数理解析研究所講究録
第 764 巻 1991 年 23-36

23

THEORY OF SUPER-ISOLATED SINGULARITIESAND ITS APPLICATIONSOSAMU SAEKI修

佐伯

1. Introduction.

In this paper, we consider the following problem.

PROBLEM 1.1. Is there a holomorphic function germ$f$:$C^{n},$$0arrow C,$ $0$

with anisolated singularity at the origin which cannot be connected to a real germthrough a topologically constant deformation?
Here, a function germ is real if it takes real values ona holomorphic function germ$f$:$C^{n},$$0arrow C,$ $0$$R^{n}\subset C^{n}$. Note thatwith an isolated singularity at theorigin is connected to a real germ through a topologically constant deformation
provided that $n=2$ or that$f$has a non-degenerate Newton principal part ([3,11, 15]).

Our purpose of this paper is to give holomorphic function germs of threevariables which are candidates for answering the above problem positively, i.e.,
which are probably not connected to real germs through topologically constantdeformations. Our idea is to use the theory of super-isolated singularities ([8])
to reduce the problem to that of the tangent cones, which are projective curves

in$CP^{2}$in this case. More precisely, we divide the problem into the following

two steps.

1

24
PROBLEM 1.2. Find a class of holomorphic function germs
that a topologically constant deformation starting with

$f$

$f$

: $C^{3},0arrow C,$ such$0$

always induces a topo-

logically constant deformation of the tangent cones.

PROBLEM 1.3. Find a projective curve in$CP^{2}$

which cannot be connected,

through a topologically constant deformation, to a curve defined by a real polynomial.

In this paper, we answer Problem 1.3 (\S 3). As to Problem 1.2, we try toshow that the class of super-isolated singularities (SIS) ([8]) is such a class (\S 2).
However, there is still a hole to fill in. In the author’s lecture at R.I.M.S. on

March 26, 1991, he claimed that Problem 1.2 had been solved; however, it was
not correct, since the statement of Theorem 2.10 was not correct.
A large part of this paper is a survey of Luengo’s work, especially that of\S 2. In fact, the above idea of using the theory of super-isolated singularities
to attack Problem 1.1 is due to him. Furthermore, he even claims that he has

recently solved Problem 1.1 $(n=3)$ completely. Since we unfortunately do notknow his proof, we will not discuss it in this paper.
This paper is organized as follows. In \S 2, we recall some of Luengo’s workon super-isolated singularities. In \S 3, we give an example of a plane projective
curve which cannot have the same topological type as a curve defined by areal polynomial. In fact, we will construct an arrangement of lines with thisproperty. Furthermore, at the end of \S 3, we discuss the relation to our original2

25
problem concerning the topological types (the right equivalence and the right-left
equivalence) of holomorphic function germs with isolated singularities (cf. [15]).

In \S 4, we discuss other applications of the theory of super-isolated singularities.

The author wishes to express his sincere gratitude to Prof. Luengo for hiskind help.

2. Super-isolated singularities.

This section is a survey of Luengo’s work on super-isolated singularities. Thus
we omit most of the proofs. For details, see [8].
DEFINITION (LUENGO [8]). Let (V, )$0$$\subset(C^{3},0)$be the germ of a hypersurface

singularity. We say that (V, ) is a super-isola ed singul arity (SIS) if$0$along $C=\pi^{-1}(0)$ , where

$\pi$

:

$\tilde{V}$$t$$\tilde{V}arrow V$

is smoothis the monoidal transformation with center

0. Note that a SIS is always an isolated singularity.
We can define a SIS of an arbitrary dimension. However, we consider only
two-dimensional ones in this paper.
Let $f\in C\{X, Y, Z\}$ be the defining function of$m=mult(V, 0)$ (the multiplicity of $V$ at

$0$

$V$

, i.e. $V=f^{-1}(0)$ , and let). Then we can decompose$f$into thesum of homogeneous polynomials$f=f_{m}+f_{m+1}+\cdots$

where$f_{i}$

is of degree . Let$i$center . Then we have$0$$\tilde{\pi}$:$C^{3}-arrow C^{3}$,be the monoidal transformation with$\tilde{V}=\sim\pi^{-1}(V),$ $\pi=\tilde{\pi}|\tilde{V}$$3$

and $C=\pi^{-1}(0)\subset CP^{2}=$26$\overline{\pi}^{-1}(0)$.

is called the tangent cone of$C$with the curve

points of$C$$f_{m}^{-1}(0)\subset CP^{2}$.

and it is known that $C$ is identified$V$

Note that$\tilde{V}$is tangent to$CP^{2}$at the singular$(C)\cap f_{m+1}^{-1}(0)=\emptyset$, where Sing $(C)$.

$arrow^{\pi}$

LEMMA 2.1. (V, ) is a $SIS$ ifand only if Sing$0$is$tl_{1}e$singular point set of the projective urve $C\subset CP^{2}$ .
$c$

REMARK 2.2. Using this lemma, we can construct a SIS from any projective

curve

$C$

in

$CP^{2}$

with isolated singularities. For example, it is constructed asfollows. Let $h\in C[X, Y, Z]$ be the homogeneous polynomial of degree $m$ which
defines$C$and let $l\in C[X, Y, Z]$ be homogeneous of degree 1 such that Sing $(C)\cap$$l^{-1}(0)=\emptyset$. Then $f=h+l^{m+1}$ defines a SIS in$C^{3}$. The topological type of$f$does not depend on the choice of . In fact, as we see later (Remark 2.13), the$l$topological type of a SIS is determined by its tangent cone
4

$C$

.

27
PROPOSITION 2.3(IOMDIN $[5|$ ). We have$\mu(V, 0)=(nz-1)^{3}+\sum_{p\in Sing(C)}\mu(C,p)$where $m=mult(V, 0)$ , $\mu(V, 0)$ is the Miln or $numb$ er of
local Miln or number of

$C$

$V$

,

at

$0$

and $\mu(C, p)$ is the

at .
$p$

REMARK 2.4. Siersma [16] and Stevens [17] have independently obtained aformula for the characteristic polynomial of the monodromy ofpressed in terms of the multipllcity

local monodromy of

$C$

$m$

$m_{i}$

SIS. It is ex-

and the characteristic polynomials of theat its singular points.

Let $C=\pi^{-1}(0)=C_{1}\cup\cdots\cup C_{r}$ . where
of $C$ , and let$a$be the degree of$C_{i}$$C_{i}$

are the irreducible components.

LEMMA 2.5. If we consider $C=C_{1}\cup\cdots\cup C_{f}$ as embedded in$\overline{V}$, then we have$(i\neq j)$$C_{i}\cdot C_{j}=m_{i}m_{j}$$C_{i}\cdot C_{i}=-m(m-m_{i}+1)(\leq-2)$ .

COROLLARY 2.6.

$\pi$

:

$\tilde{V}arrow V$

is the$m$in imal resol tion of .
$u$

$V$

COROLLARY 2.7. $Z=C_{1}+\cdots+C_{r}$ is the fundamen $tal$ cycle of the (minimal)
resolu tion

$\pi$

:

$\tilde{V}arrow V$, and $Z\cdot Z=-m$ .

REMARK 2.8. The concept of the fundamental cycle was introduced by Artin[1]. Note that it is uniquely determined by the intersection matrix

the resolution.

5$(C:\cdot C_{j})$of28
LEMMA 2.9 (WAGREICH [18]). Let$\alpha$:$\overline{W}arrow W$

be a resolut ion of a normal

two-dimensional singularity $(W, 0)$ . Then mult $(W, 0)\geq-Z\cdot Z$ , where$fu$

ndamental cycle of the resol ut ion$\alpha$$Z$

is the.

Now we can state one of our main theorems of this section.

THEOREM 2.10. Let$p$: $Barrow T$ be a topologically constant (analytic) defor-

mation of a $SIS(V, 0)$ with the smooth baseand let

$T_{\}}$: $Tarrow B$ be th$\sigma$$e$section $wh$ ich picks the singular points. Then $T’=\{t\in T;mult(V_{t}, \sigma(t))=$and$mult(V, 0)$$rvh$$(V_{t}, \sigma(t))$

is a $SIS.$ } is a (non-empty)$Z$

ariski open set in$T$,

ere $V_{t}=p^{-1}(t)$ .

Ou tline of Proof.

By assumption,

has the same topological type$(V_{t}, \sigma(t))$

as (V, ). Thus, by Neumann [10], they have homeomorphic resolutions. Then$0$

they have the same fundamental cycle and Lemma 2.9 and Corollary 2.7 imply

that mult$(V_{t}, \sigma(\mathfrak{i}))\geq mult(V, 0)$$mult(V, 0)$ .

. If is sufficiently close to$t$

Thus mult $(V_{t}, \sigma(t))=mult(V, 0)$ for all

Furthermore, by Lemma 2.1, (V,$\sigma(t)$

$t$

$0,$

$mult(V_{t}, \sigma(t))\leq$sufficiently close to$0$.

) is a SIS if is sufficiently lose to . Il
$t$

$c$

$0$

REMARK 2.11. In the author’s lecture at R.I.M.S. on March 26, 1991, he claimed
that $T=T$ in Theorem 2.10. However, it was not correct, since the above proof
does not guarantee that $T’=T$ .

Let $F=F(X, Y, Z,t)=0$ be the defining equation of

6$B$. Then by the29
above theorem we can write$F=F_{m}+F_{m+1}+\cdots$ ,where$F_{i}$

is homogeneous of degree with respect to $X,$
$i$

$Y$

and

$Z$

. Thus we havea family of plane projective curves as follows.
$D=F_{m}^{-1}(O)(\subset CP^{2}\cross T’)$$\downarrow\overline{p}$$T’$Note that$(V_{t}, \sigma(t))$

THEOREM 2.12.

omorph isms
Proof.

$\varphi_{t}$$\overline{p}$is a SIS with tangent cone

$C_{t}=\overline{p}^{-1}(t)$.

is equisingular, $i.e$ . there exists a contin uous family of home-: $CP^{2}arrow CP^{2}(t\in T’)$ such that $\varphi 0=id$ and$\varphi_{t}(C_{0})=C_{t}$.

Since $p|p^{-1}(T’)$ : $p^{-1}(T’)arrow T’$ is a topologically constantdeformation, it is -constant. Furthermore, by Theorem 2.10, it is equimultiple.
$\mu$Thus, by Proposition 2.3, the total Milnor number of$t\in T’$$\overline{p}^{-1}(t)$is independent of. It is known that such a family of plane projective curves is equisingular.

Il
REMARK 2.13. For a SIS (V, ) with multiplicity$0$$\mu^{(3)}=\mu(V, 0)$$m$, we have(cf. Proposition 1.2)$\mu^{(2)}=(m-1)^{2}$$\mu^{(1)}=m-1$.

Thus a family of SIS’s is topologically constant if and only if it is$\mu^{*}$-constant.

The author has been informed that Luengo has obtained the following.

7

30
THEOREM 2.14. Let (V, ) and (V’, ) $\subset(C^{3},0)$ be isolated hypersurface$0$$si_{11-}$$V$

and $V’$ are homeomorphic and (V, ) is a

$0$

$gu$

larities. If the link 3-manifolds of

$SIS$

$0$

, then (V’, ) is also a $SIS$ and their tangent cones $C$ and C’ have the same
$0$

local topological type, $i.e$ . there exist open sets $U\supset C$ and $U’\supset C’$ in $CP^{2}$ anda homeomorphism$\varphi$: $Uarrow U’$ such that $\varphi(C)=C’$ .

If this theorem is true, we see immediately that $T’=T$ in Theorem 2.10,

which solves Problem 1.2 in \S 1.

3. Topologically non-real curves in$CP^{2}$.

In this section, we prove the following.
PROPOSITION 3.1. There exists a (reduced) plane projective curve $C(\subset CP^{2})$such that if a plane projective curve C’ has th same topological type as$e$

if there exists a homeomorphism$\varphi$$C$, i.e.

: $CP^{2}arrow CP^{2}$ with $\varphi(C)=C’$ , then $C’$cannot be defined by any real polynomials.
Before proving Proposition 3.1, we discuss how to construct a potential

example of a topologically non-real germ of a holomorphic function. Using a

curve

$C$

as in Proposition 3.1, we can construct, as in Remark 2.2, a germ of aholomorphic function $f:C^{3},0arrow C,$$0$with an isolated singularity at the origin

whose tangent cone is identified with $C$ . Then, assuming $T=T$ in Theorem
2.10, we see that$f$cannot be connected to a real germ through a topologicallyconstant deformation, since its tangent cone

8

$C$

is non-real. Even if $T’\neq T$

31
in Theorem 2.10, we see that$f$cannot be connected, through a topologicallyconstant deformation, to a real germ which is a SIS.
Before we proceed to the proof of Proposition 3.1, we must note that Luengo

has independently found a non-real curve. His example is an irreducible rational

curve of degree 11 with exactly one singularity, which is of type $x^{4}+y^{31}=0$ . Heproves that this curve is topologically non-real, using his theory as in [7] withthe help of a computer. Since his example seems difficult, we give here anotherexample whose non-realness we can prove seemingly much easier.

Proof of Proposition 3.1.

We will construct an arrangement $A(\subset CP^{2})$which cannot have the same topological type as a curve defined by a real poly-

nomial. We note that an arrangement is a reduced curve in$CP^{2}$all of whose

irreducible components are lines.
First consider the arrangement$z^{3})(z^{3}-x^{3})=0$ .

$A_{0}$We see easily thatdefined by the equation $(x^{3}-y^{3})(y^{3}-$$A_{0}$

consists of 9 lines and that it has 12

singularities, all of which are triple points.

LEMMA 3.2 (MELCHIOR [9]). Let $A’$ be an arrangement each of whose component is defined by a real olynom $ial$. Suppose $A’$ has more than 1 singular$p$points. Then $A’$ has more than 2 double points.

Lemma 3.2 can be proved by an easy calculation of the Euler characteristicof $RP^{2}$ by means of the cell decomposition associated with $A’$ . For details see$[2, 4]$ .

9

32
In view of Lemma 3.2, we see thatcannot have the same topological$A_{0}$

type as an arrangement each of whose component is defined by a real polynomial,dthough$A_{0}$

itself is defined by a real polynomial.

Now we construct a desired arrangement by adding several lines to$q_{0)}\cdots q_{11}\in A_{0}$be the singular points ofare triple points of$A_{0}$$A_{0}$. Remember that all these points$l_{1,1}$

is a line which passes throughdoes not pass through the other singular points of$l_{i,1},$$\cdots l_{i,i}$. Let. Define the arrangements $A_{i}(i=1, \cdots , 11)$ inductively

as follows. Set $A_{1}=A_{0}\cup l_{1,1}$ , where

where$A_{0}$$A_{0}$. Set$A;=A_{i-1} \cup\bigcup_{j=1}^{i}l_{i,j}$

are distinct lines each of which passes throughpass through the other singular points of$A_{i-1}$but$q_{1}$$q_{i}$,

but does not.

Now we show that $A=A_{11}$ is a desired non-real plane projective curve.

Suppose that a plane projective curve $C$ defined by a real polynomial has thesame topological type as
Since$\gamma$$C$$A$. Then we see easily that$C$

is also an arrangement.

is defined by a real polynomial, $C$ is invariant under the conjugation: $CP^{2}arrow CP^{2}$ ; i.e., $\gamma(C)=C$ . On the other hand, for $3\leq\forall_{m}\leq 14,$ $C$ hasexactly one m-fold point
homeomorphism from$A$

that $p_{m}\in RP^{2}\subset CP^{2}$$p_{m}$. ($\{q_{0}, \cdots q_{11}\}$$(3 \leq m\leq 14)$. Hence, the subarrangement$A_{0}$. This means that every component of

Since$C_{0}$$\{p_{3}, \cdots p_{14}\}$by a

to $C.$ ) Thus we must have $\gamma(p_{m})=p_{m}$ , which implies

which corresponds to the subarrangement$RP^{2}$

corresponds to, which is homeomorphic to$A_{0}$

of$C_{0}$$A$$C_{0}$

of $C$

has all its singular points on

is defined by a real polynomial.

, has 12 singular points none of which isa double point, this contradicts to Lemma 3.2. This completes the proof.

10

$||$

33
REMARK 3.3. The arrangement which we constructed is of degree 75 and has

2333 singular points. If we construct a SIS (V, ) from this arrangement, we have$0$$\mu(V, 0)=408363$ , by Proposition 2.3.

Even if we use the example of Luengo,

which is of degree 11, we have $\mu=1090$ , which is very large. We do not know

if there exists a non-real curve of degree less than 11.
Before ending this section, we must mention our original problem which
led us to consider Problem 1.1. In [15], we defined the topological right equivalence and $right$ -left equivalence between two germs of holomorphic functions withisolated singularities. The right equivalence implies the right-left equivalence.
However, we do not know if the converse is true. This is our original problem.
By King [6], if we can prove that, for any$\overline{f}$, which is defined by$\overline{f}(z)=\overline{f(\sim)}$$f$:$C^{n},$$0arrow C,$ $0,$$f$

and its conjugate, are right equivalent, then we can deducethat the right-left equivalence implies the right equivalence. Furthermore, we

see easily that if$f$deformation, thenfunction germ$f$is connected to a real germ through a topologically constant
$f$

and

$\overline{f}$are right equivalent. If we consider the holomorphic: $C^{3},0arrow C,$$0$(SIS) constructed from a non-real plane projec-

tive curve as in this section, we do not know if it can be connected to a real

germ. Thus there is a possibility that
we can show thatit implies that

$f$

$f$

and

$\overline{f}$

$f$

and

$\overline{f}$may not be right equivalent. If

are not right equivalent (by another method), thencannot be connected to a real germ through a topologicallyconstant deformation.

11

34
4. Other applications.
In [8], Luengo introduced the concept of a SIS to show that the$\mu$-constantstratum in the miniversal deformation of an isolated hypersurface singularity is
not necessarily smooth. His idea was to reduce the problem to that of the tangent

cones. He showed that for a SIS, the topologically constant stratum is (locally)
isomorphic to the equisingular stratum of the tangent cone and he constructeda plane projective curve whose equisingular stratum is not smooth. After that,

Stevens [17] showed that, in this case, the topologically constant stratum is acomponent of the -constant stratum, thus solving the above problem.
$\mu$Here, we give another application of the theory of super-isolated singulari-ties concerning the following two conjectures.

CONJECTURE 4.1 (YAU [20]). Let (V, ) and $(W, 0)\subset(C^{3},0)$ be isolated hy$0$

persurface singularities, i.e. $V=f^{-1}(0)$ and $W=g^{-1}(0)$ for some holomorphic

function germs $f,$$g$

: $C^{3},0arrow C,$$0$with isolated singularities at the origin.

Then (V, ) and $(W, 0)$ are topologically equivalent, $i.e.$ , there exists a germ
$0$of a homeomorphism$\varphi$$\pi_{1}(K(V))\cong\pi_{1}(K(W))$

link 3-manifolds of$V$: $C^{3},0arrow C^{3},0$ such that $\varphi(V)=W$ , if and only if

and $\Delta_{V}(t)=\Delta_{W}(t)$ , where $K(V)$ and $K(W)$ are theand $W$ respectively, andacteristic polynomials of the monodromy for$V$$\Delta_{V}(t)$and$\Delta_{W}(t)$are the char-

and $W$ respectively.

CONJECTURE 4.2 (cf. O’SHEA [13, p.124]). If

$f$

and

$g$

:

$C^{n},$$0arrow C,$ $0$

have

the same topological type, then they are embedded in a topologically constant12

35
family.

We note that the both conjectures are true for weighted homogeneous isolated hypersurface singularities in$C^{3}$([14, 19]).

PROPOSITION 4.3. There exist holomorphic $funct$ ion germ $sf$ and$C,$ $0$$g$: $C^{3},0arrow$with isolated singularities at the origin which do not satisfy either Conjec-

ture 4.1 or Conjecture 4.2; i.e., either of them is false.

Proof.
and$C\circ\sim$

Zariski [21] showed that there exist plane projective curves$C_{1}$of degree 6 such that(i) $C_{1}\cdot(i=1,2)$ has exactly six singularities, all of which are cusps,(ii) $\tau_{\iota_{1}}(CP^{2}-C_{1})\not\cong\pi_{1}(CP^{2}-C_{2})$ .
$C_{1}$

has the six cusps on a conic, while

Construct holomorphic function germs
as in Remark 2.2.

$f_{1}$

and$f_{2}$$C_{2}$$f_{i}$

does not.
: $C^{3},0arrow C,$ $0(i=1,2)$ from

$C_{i}$

define SIS’s and we see that $K(V)$ is diffeomorphic

to $K(W)$ and that $\Delta_{V}(t)=\Delta_{W}(t)$ (cf. [16, 17]), where $V=f_{1}^{-1}(0)$ and$W=f_{\underline{9}}^{-1}(0)$$f_{2}$. However, in view of Theorem 2.12,$f_{1}$

cannot be connected tothrough a topologically constant deformation, since their tangent cones havedifferent topological types. II

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Japan

Department of Mathematics, Faculty of Science, Yamagata University, Yamagata 990,

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