数理解析研究所講究録1033 巻 1998 年 27-33

27

Fundamental groups of complements to hypersurfaces

Ichiro Shimada (Hokkaido University)

1. Statement of results
In this talk, we will present a generalization of Zariski’s hyperplane section theorem.

Let be a hypersurface in a complex projective spaceof dimension. We takea linear planeinin a general position with respect to . Zariski’s hyperplane section
theorem asserts the following isomorphism:$S$$\geq 2$$\mathrm{P}^{n}$$\mathrm{P}^{2}$$S$$\mathrm{P}^{n}$$\pi_{1}(\mathrm{P}^{2}\backslash (\mathrm{P}^{2}\cap S))$$\cong$$\pi_{1}(\mathrm{P}^{n}\backslash S)$.

This enables us to calculate the fundamental group of the complement to a hypersurface
by van-Kampen Zariski method. This theorem was stated by Zariski in [Z], but the proof
had a gap. The first rigorous proof was given by Hamm and L\^e in [H-L]. They used theMorse theory.
Now we are going to consider the following situation. Let be a complex homogeneous
variety on which a connected affine algebraic group acts transitively. The stabilizer groupof a point of is assumed to be connected. Let : $Xarrow U$ be a morphism from a
non-singular connected algebraic variety $X$ . We do not assume that is proper. For an, letelement: $Xarrow U$ be the composite of with the action : $Uarrow U$ of on. Suppose that we are given a non-zero reduced effective divisor $D$ of .
$U$

$G$

$H_{p}$

$p$

$U$

$f$

$f$

$\gamma\in G$$f$$\gamma f$$U$$\gamma$$\gamma$$U$$\cdot$Now we consider the following three conditions of .
$f$(C1) The image of is of dimension at least 2.
(C2) The locus of all points of $X$ at which the tangential map ofcodimension at least 2;$f$$\dim\{x\in X ; \dim f_{*,x}(T_{x}x)=0\}$(C3) A morphism$f$is of rank zero is of$\leq\dim X-2$ .

is said to be a nonsingular projective completion of if is anonsi,ngular algebr.aic variety which contains $X$ as its Zariski open dense subset and. Now the third condition
is a projective morphism which coincides with onof such that, if, there is a non-singular projective completion
with codimension 1 in ,is an irreducible component of the boundaryis at least one.
$\overline{f}:\overline{X}arrow U$$f$$\overline{X}$$\overline{f}$$f$$X\subset\overline{X}$$\overline{f}:\overline{X}arrow U$$\mathrm{i}\mathrm{s}_{-}\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}$$W:=\overline{X}\backslash X$$f$$W_{k}$$\overline{X}$$\mathrm{t}\mathrm{h}\dot{\mathrm{e}}\mathrm{n}\dim\overline{f}(W_{k})$in terms of
Our purpose is to calculate the fundamental groupaclear
answer to this
.
can
give
We
when is chosen generally fromandproblem in the following situations.
$\pi_{1}(^{\gamma}f^{-1}(U\backslash D))$$\pi_{1}(U\backslash D)$$\gamma$$G$$\pi_{1}(X)$28

I. Projective spa cesLetbe a projective space
linear transformations.
$U$

Theorem (P). Suppose thatthe morphism$f$$\mathrm{P}^{n}$with $n\geq 2$ , andsatisfies the th$\mathrm{r}ee$

the group$G$$\mathrm{G}\mathrm{L}(n+1)$

of generalconditions. Then, for a general$\gamma\in G$,$\gamma f^{-1}(^{\mathrm{p}^{n}}\backslash D)arrow(\mathrm{P}^{n}\backslash D)\cross X$

given by$x\mapsto(^{\gamma}f(x), x)$

induces a surjective homomorphism on the fundamental groups$\pi_{1}(^{\gamma}f^{-}1(\mathrm{p}^{n}\backslash D))arrow\pi_{1}(\mathrm{p}^{n}\backslash D)\mathrm{X}\pi 1(x)$and its kernel is isomorphic to the cokernel of the homomorphismby .

,$\pi_{2}(X)arrow\pi_{2}(\mathrm{P}^{n})$indu $ced$

$f$

Since

$\pi_{2}(\mathrm{P}^{n})$is an infinite cyclic group, the kernel is always a cyclic group.

When $X$ is a projective plane and is a linear embedding, this theorem is nothing
but Zariski’s hyperplane section theorem.
$f$

II.

$A$

ffine spaces

with $n\geq 2$ , and let
Let be an affine space.
, which is a subgroup ofphisms of$U$$\mathrm{A}^{n}$$\mathrm{A}^{n}$$G$

be the group of all affine automor-$\mathrm{G}\mathrm{L}(n+1)$

Theorem (A). Suppose th at satisfies the conditions (1), (2) and (3) above. Then,
induces an, the $nat$ ural morphism
for a general$f$$\gamma\in G$.hism$\gamma f^{-1}(\mathrm{A}^{n}\backslash D)arrow(\mathrm{A}^{n}\backslash D)\cross X$$\mathrm{i}somo\mathrm{r}_{\mathrm{P}}$$\pi_{1}(^{\gamma}f^{-}1(\mathrm{A}^{n}\backslash D))\cong\pi_{1}(\mathrm{A}^{n}\backslash D)x\pi_{1}(X)$.

III. Grassmanni an varieties
It is natural to expect that theorem of this type holds for o.ther homogeneous varieties.
However, even when we consider simple examples like Grassmannian varieties, we have to
put some additional conditions on the morphism .
Let be the Grassmannian variety Grass $(r, m)$ of all -dimensional linear subspacesof an $m$-dimensional linear space , where $2\leq r\leq m-2$ . On this variety, the generallinear group
acts transitively with connected stabilizer subgroups. As before,
let $D$ be a non-zero reduced effective divisor of .
$f$

$U$

$r$

$V$

$G=\mathrm{G}\mathrm{L}(V)$$U$: $Xarrow U$ satisfies the conditions (2) and (3) and moreover
sequence
an$\dim f(X)\geq\max(r, m-r)+1$ . Then, for a general, we
Theorem (G). Suppose that$f$$h\mathrm{a}ve$$\gamma\in G$$e\mathrm{x}\mathrm{a}ct$$1arrow \mathrm{C}\mathrm{o}\mathrm{k}\mathrm{e}\mathrm{r}(\pi_{2}(X)arrow\pi_{2}(U))arrow\pi_{1}(^{\gamma}f^{-1}(U\backslash D))arrow\pi_{1}(X)\cross\pi_{1}(U\backslash D)arrow 1$

There is an. such t..hat $\dim f(X)=2$ and the exact sequence does not holds.

$\mathrm{e}\mathrm{x}\mathrm{a}\mathrm{m}_{\mathrm{P}^{1\mathrm{e}}}$Example. Let be the Grassmannian variety Grassand three lines
space . We choose a point$U$$\mathrm{P}^{3}$.

$P\in \mathrm{P}^{3}$of all lines in a projectiveinand passing through$(\mathrm{P}^{1}, \mathrm{P}\mathrm{a})$$l_{1},$$l_{2}$$l_{3}$$P$29such a way that there are no planes containing three of them. We take as $D$ the reduced
divisor of whose support is given by$U${ $p\in U$ ;is the line corresponding to $p\in U$ . Let$XarrowU$
the inclusion of the nonsingular subvariety
:

where
$f$

$L(p)\cap(l_{1^{\cup l\cup l)\neq\}}}23\emptyset$$L(p)\subset \mathrm{P}^{3}$

$X$

$:=$

$\{ p\in U;.

,$Q\in \mathrm{P}^{3}$be another point, andQ\in L(p)\}$of , which is isomorphic to a projective plane. The fundamental groupisomorphic to. Indeed, letbe a plane such that $P\not\in H$ , and letintersection point of with $H$ . Let
be the dual projective plane of $H$ , and$H$
the locus of all lines on passing through . The projection$U$isbe the$\pi_{1}(U\backslash D)$$H\subset \mathrm{P}^{3}$$\mathbb{Z}^{2}$$H^{\vee}$$l_{i}$$P_{i}$$L_{i}\subset H^{\vee}$$P_{i}$$\mathrm{P}^{3}\backslash (l_{1^{\cup}}l_{2s}\cup l)$$arrow H\backslash \{P_{1}, P_{2}, P_{3}\}$

with the center $P$ induces a locally trivial morphism$U\backslash Darrow H^{\vee}\backslash (L_{1}\cup L_{2}\cup L_{3})$,every fiber of which is isomorphic to . Since
and are not on any plane, the three
lines
and
do not pass. through a common point. Hence we have$\mathrm{A}^{2}$$L_{1},$$L_{2}$$l_{1},$$l_{3}$$l_{2}$$L_{3}$$\pi_{1}(U\backslash D)\cong\pi_{1}(H^{\vee}\backslash (L_{1}\cup L_{2}\cup L_{3}))\cong \mathbb{Z}^{2}$

On the other hand, for a general $\gamma\in G=$ GL (4),free group
generated by two elements. Indeed, let
the projection$arrow H’$:.

is isomorphic to the
be a general plane. Then$\pi_{1}(^{\gamma}f^{-1}(U\backslash D))$$H’\subset \mathrm{P}^{3}$$F_{2}$$\mathrm{P}^{3}\backslash \{\gamma(Q)\}$$p_{\gamma}$with the center$\gamma(Q)\in \mathrm{P}^{3}$

induces an isomorphism$\gamma f^{-1}(U\backslash D)\cong H’\backslash (p_{\gamma}(l1)\cup p\gamma(l2)\cup p_{\gamma}(l3))$

Since
obtain$p_{\gamma}(l_{1}),$$p_{\gamma}(l2)$and$p_{\gamma}(l_{3})$are three lines on $H’$ passing through the point$\pi_{1}(H’\backslash (p_{\gamma}(l_{1})\cup p\gamma(l2)\cup p_{\gamma}(l_{3})))\cong F_{2}$It is$\mathrm{o}_{\vee}\mathrm{b}$vious that

$F_{2}$.

cannot be an extension of$\mathbb{Z}^{2}$$p_{\gamma}(P)$

, we

.

by a cyclic group.

2. Corollaries
Theorem (A) has the following corollary. Let be a non-singular connected surface
be theonto the affine plane. Let
equipped with a finite morphism :and $W$ a curve on .
branch locus of . Let $D$ be a reduced curve on$S$$\overline{f}$$Sarrow \mathrm{A}^{2}$$B\subset \mathrm{A}^{2}$$\mathrm{A}^{2}$$\overline{f}$

$S$

30

Corollary. We denote by
the reduced divisor whose support is the union of the
branched curve $B$ and the imageof W. Suppos that $E$ intersects $D$ at distinct$\deg D\cdot\deg E$ points. Then the fundamental groupis isomorphic to$E\subset \mathrm{A}^{2}$$\overline{f}(W)$$\mathrm{e}$$\pi_{1}(S\backslash (W\cup\overline{f}^{-1}(D)))$$\pi_{1}(s\backslash W)\mathrm{X}\pi 1(\mathrm{A}2\backslash D)$.

Indeed, the condition

Card $(E\cap D)=\deg D\cdot\deg E$
means that $E$ and $D$ intersect transversely at their non-singular points, and that they do
not have any intersection points at infinity. Hence, under this condition, the homeomorphism type of the spacedoes not change even when the morphism isperturbed toby a general affine automorphism of the affine plane. Hence, applying
Theorem (A) to the restriction :, we obtain the corollary.
of to$S\backslash (W\cup\overline{f}^{-1}(D))$$\overline{f}$$\gamma\overline{f}$$\gamma$$f$$S\backslash Warrow \mathrm{A}^{2}$$S\backslash W$$\overline{f}$In particular, when
, we obtain the invariance theorem of theandfundamental group of the complement to affine curve. Let:
be a finite
morphism.
$S=\mathrm{A}^{2}$$W=\emptyset$$\overline{f}$Corollary. Suppose that the branch locus ofpoints. Thenis isomorphic to$B$$\pi_{1}(\mathrm{A}^{2}\backslash \overline{f}^{-1}(D))$$\overline{f}$intersects$\pi_{1}(\mathrm{A}^{2}\backslash D)$$D$.

$\mathrm{A}^{2}arrow \mathrm{A}^{2}$at distinct $\deg D\cdot\deg B$

On the other hand, when is the identity, this corollary gives Oka-Sakamoto’s producttheorem ([O-S]).
$\overline{f}$3. Sketch of the proof

The method of the proof is rather elementary. Most part of the proof consists ofsimple dimension counts. And we hope that the same method can be applied to otherhomogeneous varieties. The main ingredient of the proof is the following:.

Theorem
Let $F$ be a nonsingular connected projective variety. Let
be a
reduced effective divisor of the product spaceof an affine space with F. For a
point
, let
denote the scheme-theoretic intersection of with $\{a\}\cross F$ , whichis regard as a subscheme of F. Suppose that theof allsuch thatis$F$
not a reduced divisor of is of codimension
in, the. Then, for a generalinclusionin ces an isomorph.
$([\mathrm{S}1])$$Z$$\mathrm{A}^{N}\cross F$$a\in \mathrm{A}^{N}$$Z$$Z_{a}$$a\in \mathrm{A}^{N}$$loc\mathrm{u}s_{\cup}^{-}-$$\mathrm{e}d$$\geq 2$$\{a\}\cross F^{\mathrm{L}}arrow \mathrm{A}^{N}\cross F$$Z_{a}$$a\in \mathrm{A}^{N}$$\mathrm{A}^{N}$$\mathrm{i}sm\pi_{1}(F\backslash Z_{a})\cong\pi_{1}((\mathrm{A}^{N}\cross F)\backslash Z)$$d\mathrm{u}$The proof of this theorem has been already published in [S1]. Roughly speaking, thistheorem is shown by regarding the first projection
from the complement
to to the affine space
as something like a local trivial fiber space. Of course, there
are some points
such thathas worse singularity than that of the divisor
over a general point . Hence the projection
is not locally trivial over
. Nevertheless, the fundamental group of algebraic variety is not affected by changing
a locus of codimension at least 2. Hence,er the conditionof codimension, the first piece of the homotopy exact sequencein$(\mathrm{A}^{N}\cross F)\backslash Zarrow \mathrm{A}^{N}$$Z$$\mathrm{A}^{N}$$a’\in \mathrm{A}^{N}$$Z_{a’}$$Z_{a}$$(\mathrm{A}^{N}\cross F)\backslash Zarrow \mathrm{A}^{N}$$a$$\mathrm{A}^{N}$$\mathrm{u}\mathrm{n}\acute{\mathrm{d}}$$\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}_{\cup}^{-}-\mathrm{i}_{\mathrm{S}}$$\mathrm{A}^{N}$$\pi_{2}(\mathrm{A}^{N})=0arrow\pi_{1}(F\backslash Z_{a})arrow\pi_{1}((\mathrm{A}^{N}\cross F)\backslash Z)arrow\pi_{1}(\mathrm{A}^{N})=0$

still holds.

$\geq 2$

31

Now we will show how to derive the generalized Zariski’s hyperplane section theorems
from Theorem
in the case of projective spaces and Grassmannian varieties$([\mathrm{S}1])$

First note that we may assume thatis purely of codimension one. Because
removing the locus of codimension larger than one from the non-singular algebraic variety
does not affect the topological fundamental group. Therefore we can ignore the irreducible
component of $W$ with codimension larger than one.
Let be the Grassmannian variety Grass $(r, V)$ of all -dimensional linear subspaces
in a linear space , where $1\leq r\leq\dim V-2$ . This setting covers both of the projective
spaces and the Grassmannian varieties. The point of the proofs of Theorems (P) and (G)is to apply Theorem
to the case
. There is a rational map$W:=\overline{X}\backslash W$

$U$

$r$

$V$

$\mathrm{A}^{N}=\mathrm{E}\mathrm{n}\mathrm{d}(V)$$([\mathrm{S}1])$

End (V)$\cross U$$...arrow$$U$extending the action of the general linear groupon . The indeterminate
locus of this rational map is of codimension at least 2, so that, as far as the fundamentalgroups are concerned, we can neglect it. Let$G=\mathrm{G}\mathrm{L}(V)$$G\cross\overline{X}$be the morphism given by
rational map$arrow U\mathrm{x}\overline{X}$. This morphism can also be extended to the$(\gamma, x)rightarrow(^{\gamma}\overline{f}(x), x)$

End (V)$U$$\cross\overline{X}$$U\mathrm{x}\overline{X}$$...arrow$.

The indeterminate locus ofational map is also of codimension at least 2. Let
bethe Zariski open dense subset of End (V)on which the rational map is defined. The
point is that the morphism$\mathrm{t}\mathrm{h}\mathrm{i}_{\mathrm{S}}.$$\overline{\mathcal{X}}$$\cross\overline{X}$$\psi$:$\overline{\mathcal{X}}arrow U\cross\overline{X}$is locally trivial. The fiber is isomorphic to the space$\{\gamma\in \mathrm{E}\mathrm{n}\mathrm{d}(V) ; \gamma(L)=L\}$of all endomorphisms of which maps a fixed -dimensional linear subspace $L\in U$ onto
isomorphically. Let us denote this space by . Then is isomorphic to GL
.
$V$

$L$

$r$

$\Gamma_{0}$$(r)\mathrm{X}\mathrm{A}^{m}(m-\Gamma)$$\Gamma_{0}$Now we consider the non-zero reduced divisor$E:=D\cross\overline{X}+U\cross W$on. We regard the boundary $W$ as a reduced divisor ofexact sequence$U\cross\overline{X}$1$arrow-\pi_{2}((U\cross\overline{X})\backslash E)$

associated with
and$U\backslash D$$\psi$$arrow\pi_{1}(\mathrm{r}_{0})$$arrow\pi_{1}(\overline{\mathcal{X}}\backslash \psi-1(E))$$\overline{X}$. Then we have a homotopy$arrow\pi_{1}((U\cross\overline{X})\backslash E)$$arrow$$1$. Since the complement of the divisor $E$ is nothing but the product of, we have$X=\overline{X}\backslash W$,$\pi_{2}((U\cross D)\backslash E)\cong\pi_{2}(U\backslash D)\cross\pi_{2}(X)$$\pi_{1}((U\mathrm{X}D)\backslash E)\cong\pi_{1}(U\backslash D)\cross\pi_{1}(X)$.

32

On the other hand, let be the closure of the divisorof;in Endthat is,is the divisor on End (V)whose support is the closure of the support ofand whose restriction tocoincides with. Since the complement of inEnd (V), we haveis of codimension$Z$$\psi^{*}(E)$$Z$$(V)\cross\overline{X}$$\overline{\mathcal{X}}$$\cross\overline{X}$$\psi^{*}(E)$$\overline{\mathcal{X}}$$\mathrm{x}\overline{X}$$\psi^{*}(E)$$\overline{\mathcal{X}}$$\geq 2$$\pi_{1}(\overline{\mathcal{X}}\backslash \psi^{-1}(E))\cong\pi_{1}((\mathrm{E}\mathrm{n}\mathrm{d}(V)\cross\overline{X})\backslash Z)$.

Now we can prove the following:

Claim. There is a natural natural isomorphism between
cokernel of the boundary homomorphism :$\pi_{2}(X)arrow\pi_{2}(U)$ .
with the cokernel of$\partial$$\pi_{1}(\mathrm{r}_{0})$and$\pi_{2}(U)$such that theis identified$\pi_{2}(U\backslash D)\cross\pi_{2}(X)arrow\pi_{1}(\Gamma_{0})$$f_{*}:$In the proof of this claim, we use the assumption that $D$ is non-zero, so that thehomomorphism
induced by the inclusion is a zero map.
$\pi_{2}(U\backslash D)arrow\pi_{2}(U)$Now we have an exact sequence$1arrow \mathrm{C}\mathrm{o}\mathrm{k}\mathrm{e}\mathrm{r}(T_{2}(X)arrow\pi_{2}(U))arrow\pi_{1}((\mathrm{E}\mathrm{n}\mathrm{d}(V)\cross\overline{X})\backslash Z)arrow\pi_{1}(U\backslash D)\mathrm{x}\pi_{1}(X)arrow 1$.

As before, for an element, letdenote the scheme-theoretic intersection of, and we consider it as a sub-scheme of . Then, by the definition of ,withif
GL (V), the subscheme, and hence its complement
coincides withcoincides with;.
$\gamma \mathrm{o}\mathrm{f}\dot{\mathrm{E}}\mathrm{n}\mathrm{d}(V)$$Z_{\gamma}$

$Z$

$Z$

$\overline{X}$$\{\gamma\}\cross\overline{X}$$\gamma\in$$W+\gamma\overline{f}^{*}(D)$$Z_{\gamma}$$\gamma f^{-1}(U\backslash D)$$X\backslash Z_{\gamma}=\gamma f^{-1}(U\backslash D)$

Hence, by Theorem$([\mathrm{S}\mathrm{l}])$, we have

Claim. If the locus$\cup--:=$

is of codimension$\geq 2${$\gamma\in \mathrm{E}\mathrm{n}\mathrm{d}(V)$$Z_{\gamma}$

is not a reduced divisor ofin the affine space End (V), thenfor a general.

$\pi_{1}((\mathrm{E}\mathrm{n}\mathrm{d}(V)\cross\overline{X})\backslash Z)$$\mathrm{o}\mathrm{f}_{\cup}^{-}-$;$\overline{X}$$\pi_{1}(^{\gamma}f^{-1}(U\backslash D))$}is isomorphic to$\gamma\in \mathrm{E}\mathrm{n}\mathrm{d}(V)$Therefore the proof of theorems has been reduced to the estimation of the dimension. It is rather technical, but we can prove the following:

Claim. Suppose that is a general element of the irreducible hypersurface
GL (V). Then
is a reduced divisor of ; that is,is a proper Zariski closed subset.
$\triangle$$\gamma$$\overline{X}$$Z_{\gamma}$$:=\mathrm{E}\mathrm{n}\mathrm{d}(V)\backslash$$\cup--\cap\triangle$

Claim. Suppose that satisfies the conditions in the theorems.
codimension
in GL (V).
$f$$\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{n}_{\cup}^{-}-\cap \mathrm{G}\mathrm{L}(V)$is of$\geq 2$

These two claims showand (G) are proved.

$\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}_{\cup}^{-_{\mathrm{i}\mathrm{s}}}-$of codimension$\geq 2$For more details, please refer to the preprint [S2].

in End (V). Thus Theorems (P)33References[H-L] H. A. Hamm and D. T. L\^e, Un th\’eor\‘eme de Zariski du type de Lefschetz, Ann. Sci.
\’Ecole Norm. Sup. 6 (1973), 317-366.
[O-S] M. Oka and K. Sakamoto, Product theorem of the fundamental group of a reducible
curve, J. Math. Soc. Japan 30 (1978), 599-602.
[S1] I. Shimada, Fundamental groups of open algebraic varieties, Topology 34 (1995), 509$- 532$.

[S2] I. Shimada, Zariski’s hyperplane section theorem for morphisms to homogeneous varieties, preprint.
[Z] O. Zariski, A theorem on the Poincar\’e group of an algebraic hypersurface, Ann. Math.
38 (1937), 131- 141.

Ichiro Shimada
Department of Mathematics
Hokkaido University
Sapporo 060 JAPANshimada@math.sci.hokudai.ac.jp