数理解析研究所講究録1033 巻 1998 年 140-147

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REMARKS ON THE LOCAL GEOMETRY OF ANALYTIC MAPSMICHAL KWIECINSKI
ABsTRACT. In this lecture I present recent results of mine (some obtained
jointly with Piotr Tworzewski) concerning the question of how general fibres
of complex maps converge to special fibres. These results include in particular:
1. a relation between the number of points in the fibre of a generically discrete
map and the dimension of its singular fibres, 2. a relation between the presenceof vertical components in fibred powers of a map and openness and flatness,
3. an example of an algebraic map with nonconstructible topology.

1. INTRODUCTION,
We will be interested in complex analytic maps of analytic spaces :with $\dim \mathrm{Y}>1$ . In the case of . aps with values in curves, both algebraic properties
condition) are(like flatness) and topological properties (like openness or Thom’s
relatively easy to understand. This opens the door to the study of finer algebraic
and topological properties and has produced a considerable amount of results.
However, when the target space has big dimension, very few general results areavailable. Indeed, even such basic invariants as the dimensions of different fibres arefar from being well understood. The deep reason behind this is that maps whichmaps may have a wild topological behaviour (see example atare not Thom’s
the end of this article). Below, we present some recent results concerning the local
topology and local algebra of maps, which are non-trivial in the case $\dim Y>1$ .
The results in sections 2 and 3 as well as Theorem 4.1 were obtained jointly with
Piotr Tworzewski in [15]. Details concerning Theorems 4.2 and 5.1 can be foundin the author’s paper [13] and Example 7.1-in [14].
$f$$Xarrow \mathrm{Y}$$\mathrm{m}$$a_{f}$$a_{f}$

2. FIBRES OF GENERICALLY DISCRETE MAPS
Let us start by recalling a well known result which says that if $Y$ is smoothand if is a birational map which has an isolated fibre of positive dimension, thenthat fibre is of codimension 1 in the source space. Here, ”isolated” means that theneighbouring fibres are all at most zero-dimensional. Think of a blowup map as an
example. Let us rephrase this result:, withsmooth. Suppose that
Consider a generically discrete map :$f^{-1}(y_{0})$thenit has an isolated fibre of positive dimension. If
there are at least 2 points in the general fibre.
In the above statement, two integers appear: both equal 2. Our first question
How far canwill be: what if we change the first of the two integers to 3, 4, 5,
we change the second integer? The answer is the following theorem.
$f$

$f$

$Xarrow \mathrm{Y}$$\mathrm{Y}$$\mathrm{c}\mathrm{o}\dim_{X}f^{-1}(y_{0})\geq 2$... .

The author is supported by the Japan Society for the Promotion of Science and a Monbusho
grant.

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Theorem 2.1. [15] Let $f:Xarrow Y$ be an analytic map of analytic spaces, being
a complex manifold and both $X$ and being of pure dimension . Assume that theimage of every irreducible component of $X$ has nonempty interior in Y. Suppose
there is a point $y_{0}\in Y$ , such that $\dim f^{-1}(y_{0})=w_{0}>0$ and $\dim f^{-}1(y)\leq 0$ , for. Then there exists an open subset of , such that for all $y\in U$$\mathrm{Y}$

$Y$

$d$

$y\neq y_{0}$$U$$\mathrm{Y}$,$\# f^{-1}(y)\geq[\frac{d-1}{w_{0}}]$where the squarre brackets denote the integer part of a rational number.
All topological notions will always refer to the transcendental topology. An
analytic space may have singularities.
In the algebraic case, when one assumes thatis irreducible, can be chosen
to be dense in Y. In the analytic case, even $f^{-1}(U)$ need not be dense in $X$ . This
is why we do not use the word ”generic” in the actual statement of the Theorem.
The bound of Theorem 2.1 is sharp, as can be seen from the following example.
Example 2.2. Consider
with coordinatesandwith homogeneous coordinates. Let $X$ be the hypersurface indefined by theequation $x_{0}\lambda^{d1}-+x_{1}\lambda^{d-2}\mu+\cdots+x_{d-1\mu^{d1}}-$ and let :be the restrictionof the first projection. Then the fibre of at is of dimension one; all the otherfibres are zero-dimensional and the generic fibre consists $d-1$ points (its smallest
possible cardinality by Theorem 2.1).
$U$$\mathrm{Y}$$\mathbb{C}^{d}$$x_{0},$$(\lambda :$\ldots,$$x_{d-1}$\mu)$$\mathbb{C}\mathrm{P}^{1}$$\mathbb{C}^{d}\cross \mathbb{C}\mathrm{P}^{1}$

$f$

$f$

$Xarrow \mathbb{C}^{d}$$0$

To generalize Theorem 2.1 to the case of a map with non-isolated singular fibres,
we introduce the following notion of an equidimensional partition, which is weaker
than that of a stratification of a map.

Definition 2.3.

Let:

be an analytic map of analytic spaces.
$X$ is called ancountable partition
equidimensional partition (for )
of
$p\in
P$
each
:
if for1.
is a nonempty irreducible locally analytic subset of $X$ ,2. the restriction: $X_{p}arrow Y$ is equidimensional ($i.e$ . all of its nonempty
fibres are of pure dimension and of the same dimension).
$f$$Xarrow \mathrm{Y}$$A$$\{.X_{p}\}_{p\in P}$$f$$X_{p}$$f|_{X_{\mathrm{p}}}$Standard arguments in stratification theory provide us with the following proposition.
Proposition 2.4. For any analytic map
sional partition of $X$ .

$f$: $Xarrow Y$ , there exists an equidimen-

Remark 2.5. From an equidimensional partition as above, we can read off thefollowing numerical data1.
the dimension of any fibre of,
2. $k_{p}.=\dim x_{p}$ .
$w_{p}$-$f|_{X_{\mathrm{p}}}$The generalization of Theorem 2.1 to the case of a generically finite map with
non-isolated singular fibres of arbitrary dimensions is the following.

Theorem 2.6. [15] Let$f$:$Xarrow \mathrm{Y}$be an analytic map of analytic spaces, $Y$$anequdimens\dot{i}onalpartt_{\dot{i}}onobe\dot{i}nga_{i}manif_{ol}dandbo_{i}thXandYbeingofp_{O}uredimens_{1}i_{\mathit{0},(}nd.Let\{X\}_{p}fXforf.SuppSethatf-f(_{X))i_{Sa}}diScretesetp\in Pbe$

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The proof that theorem 3.4 implies theorem 2.6 essentialy relies on noticingthat for generically discrete maps
isthan the number of points in thegeneric fibre. This follows from an equivalent description of the invariantas themaximal number of points of a special fibre, that can be simultaneously approached
by points in one sequence of generic fibres.
The very rough idea of the proof of theorem 3.4 is as follows. Inequality ”: ifsome fibres have bigger dimensions than others then those dimensions grow faster inthe fibred powe.rs and eventually produce isolated vertical components. Inequality”
is smooth and hence we know the number of local equations for the diagonalinand thus also for
in $X\cross\cdots\cross X$ . Thisus tocalculate that isolated vertical components do not appear too soon, otherwise theirdimensions would be too small.
$\phi(f)$$\mathrm{s}\mathrm{m}\mathrm{a}\mathrm{u}_{\mathrm{e}}\mathrm{r}$$\phi$$,,\leq$$,,\geq$$:\mathrm{Y}$$\mathrm{Y}\cross\cdots\cross Y$$\mathrm{a}\mathrm{U}\mathrm{o}\mathrm{w}\mathrm{s}$$X\cross\cdots\cross X\mathrm{Y}\mathrm{Y}$4. OPENNESS AND FLATNESS
A natural question to as about the invariant introduced in the preceding
sectionis when is its value infinite. Under the mild assumption that is locally irreducible,
the asnwer is thatis infinite iff is an open map. Without using the invariant, the statement reads as follows:$\mathrm{k}$$\mathrm{Y}$$\phi(f)$$f$$\phi$

Theorem 4.1. [15] Let
be an analytic map of analytic spaces. Supposethat is locally irreducible. Then the following conditions are equivalent:
is open,for any $i\geq 1$ , the canonical map $X\cross\cdots\cross Xarrow Y$ has no isolated vertical$f:Xarrow \mathrm{Y}$

$Y$

$f$

$\bullet$$\bullet$$-\mathrm{Y}itimesY$

components.

The proof of the above theorem relies mainly on the equivalence between openness and equidimens.ionality, stratifications and dimension counts.
The natural question now is to see what happens if in the second condition wealso take into account the embedded components of the fibred powers. It turns outthat we then get a characterization of flat maps :Theorem 4.2. [13] Let :be an analytic map of analytic spaces. Suppose
thatis reduced and locally irreducible ($i.e$ . every local
is an integral
domain). Then the following $condit\dot{i}ons$ are equivalent:$f$$Xarrow \mathrm{Y}$$\mathrm{Y}$$7\dot{n}ng\mathcal{O}\mathrm{Y},y$$\bullet$$\bullet$is fiat,for any $i\geq 1$ , the canonical map$f$$X\cross\cdots\cross Xarrow \mathrm{Y}$has no (isolated or$\sim \mathrm{Y}itimes\mathrm{Y}$embedded) vertical components.

Rec-all that a component of the source space of a map (be it isolated or embedded)
is called a vertical component iff its image has empty interior in the target spacewith the transcendental topology. If no such component exists, we say that themap has no vertical components.
Any example of an open non-flat map shows the difference between the above
theorems. Also the methods of proof are quite different.
The hard part of the proof of theorem 4.2 is to show that some fibred powerof a non flat map must have a vertical component (isolated or embedded). Our
main tool is Hironaka’s characterization of flatness ([7], 6, Proposition 10, see also
$g$

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Bierstone and Milman [3] , which roughly says that a map is flat at a point of thesource space if and only if a standard basis of the germ of the fibre passing throughthat point is a restriction of a standard
is of the germ of the source space at$\xi\in X$ , Hironaka’s map
point.
point
map
that
If a
is not flat at someis notinjective. We pick a power series in its kernel and consider the ideal generated by
the coefficients of that power series. By Noetherianity, this ideal is generated bysome minimal finite set of these coefficients:
. Now
can
be regarded as the germ of a function on the i-th fibred power of at the point. The construction of together with some calculations and standard
faithful fl.atness arguments show that is a nonzero torsion element over the local. Some not too difficult commutative algebra shows that the presence
ring
of such a torsion element implies the existence of a vertical component.
$)$$\mathrm{b}\mathrm{a}s$$\kappa$$a_{1},$$\ldots,$$a_{i}$$a=a_{1}\wedge\cdots\wedge a_{i}$$f$$(\xi, \ldots,\xi)$

$a$

$a$

$\mathcal{O}_{Y,f(\xi)}$

5. COMMUTATIVE ALGEBRAIC REFORMULATIONSIn the standard dictionary between affine algebraic geometry and commutative
algebra, the existence of vertical components (resp. isolated vertical components) offibred powers corresponds to the existence of torsion in tensor powers (resp. tensor
powers quotiented by the nilradical). Thus our theorems have commutative algebraic reformulations. In particular the following theorem is an affine, commutative
algebraic analogue of theorem.
$\dot{4}.2$

Theorem 5.1. Let $R$ be a finitely generated -algebra and a normal domain. Let
be a finitely generated -algebra. Then the following are equivalent:is R-flat,
is a torsion-free R-module.
for any $i\geq 1$ , the -th tensor power$\mathbb{C}$

$A$

$R$

$\bullet$$A$$\dot{i}$$\bullet$$. \cdot\sim A\bigotimes_{R}\cdots\bigotimes_{R}Atimes$

In fact, in the above theorem, ”normal” can be replaced by the weaker condition
that each localization at a maximal ideal is analytically irreducible.
The study of torsion in tensor products of modules (in particular in tensor powersof modules) was initiated by Auslander [2] and recently
by Huneke andWiegand [9]. In particular, Theorem 3.2 of [2] says that finitely generated modules
with torsion-free tensor powers over unramified regular local rings are free and hence
implies our theorem 5.1 in the case when $A$ is finitely generated as an R-module.
After some work it also implies theorem 4.2 for proper maps with finite fibres.
Another special case of theorem 5.1, when is a symmetric algebra, was studied
by the author in [11] and applied in [12] to produce new bounds on codimensionsof determinantal varieties.
$\mathrm{r}\mathrm{e}\mathrm{v}\mathrm{i}\gamma \mathrm{e}\mathrm{d}$$A$

6. AN OPEN PROBLEM
A natural problem for further research is to determine a value of in theorems
4.2 and 5.1, for which it is enough to look at a fibred (resp. tensor) power in orderto determine flatness. In the cases studied in [2] and in Theorem 4.1, $i=\dim \mathrm{Y}$ issufficient. The first question to answer would be if it is also sufficient in theorems
4.2 and 5.1.
$i$

7. A COMPLEX MAP WITH NONCONSTRUCTIBLE TOPOLOGY
In this section we explain why studying the topology of algebraic maps, whosetarget space is of big dimension, is so hard. Indeed, unlike the topology of complex

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spaces, where the theory of Whitney stratifications is available, we have no general theory describing the topology of maps. Stratifications of maps give us some
topological triviality above each stratum, but no information at all about how the
topology behaves as we approach the boundary of these strata. Stratifications satis
Thom’scondition do give such information, but then not all algebraic
maps have such stratifications.
Among the few general results describing the behaviour of fibres of analytic
with high dimensional target space, let us cite some notable exceptions: Hironaka’s
flattening Theorem [7], its local version by Hironaka, Lejeune-Jalabert and Teissier[8], and analogous results by Sabbah [18] (concerninginstead of flatness) andTeissier [19] (concerning traingulability). Also should be cited the Relative Lefschetz Theorems of Goresky and MacPherson ([5], Part II, Chapter 1), which givebounds on the homotopy type of a complex analytic space in terms of dimensions$q_{\mathrm{i}}\mathrm{n}\mathrm{g}$$a_{f}$$\mathrm{m}\mathrm{a}\mathrm{p}\mathrm{s}_{-}$$a_{f}$of the singular fibres.
Below, we provide a simple example of a complex algebraic map :which(even locally) has an infinite number of different local topological types at points ofX. By local topological type, we mean the right-left topological equivalence class ofa germ of . There are related examples of Thom [20] and Nakai [16]. The differencewith our’s is that they treat the varying of topological type in parametrized families
of maps. It is not clear if one can obtain the type of example we are looking for
directly from those examples. Nevertheless, our example is inspired by Thom’s(which is real and global).
$f$$Xarrow \mathrm{Y}$$f$

Example 7.1. [14] Let $X$ be the hypersurface $x_{1}x_{2}=0$ in$(x_{1}, x_{2}, z,t)$ . Let
and define the map : $Xarrow Y$ by$Y=\mathbb{C}^{3}$$\mathbb{C}^{4}$

with variables$f$$f(X_{1},x_{2}, z,t)=(x_{1}+x_{2}, (x_{1}+t_{X_{2}})Z,t)$.

To understand the above map, keep in mind that on $X$ eitherorvanishes.
Claim. [14] In any neighbourhood of any point $(0,0, \mathrm{o},t)\in X$ , with $|t|=1$ , themap has infinitely many different local topological types. More precisely, suppose$\alpha_{2}\in[0,1),$
thatis irrational,
and
. Let
and
. Then has different topological types at $(0,0,0, t_{1})$ and at $(0,0,0, t_{2})$ .
The following tool is essential in the proof of the claim. It is a concrete realizationof an idea of Thom.1 On each fibre $f^{-1}(y)$ , we define an invariant relation(a subset of $f^{-1}(y)\cross f^{-1}(y)$ ). Let $U=\{x\in X : \dim_{x}f^{-1}(f(X))=0\}$ .
$x_{1}$$x_{2}$$f$$\alpha_{1},$$\alpha_{1}\neq\alpha_{2}$$\alpha_{2}$$t_{2}=e^{2\pi\alpha_{2}i}$$\alpha_{1}\neq 1-\alpha_{2}$$t_{1}=e^{2\pi\alpha_{1}i}$$f$$\mathcal{R}(y)$Definition 7.2. For$y\in \mathrm{Y}$, let $R(y)=f^{-1}(y)\cross f^{-1}(y)\cap$ closure of $(U\cross U)$ ,$Y$

where the closure is taken in the fibred product $x\cross XY$ induced by the map .
$f$Definition 7.3. Let and
be topological spaces, $\in A,$ $b\in B.$ Let and be
relations on and respectively. We say thatandare topologicallyequivalent if there exists a homeomorphism of germs :, such that in
the induced product homeomorphism: $(A \mathrm{x}A)_{()}a,aarrow(B\cross B)_{(b,b)}$ we have
$A$

$A$

$B$

$a$

$B$

$\delta$$\rho$$(\rho, a)$$(\delta, b)$$g$$A_{a}arrow B_{b}$$g\cross g$$(g\cross g)^{-1}(\delta b,b))(=\rho(a,a)$.

Proposition 7.4. If has the same local topological type at two points and $bofX$
then $(R(f(a)), a)$ andare topologically equivalent. (Here the relations
are considered on the fibres $A=f^{-1}(f(a))$ and $B=f^{-1}(f(b)))$ .
$f$

$a$

$(\mathcal{R}(f(b)), b)$

1In [20] Thom writes ”...dans un voisinage d’une strate \’eclat\’ee, l’application op\‘ere des identifications qui se traduisent par une correspondance dans la strate (S) elle-m\^eme;...’’.

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Since Definition 7.2 involves only local topological objects, Proposition 7.4 isobvious.
The proof of the claim then consists in determining the invariant relation.
Notice that
, with coordinate . Let $(z, z’)$ be corresponding coordinates iri. Then, the relation $R(f(at))$ is thehypersurface $H_{t}(z, z)’=0$ , with$\dot{\mathcal{R}}(f(a_{t}))$$f^{-1}(f(a_{t}))\cong \mathbb{C}$$z$$f^{-1}(f(a_{t}))\cross f^{-1}(f(a_{t}))\cong \mathbb{C}^{2}$$H_{t}(_{Z,Z’})=(z-Z)’(z-tZ’)(tz-Z’)$.

To establish this, remark that the set $U$ from Definition 7.2 is in this case thecomplement of $\{x_{1}=x_{2}=0\}\cup\{t=x_{1}=0\}$ . Then, one can use computer algebratote the appropriate closure or do it directly (look at limits of pairs of pointsof with same image by ).
By Proposition 7.4 it is enough to show that $(R(f(a_{t})1), 0)$ and $(R(f(a_{t})2), 0)$
are not topologically equivalent. Assume the contrary. This means that there exists
a homeomorphism of germs :, such that the product homeomorphism:
maps the germ of the hypersurface $H_{t_{1}}(Z, Z’)=0$ to the germof $H_{t_{2}}(Z, Z’)=0$ . Of course, the diagonal is mapped to the diagonal and every
other irreducible component into an irreducible component (since after removing theorigin they become connected components). Suppose that the component $z’=t_{1}z$is mapped to $z’=t_{2}z$ (the other case being similar). This means that satisfies
the following identity on:$\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{c}\mathrm{u}\mathrm{l}\dot{\mathrm{a}}$

$U$

$f$

$g$

$g\cross g$$\mathbb{C}_{0}arrow \mathbb{C}_{0}$$\mathbb{C}_{0}^{2}arrow \mathbb{C}_{0}^{2}$$g$$\mathbb{C}_{0}$$e^{2\pi}z\alpha_{1}i=g^{-1}(eg(2\pi\alpha_{2}i)z)$.

Replace by a representative and let be a small circle centered at . Now is
closed under multiplication by and therefore, $g(S)$ is closed under multiplication
by . Sinceis irrational, $g(S)$ is also a circle. Thus, the above identity holds after
restricting andto circles. This implies that rotations of the circle byand byhave the same rotation number (see e.g. $[1],[10],[17]$ ) and contradicts
the assumption that
.
$S$

$g$

$0$

$S$

$t_{1}$$t_{2}$$\alpha_{2}$$g$$g^{-1}$$2\pi\alpha_{1}$$2\pi\alpha_{2}$$\alpha_{1}\neq\alpha_{2}$REFERENCES[1] V.I. Arnold, ”Geometrical methods in the theory of ordinary differential equations”, Springer
Grundlehren der mathematischen Wissenschaften 250, 1988.
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647.

[3] E. Bierstone, P.D. Milman, ”The local geometry of analytic mappings”, Dottorato di Ricerca
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[7] H. Hironaka, Stratifications and Flatness, in ”Real and Complex Singularities”, Proc. Oslo
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147[12] M. Kwiecitski, Bounds on codimensions of Fiuing ideals, Journal of Algebra, 194,378-382(1997).
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, P. Tworzewski, Finite sets in fibres of holomorphic maps ,.
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[18] C. Sabbah, Morphismes analytiques sans e’clatement et cycles e’vanescents, Ast\’erisque 101102 (1983), 286-319.
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[20]’ La stabilite’ topologique des applications polynomiales, Enseign. Math. 8 (1962),$\mathrm{K}\mathrm{w}\mathrm{i}\mathrm{e}\mathbb{C}\dot{\acute{\mathrm{m}}}\mathrm{S}\mathrm{k}\mathrm{i}$$\mathrm{a}-\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}/9606008$$\mathrm{R}.\mathrm{T}\mathrm{h}\mathrm{o}\mathrm{m}24-33.$

TOKYO METROPOLITAN UNIVERSITY, DEPARTMENT OF MATHEMATICS, MINAMI- OHSAWA 1-1,
HACHIOJI-SHI, TOKYO 192-03, JAPAN.

on leave from: UNIWERSYTET
059 KRAKO’w, POLAND.
$E$, INSTYTUT MATEMATYKI, UL. REYMONTA 4, 30-$\mathrm{J}\mathrm{A}\mathrm{G}\mathrm{l}\mathrm{E}\mathrm{L}\mathrm{L}\mathrm{O}\acute{\mathrm{N}}\mathrm{S}\mathrm{K}1$-mail address: michalQmath. metro-u. ac. jp