An arc-analytic function with nondiscrete singular set

ANNALES

POLONICI MATHEMATICI LIX.3 (1994)

An arc-analytic function with nondiscrete singular set

by Krzysztof Kurdyka (Le Bourget-du-Lac and Krak´ow)

Abstract. We construct an arc-analytic function (i.e. analytic on every real-analytic arc) in R ² which is analytic outside a nondiscrete subset of R ² .

Let M be a real-analytic manifold. A function f : M → R is called arc-analytic iff for every analytic arc γ : ]−ε, ε[ → M the composition f ◦ γ is analytic (see [K1]). For every function f on M , let Sing f denote the set of points of nonanalyticity of f . If f is an arc-analytic function with subanalytic graph, then Sing f is a subanalytic subset of M (see [T] or [K2]); moreover, dim(Sing f ) ≤ dim M − 2 (see [K1]). Actually, in this case a stronger result is true: there exists a locally finite composition π of local blowing-ups of M such that f ◦ π is analytic (see [BM], [P]). Recently examples of arc-analytic functions with nonsubanalytic graphs were given ([K3] and [BMP], where a discontinuous example is given).

Suppose that f : R ² → R is an arc-analytic function. Professor Siciak asked whether Sing f is always discrete. By the previous remarks this is the case if, for example, f has subanalytic graph, because Sing f , being subanalytic of dimension 0, contains only isolated points.

In this note we construct an arc-analytic function f : R ² → R such that Sing f is nondiscrete and f is unbounded at each point of Sing f . Our construction is based on an idea of [K3].

The author is grateful to Professor Siciak for stating the problem.

Let a ν ∈ R, ν ∈ N, ν ≥ 1, be a convergent sequence in R. Define a 0 = lim

ν→∞ a ν , Z 0 = {(a ν , 0) ∈ R ² : ν ∈ N, ν ≥ 0} .

We will construct an arc-analytic function f : R ² → R such that Sing f = Z 0 , and f is unbounded at each point of Z 0 .

1991 Mathematics Subject Classification: Primary 32B20; Secondary 32B30.

Key words and phrases: arc-analytic, blow-up, projective limit, strict transform.

252 K. K u r d y k a

We will blow up every point of Z 0 infinitely many times. To get a formal construction we take the projective limit of the following system.

Set c ν = (a ν , 0), P 0 = {y = 0} ⊂ R ² .

(i) Let X 0 = R ² , and let π 1,0 : X 1 → X ₀ be the blowing-up of c 0 in X 0 . Let P 1 be the strict transform of P 0 . We put

c ¹ ₀ = π _1,0 ⁻¹ (c 0 ) ∩ P 1 and c ¹ _ν = π ⁻¹ _1,0 (c ν ) for ν ≥ 1 (we assume that c ν 6= c _µ for ν 6= µ).

(ii) Suppose we have already constructed π n,n−1 : X n → X _n−1 , and P n

is the strict transform of P n−1 by π n,n−1 . Suppose we also have a sequence c ⁿ _ν ∈ P _n such that π n,n−1 (c ⁿ _ν ) = c ⁿ⁻¹ _ν , ν ∈ N. We define π n+1,n to be the compositon

X _n ^{n p}

n−1

−→ X

_n ^{n−1 p}

−→ . . .

^p

−→ X

_n ^{1 p}

−→ X

_n ⁰ = X n

where each p ⁱ _n : X _n ⁱ⁺¹ → X _n ⁱ , i = 0, . . . , n − 1, is the blowing-up of (p ⁱ⁻¹ _n ◦ . . . . . . ◦ p ⁰ _n ) ⁻¹ (c ⁿ _i ) in X n . We put X n+1 = X _n ⁿ and

π n+1,n = p ⁿ⁻¹ _n ◦ . . . ◦ p ⁰ _n .

Finally, let P n+1 be the strict transform of P n by π n+1,n and let c ⁿ⁺¹ _ν = (π n+1,n ) ⁻¹ (c ⁿ _ν ) ∩ P n+1 , ν ∈ N .

We define Z n = {c ⁿ _ν ∈ X n : ν ∈ N}. For every n ∈ N we put π ^n,n = id X

, and for m ≤ n we put

π n,m = π n,n−1 ◦ . . . ◦ π _m+1,m .

Hence we have constructed a projective system π n,m : X n → X _m , m ≤ n, with a subsystem Z n → Z _m . Then there exist topological spaces X =

←− lim X n , Z = lim ←− Z n and continuous mappings pr _n : X → X n such that pr _m = π n,m ◦ pr _n for m ≤ n.

Set L = X \ Z. Clearly L is a Hausdorff σ-compact topological space.

We will define a structure of a real-analytic manifold on L.

Let x = (x n ) _n∈N ∈ L. Then there exists an open neighbourhood U of x in L and n 0 ∈ N such that if y = (y ⁿ ) ∈ U then y k = pr _k (y) 6∈ Z k for k ≥ n 0 . Hence pr _k | _U : U → pr _k (U ) is a homeomorphism. Notice that pr _k (U ) is an open subset of the real-analytic manifold X k . Clearly the family of all such mappings defines a structure of a real-analytic manifold on L. Moreover, each pr _k is analytic on L. Notice also that

pr _n : L \ pr ⁻¹ _n (Z n ) → X n \ Z _n

is an analytic diffeomorphism. In the sequel we need pr ⁻¹ ₀ : R ² \ Z ₀ → L,

which we denote by q. The mapping q has the following property:

An arc-analytic function 253

Lemma. Let γ : ]−ε, ε[ → R ² be an analytic arc such that γ(t) 6∈ P 0 = {y = 0} for t 6= 0. Then the mapping

q ◦ γ : ]−ε, 0[ ∪ ]0, ε[ → L extends to an analytic mapping from ]−ε, ε[ to L.

P r o o f. If γ(0) 6∈ Z 0 then the assertion is trivial. Suppose that γ(0) = c ν

for some ν 0 ∈ N. The order of contact of γ(]−ε, ε[) and P ⁰ = {y = 0} at γ(0) is finite. If we blow up the point γ(0), then either the strict transforms of those curves are disjoint or the order of their contact decreases by 1. Hence for some n ∈ N,

t→0 lim (π n,0 ) ⁻¹ ◦ γ(t) = e γ n (0) 6∈ P n .

Clearly e γ n = (π n,0 ) ⁻¹ ◦ γ has an analytic extension through 0. Since q ◦ γ = pr ⁻¹ _n ◦ (π _n,0 ) ⁻¹ ◦ γ

and pr ⁻¹ _n is analytic outside Z n (recall that Z n ⊂ P _n ), it follows that q ◦ γ extends to a function analytic at 0. This ends the proof of the lemma.

Recall that our analytic manifold L has a countable basis of topology, hence by the Grauert Embedding Theorem ([G]) there exists a proper ana- lytic embedding ϕ : L → R ^N for some N ∈ N.

Take now a countable subset A of P 0 \ Z ₀ which is discrete in R ² \ Z ₀ and A \ A = Z 0 . Notice that for every n ∈ N the set π n,0 ⁻¹ (A) is also discrete in X n \ Z n ; moreover, pr _n maps homeomorphically L \ pr ⁻¹ _n (Z n ) onto X n \ Z n . Hence q ⁻¹ (A) is also discrete in L = lim ←− X n \ lim ←− Z n . Thus the set e A = ϕ(q ⁻¹ (A)) is discrete in R ^N , since ϕ is proper.

We claim that there exists a discrete subset e B of ϕ(L) such that if we set B = pr ₀ ◦ ϕ ⁻¹ ( e B) then

B ∩ P 0 = ∅ and B \ B = Z 0 .

To get such a e B let us arrange the elements of e A in a sequence e a k , k ∈ N.

Notice that ϕ(pr ⁻¹ ₀ (P 0 )) is nowhere dense in ϕ(L). Hence there exists a sequence

e b k ∈ ϕ(L) \ ϕ(pr ⁻¹ ₀ (P 0 ))

such that k e a k −e b k k < 1/k. We put B = {b ₀ , b 1 , . . .}, where b k = pr ₀ ◦ϕ ⁻¹ (e b k ).

Let us write, in coordinates in R ² , b k = (x k , y k ). Notice that y k 6= 0 for all k ∈ N.

Now take an analytic function e h : R ^N → R such that eh(eb k ) = y ⁻² _k . Such an e h exists since ϕ(L) is closed in R ^N , hence e B is discrete in R ^N . Now put h = e h ◦ ϕ ◦ q and observe that h is analytic in R ² \ Z ₀ . Finally, put

f (x, y) =  yh(x, y) if (x, y) 6∈ Z 0 ,

0 if (x, y) ∈ Z 0 .

254 K. K u r d y k a

To see that f is arc-analytic take an analytic arc γ : ]−ε, ε[ → R ² . If γ(]−ε, ε[) ⊂ P 0 = {y = 0} then f ◦ γ ≡ 0 is analytic. Otherwise the set γ ⁻¹ (Z 0 ) is discrete in ]−ε, ε[ and by the lemma q ◦ γ extends to an analytic mapping from ]−ε, ε[ to L. Hence also h ◦ γ extends to an analytic mapping on ]−ε, ε[. Thus f ◦ γ is analytic on ]−ε, ε[.

Clearly f is analytic in R ² \ Z 0 . Observe that f (b k ) = f (x k , y k ) = y _k ⁻¹ and lim k→∞ y k = 0. Since B \ B = Z 0 , for every (x 0 , y 0 ) ∈ Z 0 we have

lim sup

(x,y)→(x

,y

)

|f (x, y)| = +∞ . This proves that Sing f = Z 0 .

R e m a r k. This example raises two questions about arc-analytic func- tions.

1) Can one find an arc-analytic function on a manifold M such that Sing f is dense in the analytic Zariski topology (i.e. every analytic function vanishing on Sing f must vanish on M )?

2) Given an arc-analytic function f : M → R, can one find a countable composition π of blowing-ups such that f ◦ π is analytic? Here countable composition might be understood as a projective limit as in our example.

References

[BM] E. B i e r s t o n e and P. D. M i l m a n, Arc-analytic functions, Invent. Math. 101 (1990), 411–424.

[BMP] E. B i e r s t o n e, P. D. M i l m a n and A. P a r u s i ´ n s k i, A function which is arc- analytic but not continuous, Proc. Amer. Math. Soc. 113 (1991), 419–423.

[G] H. G r a u e r t, On Levi’s problem and the imbedding of real-analytic manifolds, Ann. of Math. 68 (1958), 460–472.

[K1] K. K u r d y k a, Points r´ eguliers d’un sous-analytique, Ann. Inst. Fourier (Gre- noble) 38 (1) (1988), 133–156.

[K2] —, Ensembles semi-alg´ ebriques sym´ etriques par arcs, Math. Ann. 282 (1988), 445–462.

[K3] —, A counterexample to subanalyticity of an arc-analytic function, Ann. Polon.

Math. 55 (1991), 241–243.

[P] A. P a r u s i ´ n s k i, Subanalytic functions, Trans. Amer. Math. Soc., to appear.

[T] M. T a m m, Subanalytic sets in the calculus of variations, Acta Math. 146 (1981), 167–199.

LABORATORIE DE MATH ´ EMATIQUES INSTITUTE OF MATHEMATICS

UNIVERSIT ´ E DE SAVOIE JAGIELLONIAN UNIVERSITY

CAMPUS SCIENTIFIQUE REYMONTA 4

73376 LE BOURGET-DU-LAC CEDEX, FRANCE. 30-059 KRAK ´ OW, POLAND E-mail: KURDYKA@UNIV-SAVOIE.FR E-mail: KURDYKA@IM.UJ.EDU.PL

Re¸ cu par la R´ edaction le 27.5.1993