AN ELEMENTARY EXPOSÉ ON EQUISINGULARITIES

TZEE-CHAR KUO AND LAURENTIU PAUNESCU

Abstract. Take two convergent power series in n variables. When should they be declared “equivalent”? This is the equisingularity problem. For the case of n real variables the notion of blow-analytic equivalence has led to interesting results. The main ideas and results can be fully explained in the two variables case. This we do in §2. The notion of blow-up plays a vital rôle, hence we give an elementary description of it in §1. This notes can be understood by non-specialists.

1. Blow-up

Take a Möbius band, M, as obtained by rotating a line around the centre circle. A natural map β : M → ℝ can be visualized as follows.

D’ C’ C  C’ C D’ D A  A’ B  B’  A  A’ B  B’ D

Every generating line is mapped to a line through the origin O. The entire centre circle collapses to the origin; elsewhere, β is bijective.

We call β (or more properly β^{−1}) the blow-up of ℝ^{2} at the origin O, or with centre O. A neighbourhood of O is magnified, infinitely many times, to a neighbourhood of the centre circle. The precise definition is given below.

Consider the canonical line bundle over the real projective line

π_{1} : E → ℝP^{1} (l, x) ↦ l, E := {(l, x)|x ∈ l ∈ ℝP^{1}}.

Recall that a point, l, of ℝP^{1} is just a line in ℝ^{2} through O.

Topologically, E is a Möbius band, ℝP^{1} is a circle. The restriction to E of the second projection

1991 Mathematics Subject Classification. 32S05, 32S15, 14H20.

The authors are supported by ARC Large Grant A699045676.

We like to thank Craig Jin for his help in computer graphing.

β := π_{2}|E : E → ℝ^{2}, (l, x) ↦ x,

is called the blow-up of ℝ^{2} at O.

Note that β^{−1}(O) = ℝP^{1} × {O} is the zero cross-section of the bundle, outside of which β is a bianalytic isomorphism.

Intuitively, E is obtained by separating lines in ℝ^{2} through O and spread them over ℝP^{1} in a natural manner; what β does is just putting each line back to where it used to be in ℝ^{2}, as indicated earlier.

The above readily generalises to any 2-dimensional real analytic manifold 𝕄. Take a point P ∈ 𝕄. A neighbourhood of P is a copy of ℝ^{2} with P = O, whence we can define the blow-up of 𝕄 at P:

β : (M, β^{−1}(P)) → (𝕄, P),

which is an analytic isomorphism outside β^{−1}(P).

We can also compose them. Take β_{n} : M_{n} → M_{n−1}, 1 ≤ n ≤ N, M_{0} := 𝕄, and consider the composition:

β = β_{1} ∘ β_{2} ∘ ⋯ ∘ β_{N} : M_{N} → 𝕄.

We shall discuss the generalisations to higher dimensional, and the complex cases in §3. Now consider the map

β_{1}^{∗} : ℝ^{2} → ℝ^{2}, (X, Y) ↦ (x, y) = (X, XY).

Each horizontal line Y = c, c a constant, has image y = cx. The Y -axis (X = 0), collapses to the origin, while X = c, c ≠ 0, has image x = c. The rectangle in the (X, Y)-plane is mapped to a sector around the x-axis, as indicated below.

B Y  X  x y B’ D’  D’  B’ D B A  A’ D  A  A’

Philosophy: When an object lies in the sector, (β_{1}^{∗})^{−1} would magnify it into an object in the rectangle, often making it simpler. For example, the cusp y^{2} = x^{3} is magnified to X^{2}(Y^{2}−X) = 0, consisting of a parabola plus the Y -axis, as indicated below.

Objects that are tangent to the y-axis, however, need a different magnification:

C Ỹ  X̃  x y D’  C’ C B’ D  B  D’  C’ B’ B  D
β_{2}^{∗} : ℝ^{2} → ℝ^{2}, (X̃, Ỹ) ↦ (x, y) = (X̃Ỹ , Ỹ).

Gluing up the above two rectangles yields a neighbourhood of the centre circle of E. Using the language of Differential Topology, β_{1}^{∗}, β_{2}^{∗}, are two coordinate charts for the blow-up map β.

In order to study a given function f(x, y), we often study f ∘ β_{1}^{∗} and f ∘ β_{2}^{∗}, which can be simpler.

2. Blow-analytic equisingularity

Consider the familiar function of two real variables:

m(x, y) = \frac{x^{2}y}{x^{2} + y^{2}} , m(0, 0) = 0,

which is continuous, but not C^{1}.

4 TZEE-CHAR KUO AND LAURENTIU PAUNESCU

Take the blow-up β : M → ℝ^{2} introduced earlier, and then consider the composition m ∘ β.

In either charts,

(m ∘ β_{1}^{∗})(X, Y) = \frac{XY}{1 + Y^{2}} , (m ∘ β_{2}^{∗})(X̃, Ỹ) = \frac{X̃^{2}Ỹ}{1 + X̃^{2}} ,

both are analytic functions.

Thus m ∘ β is an analytic function on M.

A function q : ℝ^{2} → ℝ is called blow-analytic if there exists a succession of blowups:

β := β_{1} ∘ ⋯ ∘ β_{N} : M_{N} → M_{N−1} → ⋯ → M_{1} → ℝ^{2},

such that the composition q ∘ β is analytic on M_{N}.

Let us also localize this notion. Take a germ q : (ℝ^{2}, 0) → (ℝ, 0). We say q is blow-analytic if there exists a

composite (as above) β, for which each β_{i} is the blow-up of M_{i−1} with centre a point in (β_{1} ∘ β_{2} ⋯ ∘ β_{i−1})^{−1}(0), such that q ∘ β is analytic in a neighbourhood of β^{−1}(0).

An important example of blow-analytic function, found in [12], is

Q(x, y) = \frac{xy^{5}}{x^{4} + y^{6}} .

The readers are invited to find out for themselves how many blow-ups are needed to make Q analytic.

We introduce a more effective way in the following. According to [11], we say that q is arc-analytic if for any two convergent power

series x(t), y(t),we have also that q(x(t), y(t)) is a convergent power series. The readers are invited to check, quite easily, that Q is arc analytic. Consider a meromorphic function, i.e. a quotient of analytic functions,

m(x, y) = \frac{n(x, y)}{d(x, y)}.

An important fact is that m(x, y) is arc-analytic if and only if it is blow-analytic. This follows immediately from the celebrated Desingularization Theorem of Hi-

ronaka. It is also a special case of the Bierstone-Milman theorem on arc-analytic functions [1].

Following [9], [10], we call two germs of analytic functions f, g : (ℝ^{2}, 0) → (ℝ, 0) blow-analytically equivalent if there exists a local homeomorphism h : (ℝ^{2}, 0) → (ℝ^{2}, 0), such that g ∘ h = f, and the components of h and h^{−1} are (germs of) blow-analytic functions.

In 1965, Whitney [13] discovered a family of germs,

w_{t}(x, y) = xy(x + y)(x − ty), 0 < t < ∞,

parametrized by t, which consists of infinitely many different C^{1}-types.

AN ELEMENTARY EXPOSÉ ON EQUISINGULARITIES 5

This family, however, consists of only one blow-analytic type [9].

In 1998, Fukui [2], proved that if f and g are blow-analytically equivalent, then they have the same multiplicity. His proof is very short and elegant, making use of the Fukui invariant set A(f). An effective method of computing the Fukui set has recently been found in [6].

In 1999, Fukui and Paunescu [5], have proved that a t-parametrised family of weighted homogeneous forms w_{t}(x, y), having an isolated singularity at (x, y) = (0, 0), consists of only one blow-analytic equivalence class. A typical example is:

x^{2} + txy^{2} + y^{4}, t > 2.

3. Higher dimensional case

It is rather straightforward to generalize the notion of blow-up.

Take 𝕂 = ℂ or ℝ. Consider 𝕂^{n} and the canonical 𝕂-line bundle over the 𝕂projective space 𝕂P^{n−1}:

π : E → 𝕂P^{n−1}, π(l, x) = l, E := {(l, x) ∈ 𝕂P^{n−1} × 𝕂^{n}|x ∈ l}.

The restriction to E of the second projection

β := π_{2}|E : E → 𝕂^{n}, β(l, x) = x,

is called the blow-up of 𝕂^{n} at the origin O.

Now, take a 𝕂-analytic manifold, M, say of dimension m, and a closed, smooth submanifold, C, say of codimension n. Consider a tubular neighbourhood of C, its normal sections can be considered as copies of 𝕂^{n}. We can blow-up each copy at its origin to yield

β : (M̃, β^{−1}(C)) → (M, C),

which we call the blow-up of M with centre C.

The blow-analytic equivalence readily generalises to the higher dimensional real analytic case (𝕂 = ℝ), but not to the complex case.

For further details see [3].

A number of puzzling examples have been discovered in recent years.

The first was by Koike in [8]. He discovered a blow-analytic homeomorphism on ℝ^{3} which is not Lipschitz.

In [12] the following simpler example is constructed. Consider

h : ℝ^{3} → ℝ^{3}, h(X, Y, Z) = (X, Y, Z − 2\frac{XY^{5}}{X^{4} + Y^{6}} ),

whose inverse is

h^{−1} : ℝ^{3} → ℝ^{3}, h^{−1}(x, y, z) = (x, y, z + 2\frac{xy^{5}}{x^{4} + y^{6}} ).

The component functions are arc-analytic, whence blow-analytic. Take a smooth arc λ(t) = (t^{3}, t^{2}, t). Then h ∘ λ(t) = (t^{3}, t^{2}, 0) is a cusp. Hence h

cannot be Lipschitz.

In [7], some blow-analytic homeomorphisms on ℝ^{2} are constructed which can transform singular curves to smooth ones; in particular, they are not Lipschitz.

In a forthcoming paper [4], a more systematic construction of this kind of homeomorphisms is developed.

Much research remains to be accomplished in this dirrection.

References

[1] E.Bierstone and P.D.Milman, Arc-analytic functions, Invent. math., 101 (1990), 411-424.

[2] T. Fukui, Seeking invariants for blow-analytic equivalence, Compositio Math. 105 (1997), 95– 107.

[3] T. Fukui,S. Koike, T.-C.Kuo, Blow-analytic equisingularities, properties, problems and progress, in “Real analytic and algebraic singularities”, Pitman Research Notes in Mathematics Series, 381, 1997, Longman, 8–29.

[4] T. Fukui, T.-C.Kuo, L. Paunescu, Miscellaneous results on blow-analytic homeomorphisms, in preparation.

[5] T. Fukui, L. Paunescu, Modified analytic trivialization for weighted homogeneous functiongerms, J. Math. Soc. Japan (to appear).

[6] S. Izumi,S. Koike, T.-C.Kuo, A formula for the Fukui invariant and stability, in preparation.

[7] M. Kobayashi, T.-C.Kuo, On Blow-analytic equivalence of embedded curve singularities, in “Real analytic and algebraic singularities”, Pitman Research Notes in Mathematics Series, 381, 1997, Longman, 30–37.

[8] S. Koike, On strong C^{0}- equivalence of real analytic functions J. Math. Soc. Japan 45 (1993), 313–320.

[9] T.-C.Kuo, The modified analytic trivialization of singularities, J. Math. Soc. Japan 32 (1980), 605–614.

[10] T.-C.Kuo, On classification of real singularities, Invent. math., 82 (1985), 257–262.

[11] K. Kurdyka, Ensembles semi-algébriques symétriques par arcs, Math. Ann., 282 ( 1988), 445-462.

[12] L.Paunescu, An example of blow-analytic homeomorphism, in “Real analytic and algebraic singularities”, Pitman Research Notes in Mathematics Series, 381, 1997, Longman, 62–63.

[13] H.Whitney, Local properties of analytic varieties, in “A Symposium in Honor of M.Morse”, (Ed. by S.S.Cairns), Princeton Univ. Press. 1965, 205–244.

T.C. Kuo and Laurentiu Paunescu: School of Mathematics and Statistics, University of Sydney, Sydney, NSW, 2006, Australia. E-mail: tck@maths.usyd.edu.au. E-mail: laurent@maths.usyd.edu.au.