SOME VIEWPOINTS ON ALGEBRAIC GEOMETRY AND SINGULARITY THEORY*

Tzee-Char Kuo

(received 14 September, 1981)

My colleague, Dr. Gagen, the secretary of the Convention, has

advised me to make this lecture as comprehensible as possible to the

general audience. Perhaps I should begin with an important idea in

modern analysis which can be exposed in Calculus.

1. Morse Theory in Calculus

Let us consider a smooth function a = f(x,y) , defined, say, on

the whole of . We know that over a critical point (•'Kq ,*/q ) »

where f (a?0 ,?/0) = f (a:0,i/g) = 0 , the tangent plane to the graph is x y

horizontal.

Theorem (See [9], p. 12, Theorem 3.1). Suppose f 1 ([a,i>]) contains

no critical points, then the level curves at height a and b are

homeomorphic:

~ f~\b) .

{In fact, they are diffeomorphic.)

The situation is best illustrated in Pig. 1.

The idea of the following proof is, as I heard, due to

C. Ehresmann. Let points on f 1(b) flow down in the direction of

steepest descent (the gradient vector); they all reach f 1(a) since

there are no critical points to stop them. This flow - let us call it

* Invited Address at the Second AustralasianMathematics Convention, University of Sydney, 11 May to 15 May, 1981.Work supported by ARGC Grant 7.L20.205.

Math. Chronicle 11(1982) Part 2 67-80.

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the Ehresmann flow - yields the desired homeomorphism which is actually

a diffeomorphism.

1 E h r e s m a n ' n Pl° w

As in every Calculus lecture, there should be some exercises.

Here is one: Exercise. Formalize the above proof. (Compare [9], p . 12).

The above theorem is a typical one in Global Analysis. The

hypothesis is mainly of local nature (every point is non-critical), the

conclusion is global (the topological structure of level curves are the

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same). A jump from local to global is always a valuable knowledge.

We wish to develop this kind of theorem in Algebraic Geometry as a

machinery for a qualitative study of varieties.

2. Algebraic Geometry

Polynomials in one or several variables form one of the most

important classes of objects in mathematics. They are simple enough

for calculation (involving only a finite number of additions and

multiplications), and yet there are enough of them to approximate

functions both locally (Taylor's Theorem) and globally (Weierstrass

Approximation Theorem). Another fact which underlines their importance

is that a Fourier Transform turns some typesof Partial Differential

Equations into a polynomial problem.

As a first step for studying polynomials, we study their zero sets.

An algebraic variety is a set of the form

V = {x = (*,, ••• * *„) I P l (■*) = *** * Pk (x ) = 0)

where are polynomials in variables X j ,•••,* .

For x € , p^ having coefficients in J? , we call V a real

variety. For x C ®n , having coefficients in (I , we call V

a complex variety.

Example. The surface in Jt3 defined by the homogeneous equation

V : z2(x2 - y2) = (x2 + y2)2

is a double cone, as shown in Fig. 2.

For later purposes it is more interesting to consider V as a

2-parametrized family of curves in K 2 ,

V^ : z2 (.t2 - y2) = (x2 + y2)2 , (z fixed).

Note that when a ^ 0, V is always a figure "8” curve; when 2 = 0,

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VQ collapses to a point. A sudden change of "type" takes place as z

reaches the value 0 .

The above example illustrates the following

Classification Problem. Define "same type" among algebraic varieties,

and find the accompanying classification theorem.

For topologists, "same type" should at least mean "same topolo­

gical type". In this respect, T. Fukuda has established the following

famous

Theorem ([2]). Polynomial maps of bounded degree from to F (or

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ttfrom G to (C) have only a finite number of different topological

types. It follows that algebraic varieties (real or complex) defined

by k equations of degree m in n variables (k, m, n are given

integers') have only a finite number of different topological types.

This theorem is of course a b 5g step forward in the classification

problem. It is, however, definitely not the best possible solution,

since x = 0 (a line)', y2 = x 3 (a cusp) and y 2 = x 5 (a sharper

cusp), in 1R2 , all have the same topological type. A mere topolo­

gical classification is too coarse. Should one wish to find C l-

classification, it is then well-known that a simple family, such as

the Whitney example yx{y - x){y - tx) , can contain a continuum family

of different C^-types.

We shall propose a way out of this difficulty in §6.

3. Complex Algebraic Geometry vs. Real Algebraic Geometry

Now, let us take a break to examine a more basic aspect of the

subject. Algebraic Geometry has been dominated almost entirely by

the complex theory. Despite its great success as a branch of Pure

Mathematics, one can not but suspect whether the results would be

really applicable to physical sciences. Our physical world is a space

over H , not C . Knowledge on real polynomials is surely more

relevant than that on complex ones. A Fourier transform turns a

Partial Differential Equation to a problem on real polynomials. An

event of historic importance took place in 1959 when S. fcojasiewicz,

in his solution of the Division Problem in the theory of Partial Differ­

ential Equations ([8]), invented what we now call the fcojasiewicz

inequalities which reflected some profound properties of real analyticity.

We believe Real Algebraic Geometry should deserve more attention than

Complex Algebraic Geometry.

Real Algebraic Geometry is also harder; compare, for instance,

the following two theorems.

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Theorem. (See [7], p.97). Every irreducible complex variety is

connected.

Theorem. (Ilarnack, see [l], Chapter 4). A real projective plane curve

of degree n can have at most - — + 1 connected components.

There are curves of degree n having this number of components.

Ilarnack's Theorem has no generalization to higher dimensional

cases yet.

Let me finish this section by quoting Rene Thom :

It might be argued that the importance accorded by analysis to

the complex field and the theory of analytic functions during the last

century has had an unfortunate effect on the orientation of mathematics.

By allowing the construction of a beautiful (even too beautiful)

theory .... it has led to a neglect of the real and qualitative nature

of things....... In the case of any natural phenomenon governed by an

algebraic equation it is of paramount importance to know whether this

equation has solutions, real roots, and precisely this question is

suppressed when complex scalars are used. As examples of situations

in which this idea of reality plays an essential qualitative role we

have the following : the characteristic values of a linear differential

system, the index of critical points of a function, and the elliptic or

hyperbolic character of a differential operator.

(R. Thom, Structural Stability and Morphogenesis, English trans­

lation by D.II. Fowler, Benjamin, 1975. p. 35.)

4. Developing Morse Theory for Algebraic Geometry, the Equisingularity

Problem

Should one decide to explore Real Algebraic Geometry, there is an

obvious advantage for him: lie need not read any Algebraic Geometry

book. For these books usually begin, in paragraph 1, Section 1,

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Chapter 1, with the hypothesis that the ground field is algebraically

closed; as far as Real Algebraic Geometry is concerned, this assump­

tion is wrong.

How to develop Morse Theory for algebraic varieties? The Example

in 2 suggests the following ideas. If, as the parameter s varies,

the singularities of V remain the "same type", then an Ehresmanna

flow exists around the singular set of V , and V and V^ will be of

the same type. We are thus led to the

Equisingular!ty Problem Define "same type" among singularities,

establish a finite classification theorem; Ehresmann flow should

exist along equivalent singularities.

This problem is being studied extensively by many authors (See

[lO]), perhaps with motivations different from ours. It is the

central problem of contemporary Algebraic Geometry and Singularity

Theory. We shall find a solution to this problem in 6, at least for

isolated singularities.

5. Blowing-ups, Hlronaka modifications

A notion of fundamental importance in Algebraic Geometry is that

of blowing-up. It can be explained step by step via simple cases, as

follows. The blowing-up of R 2 with centre 0 is the map

6 : (M2 ,C) — (R2,0)

which sends the open Mobius strip, M 2 , onto P 2 , collapsing its

centre circle, C , to 0 . To see 8 , one can consider R 2 as

being covered by lines, L , through 0, where 0 is the angle0

between the x-axis and , 0 < 0 5 n , and Ln = L = .r-axis.0 ’ u n

(See Fig. 3). Now, lift L in the 3-direction by 0, and identifyU

L q and L , we then obtain the Mobius strip. The projection map is

6 , with g 1(0) = C . Note that B is one-to-one except on C ,

which is therefore called the exceptional variety. Similarly, the

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blowing-up of It” with centre 0 is

6 : (M*, F P " ' 1 ( f ”,0)

where F P n 1 is the real projective (n-1)-space, M” being the

canonical line bundle over F P n 1 . The blowing-up of F n+^ = F * x F^

with centre F is

8 : (Mn,k, W n-' x R*) - ( r " * \ R*)

obtained by blowing-up each section F n x {£} at {0} x {£} . in

general, let M be an analytic manifold, N a closed submanifold,

the blowing-up of M with centre 11 is obtained by blowing-up the

transversal sections of N at their origins. The exceptional variety

is a bundle over N with fibre a real projective space. The complex

case is similar; for the exceptional varieties, F P n 1 is replaced

by dP*"1 .

Given an analytic manifold, M, we can apply a succession of

blowing-ups to M , each having as its centre a closed non-singular

subvariety. The composition of a finite succession

h *1 h = 3, 0 . . . 0 3 ̂ : Mj, -------► ••• —► Mj -------► M

will be called a Hironaka modification of M.

6.• Blow-analytic equivalence of singularities

Let [/]» [/'] ’• ( F n ,0) — ► ( F,0) be two germs of real analytic

functions, represented by

f : U — - F , f' : U' — ► F

where U , U' are open neighbourhoods of 0, f(0) = f'(0) = 0. We

say [/*], [/'] are blow-analytically equivalent if there exist two

ilironaka modifications

h = gj o...o 3^ : m — ► U

h' = 3j 0...0 3' ; U' — > u'

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Fig 3 Blowing - ap (J>2 at 0

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* : M a M'

such that

____1____

(with the same number k of successive blowing-ups) and a real

analytic isomorphism

(1 ) M M'

h‘

U'

r

is commutative,

(2) <f» induces a local homeomorphism

+ : (i/,0) ~ (Z/',0)

Note that we do not require £ to be analytic (nor even C 1) .

Such a requirement would be too strong to yield a finite classification

theorem.

Let us call 0 an isolated singularity of [/] if

= ... = = o only when x = 0 .dx, dx 7

1 n

Now, consider a t-parametrized family of functions

*•(*,. •••,*„; tfl) : U x R 8 — * K , F(0;t) = 0,

where i/ is an open neighbourhood of 0 in . We shall assume

F analytic in (x,t) . For fixed t, let F(x;t) be denoted by

Ft(x).

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Theorem. ([4]). Suppose 0 (II” is an isolated singularity of [F^],

t f R S . Then R 8 can be decomposed into a finite disjoint union of

connected analytic manifolds M.

R S = M x U ••• [) M l

in such a way that the germs [F,] are blow-analytically equivalent

to one another, for all t in a same M\ . In fact, F admits a

local blow-analytic trivialization over each M\ in the following

sense. Let b € M. , N^ a sufficiently small neighbourhood of b

in M^, a sufficiently small neighbourhood of 0 in (J. There

should exist a homeomorphism

xb ' ub x Nb ^

kN(0,b) a neighbourhood of (0,b) in U * R ,

and a Hironaha modification of

for which the diagrams

are commutative,

and the composition

rbH h b * id) : --- + N(0,b)

is (real) analytic.

Remark on complex case. The notion of blow-analytic equivalence

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between complex singularities [/] : (fin ,0) — ► (<E,0) can be defined

similarly, except that $ is still to be a real analytic isomorphism.

A finite classification theorem, like the above one, can be established,

avoiding the problem of moduli. The details are in [6]. Here is

a sketch.

A complex Hironaka modification on a complex manifold M is a

finite composition of (complex) blowing-ups

Pih = 01 ° ...o ----- ► • • * ----- >- M j -------»- M .

Let N be a complex submanifold of M of (complex) codimension 1.

Treating M , N as real manifolds, we can blow-up M with centre N ;

each normal section of N is blown up into a Mobius strip. In general,

let us call the blowing-up of a real manifold along a submanifold of

(real) codimension 2 a Mobius blowing-up; and call a finite succession

of such a Mobius modification.

Example. The Mobius blowing-up of <t3 along the coordinate plane

zi = 0 is M 2 x ffi2 , M 2 denoting the Mobius strip; M 2 x M 2 * M 2 is

a Mobius modification of C3 , obtained by blowing-up along the three

coordinate planes successively.

A mixed modification of M is a composition

y = /jo0 : ff — — ► M — — >- M

where h is a complex Hironaka modification and 0 a Mobius

modification.

Now, let [/], [/'] be two germs of complex analytic functions :

(ffn ,0) — ► (C,0) , we say they are mixed blow-analytically equivalent

if there exist two mixed modifications

M : AT — ► M — ► U , M' : M'— ► M' — ► U'

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<f> : M M'

such that conditions similar (1) and (2) in 6 are satisfied. A finite

classification theorem, for isolated complex singularities, is then

established in [6].

As a by-product of the above results, we find the following

Theorem. Algebraic varieties (real or complex) defined by one equation

in n variables of degree m, having isolated singularities only,

admit a finite classification under blow-analytic / mixed blow-analytic

equivalence.

The proof is contained in [4],

Note. The notion of blow-analyticity has been developed gradually in

[3], [4] and [5]. It is only slightly different from the notion of

almost analytic equivalence in [4] and [5].

7. Summary

Real Algebraic Geometry is more important than Complex Algebraic

Geometry; it should deserve more attention now. The development of

Morse Theory in Algebraic Geometry depends on the Equisingularity

Problem. Blow-analytic equivalence is the right definition for this

problem, at least for isolated singularities. We conjecture that the

above results are valid for non-isolated singularities as well. Real

analyticity works better than complex analyticity for the Equisingularity

Problem since a finite classification exists.

REFERENCES

1. J.L. Coolidge, A Treatise on Algebraic Plane Curves, Dover, 1959.

2. T. Fukuda, Types Topologiques des Polynomes, Publ. I.ll.E.S. No.46.

(with the same indexing sets) and a real analytic isomorphism

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3. T.C. Kuo, Modified Analytic Trivializations of Singularities,

J. Math. Soc. Japan. Vol. 32, No.4, 1980, 605-614.

4. T.C. Kuo, Une Classification des Singulariteo Reelles, C.R.

Acad. Sci., Paris, 228(1979), 809-812.

5. T.C. Kuo and J.N. Ward, A Theorem on Almost Analytic Equisingular-

ities, appearing in J. Math. Soc. Japan, 1981.

6. T.C. Kuo, Equivalence of Isolated Complex Singularities, Preprint,

University of Sydney, 1980.

7. S. Lefschetz, Algebraic Geometry, Princeton University Press, 1953.

8. S. Jiojasiewicz, Sur le Probleme de la Division, Stud. Math.

18(1959), 87-136.

9. J. Milnor, Morse Theory, Ann. of Math. Studies, No. 51.

Princeton University Press, 1963.

10. B. Teissier, Introduction to Equisingularity Problems, Proceedings

of Symposia in Pure Mathematics, Vol.29, 1975, 593-632.

University of Sydney

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