{x_{\gamma\iota}}(f^{-1}(f(x_{n})))$ ha.s a limit, this limit contains the tangent space of the stratum
which contains the point .
$f^{-1}(0)\cap B_{\epsilon}$$B_{\epsilon}$$x_{7b}\in B_{\epsilon}\backslash f^{-1}(0)$$x\in f^{-1}(0)\cap B_{\epsilon}$

$T$

$x$

The existence of such a stratification with Thom condition was proved by using Lojasiewicz inequality in [5], Theorem 1.2.1.

Theorem 5.11 of [11] proves that, for $1\gg\epsilon\gg|t|>0$ a fiber of andare diffeomorphic,andis diffeomorphic to. Having Theorem 0.2, we know that:induced by is a locally trivial
fibration. The fibrations
andhave diffeomorphic total spaces and diffeomorphic fibers, so they must be isomorphic.
We shall call Milnor fiber of at a fiber of any of these two fibrations.
$\varphi_{\epsilon}$$S_{\epsilon}\backslash \{f=0\}$$\partial\varphi_{\epsilon,\eta}$$(B_{\epsilon}\backslash \{f=0\})\cap f^{-1}(S_{\eta})$$(B_{\epsilon}\backslash \{f=0\})\cap f^{-1}(S_{\eta})arrow S_{\eta}$$\partial\varphi_{\epsilon,\eta}$$\varphi_{\epsilon}$$f$

1$f^{-1}(t)\cap B_{\epsilon}$$0$

Basic results

In [7] the following theorem is proved:$f$$c\infty$

72:
be a non-constant germ of complex analytic function.
$H$through , then the manifold
Let $1\gg\epsilon\gg|t|>0$ , and let
be a geneml hyperplane ofinby adding
is obtained from a tubular neighbourhood ofhandles of index .

Theorem 1.1 Let$(\mathbb{C}^{n+1},0)arrow(\mathbb{C}, 0)$$f$$\mathbb{C}^{n+1}$$0$$f^{-1}(t)\cap B_{\epsilon}\cap H$$f^{-1}(t)\cap B_{\epsilon}$$f^{-1}(t)\cap B_{\epsilon}$$n$

The proof of Theorem 1.1 uses Smale interpretation of Morse theory by adding handles.
$s$

The key construction consists in introducing the Polar curve of relatively to a general linear
function . Namely, let be a linear function. We have a germ of map:
$f$

$p$

$\ell$$\Phi=(l, f):(\mathbb{C}^{n+1},0)arrow(\mathbb{C}^{2},0)$.

The critical locus
of depends linearly on the linear form . A theorem of Bertini tells us thatof this linear system lie in the fixed points of thisthe singular points of the general element
system. Locally near , the fixed points are precisely the critical points of the function . For ageneral , the points ofare non-singular. Therefore, for a general , either
empty. We callisor it is non-singular of dimension 1. We denote by the closurc ofa relative polar curve of at .
$\Phi$$C_{\ell}$$\ell$$C\ell$$0$$f$$C\ell\backslash \{f=0\}$$C_{\ell}$$C\ell\backslash \{f=0\}$$\Gamma\ell$$f$$\Gamma_{\ell}$$0$

We can also define the relative polar curve of
into
for $1\gg\epsilon>0$ , such that:$B_{\epsilon}\backslash C(f)$

$f$

at

$0$

in the following way. Let$\nu f$be the map from$\gamma_{f}$$\mathbb{P}^{n}$$\gamma_{f}(x_{0}, \ldots, x_{n})=(\partial f/\partial z_{0}(x_{0}, \ldots, x_{n}) :...

:
Let
derivatives of$C\ell\backslash \{f=0\}$$\ell$

be the blowing-up in. The map
extends as
is a polar curve of at .

$B_{\epsilon}arrow B_{\epsilon}$$f$$\nu f(\tilde{\gamma}_{f}^{-1}(\ell))=\Gamma_{\ell}$$B_{\epsilon}$$\gamma_{f}$.

:\partial f/\partial z_{n}(x_{0}, \ldots,x_{n}))$of the Jacobian ideal of generated by the partial. Let be a general line in , then:$f$$\ell$$B_{\epsilon}arrow \mathbb{P}^{n}$$\tilde{\gamma}_{f}$$\mathbb{P}^{n}$

$0$

$f$

The rclative polar curvc of at can be empty. For instance, it is so, when is an analytic
product. When the critical locus of is locally at a non-singular curve , one can show that therelative polar curve of at is empty if and only if the hypersurface $f^{-1}(0)$ has Milnor number
constant along .
$f$

$0$

$f$

$0$

$f$

$\Sigma$$0$$f$$\Sigma$

Noticc that the critical points of the restriction of the function$|l|$to the difference of spaces,$f^{-1}(t)\cap B_{\epsilon}\backslash f^{-1}(t)\cap B_{\epsilon}\cap H$

where $H=\{\ell=0\}$ , are the points of the intersection$r_{f}\cap f^{-1}(t)\cap B_{\epsilon}$.

By taking the cone on a projective hypersurface in , Theorem 1.1 implies a Theorem of Zariski
stated in [12], which has correct proofs in [2] or in [5]:$\Delta$$\mathbb{P}^{n}$inCorollary 1.2 The fundamental group of the complement of the projective hypersurface$P$
is isomorphic to the fundamental group of the complement in a general planeof the intersection
with $P$ of the projective hypersurface .
$\Delta$$\mathbb{P}^{n}$$\triangle$

Proof. First, we observe that the Milnor fiber of the reduced cone A over the hypersurface ,defined by a homogeneous polynomial of degree , is diffeomorphic to $F=f^{-1}(1)$ , which is an.
abelian covering of degree of the complement
$F$

$f$

$d$

$\triangle$$d$$\mathbb{P}^{n}\backslash \triangle$

We have an exact sequence:$1arrow\pi_{1}(F, *)arrow\pi_{1}(\mathbb{P}^{n}\backslash \triangle, *)arrow Z/dZarrow 0$.

73
Theorem 1.1 gives that, if $H$ is a general projective hyperplane, the inclusion
into F. whereis the hyperplane above $H$ , induces an isomorphism, for $0\leq k\leq n-2$ .
Therefore, if is a general projective plane, the cone
above is a 3-dimensional affine spaceand.
$F\cap\tilde{H}$$\tilde{H}$$\pi k(F\cap\tilde{H}, *)\simeq\pi_{k}(F, *)$

$P$

$P$

$\tilde{P}$$\pi_{1}(F\cap\tilde{P}, *)\simeq\pi_{1}(F\cap\tilde{P}, *)\simeq\pi_{1}(F, *)\simeq\pi_{1}(F, *)$The space$F\cap\overline{P}$is an abelian covering of degree

$d$

of

$P\backslash \Delta$. We have the exact sequence:$1arrow\pi_{1}(F\cap\overline{P}, *)arrow\pi_{1}(P\backslash \triangle, *)arrow Z/dZarrow 0$.

Comparing this last exact sequence with the preceding one by using the inclusion maps, Theorem
1.1 gives a proof of our corollary.
Theorem 1.1 provides another proof of the Theorem of Zariski of Lefschetz type [12], stated incorollary 1.2. Using our results in [4], we shall extend this Theorem of Zariski to a more general
setting.

2Stratified Polar curve

Let

$f$

: $(X, 0)arrow(\mathbb{C}, 0)$ be a germ of non-constant complex analytic function on a reduced space$X\subset \mathbb{C}^{N}$.

A theorem of H. Hironaka in [6] shows that, for $1\gg\epsilon>0$ , one can stratify
in such a way that, for $1\gg\epsilon\gg\eta>0$ , the map:$\phi_{\epsilon,\eta}$$X\cap B_{\epsilon}$$X\cap B_{\epsilon}\cap f^{-1}(D_{\eta})arrow D_{\eta}$\‘and $f^{-1}(0)\cap$ Be

induced by

be stratified with Thom condition.

$f$

can

As it is done in [8], a fibration theorem analogous to Theorem 0.2 above can be proved:$0<e\leq e_{0}$ , there is $\eta 0>0$ , such that,
Theorem 2.1 There is, such that, for any
forany, the map:
induced by on the intersection of
the open ball centered at with radius with the inverse image in $X$ of the punctured open disc, is a locally trivial
fibration.
$\epsilon_{0}>0$$\eta,$$0<?

\int\leq\gamma)0$

$e,$

$0$

$D_{\eta}^{*}$$f$$B_{\epsilon}\cap X\cap f^{-1}(D_{\eta}^{*})arrow D_{\eta}^{*}$$\varphi_{\epsilon,\eta}$$\epsilon$$C^{0}$$:=D_{\eta}\backslash \{0\}$One notices that the statement is somehow the same as the one of Theorem 0.2, but now thefibration is , because $X$ might be singular.
$C^{0}$

Let be a linear form of the ambient spaceFirst one observes:$\ell$$\mathbb{C}^{N}$. We have a germ of map $\Phi=(\ell, f)$ :$(X, 0)arrow(\mathbb{C}^{2},0)$.

Lemma 2.2 There is a Zariski dense open space of linear forms, such that, for any, thereare an open neighbourhood of in $X$ and an open neighbourhood of in , such thatinduces a stratified map from onto which satisfies Thom condition.
$\ell\in\Omega$$\Omega$

$U$

$V$

$0$

$U$

$0$

$\mathbb{C}^{2}$$\Phi$$V$

Proof. To prove this] $emma$ , consider $1\gg\epsilon\gg\eta>0$ , such that Theorem 2.1 holds. Let us stratify
the map:$\varphi_{\epsilon,\eta}:B_{\epsilon}\cap X\cap f^{-1}(D_{\eta})arrow D_{\eta}$

such thatis a union of strata, the stratification ofsatisfies$\{\ell=0\}$
Whitney condition and Thom condition for . Consider a linear form ofissuch thattransverse in
to all the strata of
except maybe at the point.
$B_{\epsilon}\cap X\cap f^{-1}(0)$$B_{\epsilon}\cap X\cap f^{-1}(D_{\eta})$$f$$\mathbb{C}^{N}$$B_{\epsilon}\cap X\cap f^{-1}(D_{\eta})$$l$$\mathbb{C}^{N}$$\{0\}$

74
We can see that by choosing the linear form in this way, the map induces a Thom map for someneighbourhoods and . Since there are only a finite number of strata having in its closure, it
is enough to choose a linear form such that $\{P=0\}$ is transverse except at to a finite numberof strata. i.e. a linear form lying in the intersection of Zariski dense open sets of linear forms.
$\Phi$

$U$

$V$

$0$

$0$

$\ell$

Now, we can prove:

Proposition 2.3 Let $1\gg\epsilon\gg\eta>0$ . Let us stmtify :with Whitney
,
and
the
stratanumber
Let
be
in
Thom
condition.
ofstratifications
finitewhich contain in their closure. There is a Zariski open dense set S) in the space of linear forms, the restriction oftohave an empty or non-singularof , such that, forcnttical locus
1.
dimensionof$f$$S_{\alpha},$$B_{\epsilon}\cap X\cap f^{-1}(D_{\eta})arrow D_{\eta}$$\alpha\in A$$B_{\epsilon}\cap X\cap f^{-1}(D_{\eta})$$0$$C^{N}$$\ell\in\Omega$$(\ell, f)$$S_{\alpha}\backslash f^{-1}(0)$$C_{\alpha}$

Proof. Consider a stratum , such that the closure.
contains . Let be a linear form of
The critical locus
to the closure
is an analytic subset ofof the restriction ofwhich depends linearly on , so, using the same theorem of Bertini as above, for general, i.e. forin a Zariski dense open set, the singular points ofof the space of linear forms oflie inTherefore, for general,the union ofand the singular point ofis eit,her empty or non-singular of dimension 1.
$0$$\overline{S}_{\alpha}$$S_{\alpha}$$(\ell, f)$$C_{\alpha}(\ell)$$\mathbb{C}^{N}$$\ell$$\tilde{S}_{\alpha}$$\overline{S}_{\alpha}$$p$$\ell$$\ell$$\mathbb{C}^{N}$$C_{\alpha}(\ell)$$\Omega_{\alpha}$$f^{-1}(0)\cap\overline{S}_{\alpha}$$S_{\alpha}$,$C_{\alpha}(\ell)\cap(S_{\alpha}\backslash f^{-1}(0))$$\ell$, otherwise has maximal rank on
Since is stratified, eitheror, in which case, for all except at most one,
and, for a general ,all the strata
exceptthe critical locus ofis empty,
for the stratum of dimension 1 which
restricted tocontains , and in this last case the critical locus of the restriction of
to this l-dimensional
stratum is the whole stratum itself.
$S_{\alpha}\cap f^{-1}(0)=\emptyset$$S_{\alpha}\subset f^{-1}(0)$$f$$(S_{\alpha})$$\dim S_{\alpha}\geq 2$$\alpha$$(\ell, f)$$f$$\ell$$S_{\alpha}$$(\ell, f)$$0$. We have just seen that, for general,
We may suppose that, eitherorthe critical spacetois empty or non-singular of dimensionof the restriction of
1. The Zariski open dense set of the proposition is the finite intersection.
$S_{\alpha}\subset f^{-1}(0)$$S_{\alpha}\cap f^{-1}(0)=\emptyset$$(\ell, f)$$C_{\alpha}$$\ell$$S_{\alpha}\backslash f^{-1}(0)$$\Omega$$\bigcap_{\alpha\in A}\Omega_{\alpha}$

Definition 2.4 The relative polar curve of relatively to a general linear form ofis the
polar
.
union
shall
call
a
associated
to
the
We
relative
curve
closuresfiniteof.
the stratum
$p$

$f$

$\bigcup_{\alpha\in}{}_{A}\overline{C}_{\alpha}$$\Gamma_{\alpha}$$\mathbb{C}^{N}$$:=\overline{C}_{\alpha}$$S_{\alpha}$

Recall that, as above,$C_{\alpha}$

and also$\Gamma_{\alpha}$

can be empty.

In [8], we have introduced this notion in the particular caseto the germ $(X, 0)$ .
linear form of$f$

was itself the restriction of a general$\mathbb{C}^{N}$

Notice that, since is stratified, for small enough, the intersections $\{f=t\}\cap S_{\alpha}$ give a Whitney
are precisely the points ofstratification of the Milnor fiber $\{f=t\}$ . The points of$\{f=t\}\cap
S_{\alpha}$
where the restriction of tohas ordinary quadratic points.
$f$

$t$

$\Gamma_{\alpha}\cap\{f=t\}$$S_{\alpha}$$\ell$

3

Singular complement of projective hypersurfaces

and let $Y\subset X$ a
Let $X$ be a projective variety embedded in the complex projective space
projective $h_{-}ypersurface$ defined in $X$ by one homogeneous equation $f=0$ . To these data correspond.
the cones
embedded in$\mathbb{P}^{N-1}$$X,\tilde{Y}$$\mathbb{C}^{N}$

75
On the punctured cone
, we have a natural action of the punctured group of non-zerocomplex numbers. Consider the fiber $F=f^{-1}(1)\cap\overline{X}$ . Let be the natural map from$X$
onto . Let $y\in X\backslash Y$ . The inverse imageis the punctured line .
$\overline{X}\backslash \{0\}$$\mathbb{C}^{*}$$\overline{X}\backslash \{0\}$$\sigma$$\sigma^{-1}(y)$$\overline{y}^{*}$

Assume that is the restriction to $X$ of a homogeneous polynomial ofof degree . Theintersection
consists of points, where is a primitive d-th root of unity
andis one point ofF.
$f$$\mathbb{P}^{N-1}$$\sigma^{-1}(y)\cap F$$d$$\{\xi^{k}yo\}_{0\leq k\leq d-1}$

$d$

$\xi$

$\sigma^{-1}(y)\cap$$y0$Suppose that the fiber of :is connected, it is a topological covering ofof degree. Let $H$ be a projective hyperplane of$(X\backslashY)\capH$ , we. If we want to compareandcan do it by comparing and, whereis the hyperplane lying above $H$ in
.
$F$

$f$

$\overline{X}arrow \mathbb{C}$$X\backslash Y$$d$$\mathbb{P}^{N-1}$$F\cap\tilde{H}$$F$

In this situation$F$$X\backslash Y$$\tilde{H}$$\mathbb{C}^{N}$is a singular space.

Let us define the notion of rectified homotopical depth introduced by Grothendieck and understoodas in [4] (Theorem 4.1.2).

Definition 3.1 Let $X$ be a reduced complex analytic space. Let
be an analytic Whitney,X.
For eachletbe a complex link of . We say that the rectified
stratification ofhomotopical depth $rhd(X)$ of $X$ $is\geq r$ if, for each, the complex linkis $(r-\dim S_{\alpha}-2)-$$(S_{\alpha})_{\alpha\in A}$$\alpha\in A$$S_{\alpha}$$\mathcal{L}_{\alpha}$$\alpha\in A$$\mathcal{L}_{\alpha}$connected.

Remind that the complex link of a stratumof a Whitney stratification is obtained in the following
way. Let. Assume that (X, x) is locally embedded in. Let
be a slice ofat ,i.e. the intersection of $X$ with a complex submanifold ofof codimension
and transverseintoat . A complex link ofis the Milnor fiber at of a general linear form of.
restricted to$S_{\alpha}$$x\in S_{\alpha}$$\mathbb{C}^{N}$$\mathbb{C}^{N}$$\mathbb{C}^{N}$$S_{\alpha}$$x$$S_{\alpha}$$\mathcal{N}_{\alpha}$$S_{\alpha}$$x$$\dim S_{\alpha}$$\mathbb{C}^{N}$$x$$\mathcal{N}_{\alpha}$

One can prove that all complex links of a given stratum of a Whitney stratification are homeomorphic (see e.g. \S 2.3 Part II Chapter 2),
In [4] we indicate that, if $X$ is equidimensional, the dimension of $X$ bounds the rectified homotopical
depth.

We shall show:
Theorem 3.2 Let assume that the rectified homotopical depth of the projective variety $X$ is $\dim X$
and let $Y\subset X$ a projective subset defined in $X$ by one reduced homogeneous equation $f=0$ . For ageneral hyperplane $H$ the inclusion ofgives an isomorphism of the fundamental
in$(X\backslash Y)\cap H$

groups$\pi_{1}((X\backslash Y)\cap H, *)\simeq\pi_{1}(X\backslash Y, *)$for$X\backslash Y$$\dim X\geq 3$.

The proof of this theorem proceeds like what we have done for the Theorem 1.2 of Zariski ofLefschetz type ([12]). We first prove a local theorem:
Theorem 3.3 Let (X, x) a germ of reduced equidimensional germ of complex analytic space em. Let
bedded in$(\mathbb{C}^{N}, 0)$$f:(X, x)arrow(\mathbb{C}, 0)$

be a germ of non-constant complex analytic function on $(X, x)$ . Assume thathas rectified
homotopical depth. Let be a Milnor fiber of at . For a geneml hyperplane $H$ containing, the relative homotopy $\pi k(F, F\cap H, *)=0$ if $k\leq r-2$ .
$X\backslash f^{-1}(0)$$\geq r$

$x$

$F$

$f$

$0$

76
Proof of Theorem 3.3. As we have mentioned above, we are going to use our results in [4].
$H$ $:=\{\ell=0\}$ the general hyperplane that it defines, andLet be a general linear form of$:=H\cap X$ . Call $Z=f^{-1}(0)\cap X$ . Denote by
the sphere centered at with radius .
by adding
is obtained from @ (x)Theorem 2.2 of [5] asserts that
$\geq
r-1$
, where:cells of dimension$\mathbb{C}^{N},$$\ell$$Y$$x$$S_{\epsilon}(x)$$\epsilon$$\cap(X\backslash Z)\cap V_{\alpha}(Y)$$S_{\epsilon}(x)\cap(X\backslash Z)$$\epsilon$$V_{\alpha}(Y)=\{z\in X||l(z)|\leq\alpha\}$with $\alpha>0$ small enough.

Since $H$ is a gcneral hyperplane, one c\‘an show that the spaceand has the same homotopy type.
the space$S_{\epsilon}(x)\cap(X\backslash Z)\cap V_{\alpha}(Y)$retracts on$S_{\epsilon}(x)\cap(X\backslash Z)\cap Y$

has the same homotopy
The local conic theorem (see [1] Lemma 3.2) implies that,type ashas the same homotopy type as
andboundary.
havingasis the ball ofwhere$S_{\epsilon}(x)\cap(X\backslash Z)\cap Y$$B_{\epsilon}(x)\cap(X\backslash Z)\cap Y$$B_{\epsilon}(x)\cap(X\backslash Z)$$S_{\epsilon}(x)\cap(X\backslash Z)$$\mathbb{C}^{N}$$B_{\epsilon}(x)$$S_{\epsilon}(x)$Adapted to the stratified situation, the Lemma 5.9 of [11] constructs a vector field which gives a, for $1\gg\epsilon\gg\eta>0$ .
ontostratified homeomorphism of$B_{\epsilon}(x)\cap f^{-1}(D_{\eta})\cap(X\backslash Z)$$B_{\epsilon}(x)\cap(X\backslash Z)$

Now, using the fibration theorem 2.1, one can show thatand the Milnor fiber have the same homotopy type for dimensions$B_{\epsilon}(x)\cap f^{-1}(D_{\eta})\cap(X\backslash Z)\simeq B_{\epsilon}(x)\cap(X\backslash Z)$$F$$\neq 1$has:
$1arrow\pi_{1}(F, *)arrow\pi_{1}(B_{\epsilon}(x)\cap(X\backslash Z), *)arrow Zarrow 0$. For dimension 1, one.

In the hyperplane section by $H$ we have:$1arrow\pi_{1}(F\cap H, *)arrow\pi_{1}(B_{\epsilon}(x)\cap(X\backslash Z)\cap Y, *)arrow Zarrow 0$.

Onc finds that this implies, if $r\geq 3$ , the relative homotopy $\pi_{k}(F, F\cap H, *)=0$ , for $k\leq r-2$ .
Proof of Theorem 3.2. We can apply the Theorem 3.3 to obtain Theorem 3.2. We have supposedover $X$ hasthat the rectified homotopy depth of $X$ equals $\dim X$ . Let us show that the cone
rectified homotopical depth equal to $\dim X=\dim X+1$ .
$\overline{X}$, so that is a union of strata. First, notice that the punctured cones
Let us stratify $X$ by, this. By adding the trivial stratum
give a stratification of the punctured cone
.
of
the
cone
Whitney
defines astratification$Y$$(S_{\alpha})_{\alpha\in A}$$\tilde{X}\backslash \{0\}$$\tilde{S}_{\alpha}\backslash \{0\}$$\{0\}$$\overline{X}$is the Milnor fiber of the restriction of ain
The complex link of the punctured conein X. lt is clear that complex link of the punctured cone
general linear function to a slice ofin $X$ . Since it is $(\dim X-\dim S_{\alpha}-2)_{-}-$
inis homeomorphic to the conlplex link of$(\dimX+1-(\dimS_{\alpha}+1)-2)$
connected, it is equivalently-connected. So, the difference
has rectified homotopical depth $\dim X+1$ .
$\tilde{X}$$S_{\alpha}\backslash \{0\}$$S_{\alpha}\backslash \{0\}$$\tilde{S}_{\alpha}\backslash \{0\}$$\overline{X}$$S_{\alpha}$$\overline{X}\backslash Y$

Let $F$ be the Milnor fiber of the restriction of
hypcrplane:of$\overline{H}$

$f$

to

$\overline{X}$. The Theorem 3.3 implies that, for a general$\mathbb{C}^{N}$$\pi_{k}(F\cap\tilde{H}, *)\simeq\pi_{k}(F, *)$when $k\leq\dim\tilde{X}-3$ . Therefore:$\pi_{1}(F\cap\overline{H}, *)\simeq\pi_{1}(F, *)$when $1\leq\dim X-2$ or $\dim X\geq 3$ .

77Since the cone
is homogeneous, the Milnor fiber
and $F\cap H$ is homeomorphic to $F\cap H$ . Therefore:$\tilde{X}$$F$

is homeomorphic to the fiber $F=f^{-1}(1)$ ,$\pi_{1}(F\cap\tilde{H}, *)\simeq\pi_{1}(F, *)$when $\dim X\geq 3$ .
Using the covering$Farrow X\backslash Y$, we obtain:$\pi_{1}((X\backslash Y)\cap H, *)\simeq\pi_{1}((X\backslash Y), *)$when $\dim X\geq 3$ and where $H$ is the general hyperplane of$\mathbb{P}^{N-1}$

defined by$\tilde{H}$.

Following the type of arguments given above, we can prove a stronger theorem than Theorem 3.2
with a general rectified homotopical depth, but dealing with homotopy groups. We leave it to thercader.

Comments

4

Notice that, in the proof of Theorem 3.3, we did not use Stratified Morse theory, but Theorem 2.2
of [4] instead.
Using Stratified Morse theory, we may see that we add cells at the points of, whereis the part from the stratumof a relative polar curve of :at . However we cannot
prove that at these points we are adding cells with the right connectivity. We conjecture that, in$F\cap(\bigcup_{\alpha\in A}\Gamma_{\alpha})$$\Gamma_{\alpha}$$f$$S_{\alpha}$$Xarrow \mathbb{C}$$x$this particular setting where the rectified homotopical depth is maximum, this should be true.

There arc many spaces $X$ for which the rectified homotopical depth is maximum and equal to. In [9], we prove th\‘at local complex analytic hypersurfaces have maximum rectified homotopical depth. The same proof shows that local complete intersections have also maximum rectified
homotopical depth,$\dim X$

In general, one can prove that a reduced equidimensional complex analytic space has maximum
rectified homotopical depth if and only if it is a Milnor space in the sense of [10].

For instance, let$\mathcal{A}$depth.

be an affine hyperplane arrangement. It is a hypersurface. So it has maximal

One can consider as a the complement of the hyperplane atof a projective hypersurf\‘ace inthe projective space. Since the hyperplane at
is defined by one equation in , we can apply
Theorem 3.2,$\mathcal{A}$$\overline{\mathcal{A}}$$\infty$$\mathbb{P}^{N}$$\overline{A}$$\infty$

We get that the fundamental group ofgenera13-plane $E$ . We get:$\mathcal{A}$equals the fundamental group of the intersection with a

Proposition 4.1 The fundamental group of a plane arrangement inwhose planes have at least
3 distinct plane directions, such that none of them is pamllel to the intersection of the two otherplane directions, is trivial.
$\mathbb{C}^{3}$

Proof.1 We assume that the planes are in general position. In this case we give a proof by induction

on the number of planes. If the number of hyperplanes is 1, there is nothing to prove.
lWe are indebted to Mutsuo Oka for this proof.

78
Let us suppose that an arrangement $A’$ with $k\geq 1$ planes inand of the plane .
The new arrangement $A$ is the union of$\mathbb{C}^{3}$

has a trivial fundamental group.

$P$$\mathcal{A}’$is connected since it is the union of lines in general position, where
The intersection of $P$ and$k\geq 1$ . We apply Van Kampen theorem. The fundamental group ofis the amalgamated productof the fundamental group of $P$ and the fundaniental group of . Both these fundamental groups
are trivial, so the fundamental group of is also trivial.
$k$$\mathcal{A}’$$\mathcal{A}$$\mathcal{A}’$$\mathcal{A}$Now, let us assume that the planes in$P_{1}^{1},$$\mathcal{A}$$P_{n_{1}}^{1},$are not in general position. Let us order the planes$P_{1}^{2},$$P_{1}^{k},$$P_{n_{2}}^{2},$$\ldots,$$\ldots,$$\ldots,$in such a way that, for $1\leq i\leq k$ , all the planes
parallel tofor $i\neq j$ .

$P_{p}^{i}$$P_{n_{k}}^{k}$$\ldots,$,are parallel between themselves, none of$P_{m}^{i}$is$P_{m}^{j}$We make another induction on . For $k=1$ , the arrangement consists of parallel planes. We may
not speak of a fundamental group, but the arrangement is the disjoint union of planes having all,
retractswith the planes
trivial fundamental group. For $k=2$ , the arrangement
given by, the arrangement. However, forby deforlnation on
h\‘as a non-trivial fundamental group.
$k$$\mathcal{A}$$P_{1}^{1},$$\mathcal{A}_{1}$$n_{2}\geq 2$$P_{1}^{2}$$\ldots$$P_{1}^{1},$$\mathcal{A}_{2}$$\ldots,$$P_{n_{1}}^{1},$$P_{1}^{2}$$P_{n}^{1_{1}},$$P_{1}^{2},$$P_{n}^{2_{2}}$$\ldots,$

Let us suppose that the third plane direction is not parallel to the intersection of the two first. The arrangement given byvanish inones. All the generators of the fundamental group of, forand the plane
has a trivial fundamental group. By adding successively the planes
the
group
of
$2\leq j\leq n_{3}$ , using Van Kampen theorem, we prove by induction that the fundamental
is trivial.
andarrangement given by$\mathcal{A}_{2}$$P_{1}^{3}$$P_{j}^{3}$$P_{1}^{3}$$\mathcal{A}_{2}$$\mathcal{A}_{2}$$P_{1}^{3},$$P_{j}^{3}$$\ldots,$

Now it it is easy to prove by induction that arrangements with more plane directions have a trivial
fundamental group.

It remains to consider the case the third plane direction is parallel to the intersection of the two
first ones. It is easy to find that the fundamental group is not trivial. It remains like that, if all
the plane directions are parallel to the intersection of the two first ones.
With Theorem 3.2 we obtain:, for $n\geq 3$ , which has
Corollary 4.2 The fundamental group of a hyperplane arrangement in
at least 3 hyperplane directions, such that none of them is pamllel to the intersection of the two.
other hyperplane directions, is$\mathbb{C}^{n}$$tri\mathscr{O}al$in
Beware not to confuse the fundamental group of the complement of the arrangement
the
group
of
the
fundamental
computeTo
itself.
arrangement
theofand the fundamental group$A$, one has to use Corollary 1.2.
incomplement of the arrangement$\mathcal{A}$$\mathbb{C}^{n}$$\mathbb{C}^{n}$References[1] D. Burghelea, A. Verona, Local homological properties of analytic sets, Manuscripta Math. 7(1972), 55 66.
-[2] D. Cheniot, Une d\’emonstmtion du th\’eor\‘eme de Zariski sur les sections hyperplanes d’une
hypersurface projective et du theor\‘eme de Van Kampen sur le groupe fondamental ducomplementaire d’une courbe projective plane, Compositio Math. 27 (1973), 141-158.

79[3] M. Goresky-R. MacPherson, Stratified Morse theory, Ergebnisse der Mathematik und ihrer
Grenzgebiete (3) 14 Springer-Verlag, Berlin, (1988) 272 pp.

[4] H. Hamm-L\^e Dung Thr\’ang, Rectified homotopical depth and Grothendieck conjectures, The
Grothendieck Festschrift, Vol. II, 311 351, Progr. Math., 87 (1990), Birkh\"auser Boston,
Boston, MA.
-[5] H. Hamm-L\^e Dung Tr\’ang, Un theor\‘eme de Zariski du type de Lefschetz, Ann. Sci. \’Ecole
Norm. Sup. 4 (1973), 317 355.
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[6] H. Hironaka, Stratification and fiatness. Real and complex singularities, Proc. Ninth Nordic
Summer School/NAVF Sympos. Math., Oslo (1976), pp. 199265, Sijthoff and Noordhoff,
Alphen aan den Rijn, 1977.
[7] L\^e Dung Tr\’ang, Calcul du nombre de cycles \’evanouissants d’une hypersurface complexe, Ann.
Inst. Fourier 23 (1973), 261 270.
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[8] L\^e Dung Tr\’ang, Some remarks on relative monodromy. in Real and complex singularities (Proc.
Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976), pp. 397-403. Sijthoff andNoordhoff, Alphen aan den Rijn, 1977
[9] L\^e Dung Tlir\’ang, Sur les cycles vanouissants des espaces analytiques, C. R. Acad. Sci. Paris Sr.
A-B 288 (1979), A283-A285.

[10] L\^e Dung Tr\’ang, Complex analytic functions with isolated singularities, J. Algebraic Geom. 1(1992), no. 1, 83- 99.
[11] I. Milnor, Singular points of complex hypersurfaces, Ann. Math. Studies 61, Princeton U. P.,

1968.

[12] O. Zariski, A theorem on the Poincar\’e group of an algebraic hypersurface. Ann. of Math. 38(1937), 131 141.
-

L\^e

$D\tilde{u}ng$

Tka’ng

23 Route de la Malep\‘ere11240 Cailhau, France
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