A counterexample to subanalyticity of an arc-analytic function

Abstract

We construct an arc-analytic function (i.e. a function analytic on each analytic arc) whose graph is not subanalytic. Let f : U → R be a function, where U is open in R. We say that f is arc-analytic iff for each analytic arc γ : (−1, 1)→ U , the composition f ◦ γ is analytic (see [K2], [BM] for examples). If we suppose moreover that f has subanalytic graph it turns out that such an f has some interesting properties. For example if we compose f with a suitable finite composition of local blowing-ups we get an analytic function (see [BM]). In Spring 1985, during the Warsaw Semester on Singularities, after discussions with E. Bierstone, P. Milman and B. Teissier the following conjecture was stated. Conjecture. Every arc-analytic function is locally subanalytic. More precisely, given an arc-analytic function f : U → R, where U is open in R, for each x ∈ U there is a neighborhood Vx of x such that the restriction f |Vx has subanalytic graph. In this paper we give a counterexample to this conjecture. The idea of our construction was suggested by an example, due to G. Dloussky, of a mapping which is meromorphic in the sense of Stoll but not in the sense of Remmert (see 5.5 in [D]). I wish to express my gratitude to G. Dloussky for enlightening discussions. We are going to construct by induction an infinite composition of blowing-ups. Put X0 = R, P0 = {(x, y) ∈ R : y = 0}, c0 = (0, 0). We denote by π1,0 : X1 → X0 a blowing-up of X0 centered at c0. Suppose we have already constructed a blowing-up πn,n−1 : Xn → Xn−1 centered at cn−1. We denote by Pn a strict transform of Pn−1, and we put Dn = π−1 n,n−1(cn−1), {cn} = Pn ∩Dn. We take for πn+1,n : Xn+1 → Xn a blowing-up of Xn centered at cn. If n > m we put πn,m = πn,n−1 ◦ . . . ◦ πm+1,m : Xn → Xm, 1991 Mathematics Subject Classification: Primary 32B20; Secondary 32B30.