Stratifications sous-analytiques. Condition de Verdier. (Sub-analytic stratifications. Verdier condition).

*(French)*Zbl 0617.32011In 1965 the first author in his IHES preprint ”Ensembles semianalytiques” presented a scheme of construction of a semi-analytic stratification satisfying Whitney conditions and compatible with a given family of semi- analytic sets. The same scheme, as he observed at once, works for subanalytic sets as well as for conditions other than Whitney’s, provided one can prove the following key Lemma: For the condition (X) and a pair of stratas N, M, such that \(N\subset \bar M\setminus M\), the set \(\{\) \(x\in N: N,M\) do not satisfy (X) at \(x\}\) is semi-(sub)analytic in the ambient space and dense in N. (The condition (X) is then what B. Teissier calls ”a stratifying condition” in his ”Variétés polaires. II.” in Lect. Notes Math. 961, 314-491 (1982; Zbl 0585.14008)). The reviewed paper first presents this general setting and then proves the key lemma mentioned above (which follows from Theorem 1 in the paper) for subanalytic sets and the (w) condition, defined as follows:

Definition. We say that the stratas N, M with \(N\subset \bar M\setminus M\) satisfy \(\epsilon (T_ xN,T_ zM)\leq K dist(z,x)\) with a certain constant K and for x in a neighborhood of d in N, z in a neighborhood of d in M. Here \(\epsilon\) is the function introduced by T. C. Kuo and usually denoted by \(\delta\) : \(\epsilon (U,W)=\sup_{x\in U,\| x\| =1} dist(x,W)\) for U, W vector subspaces (above \(T_ xN\), \(T_ zM\) are tangent spaces).

The methods used by the authors are these of the theory of subanalytic sets without desingularization, developed by the first author in his group in 1979-85. A very elegant geometric idea of Proposition 3 is to be noted. The main theorem (Theorem 3) about the existence of a subanalytic stratification compatible with the given family of subanalytic sets was first proved by J. L. Verdier [Invent. Math. 36, 295-312 (1976; Zbl 0333.32010)]. Since his proof involved Hironaka’s theorem of desingularisation, an elementary proof was considered worth trying. Another elementary proof was given, parallelly, by the reviewer and the third author (cf. Semin. Bologna 1986). It differs from the one discussed above in the demonstration of the key lemma.

Definition. We say that the stratas N, M with \(N\subset \bar M\setminus M\) satisfy \(\epsilon (T_ xN,T_ zM)\leq K dist(z,x)\) with a certain constant K and for x in a neighborhood of d in N, z in a neighborhood of d in M. Here \(\epsilon\) is the function introduced by T. C. Kuo and usually denoted by \(\delta\) : \(\epsilon (U,W)=\sup_{x\in U,\| x\| =1} dist(x,W)\) for U, W vector subspaces (above \(T_ xN\), \(T_ zM\) are tangent spaces).

The methods used by the authors are these of the theory of subanalytic sets without desingularization, developed by the first author in his group in 1979-85. A very elegant geometric idea of Proposition 3 is to be noted. The main theorem (Theorem 3) about the existence of a subanalytic stratification compatible with the given family of subanalytic sets was first proved by J. L. Verdier [Invent. Math. 36, 295-312 (1976; Zbl 0333.32010)]. Since his proof involved Hironaka’s theorem of desingularisation, an elementary proof was considered worth trying. Another elementary proof was given, parallelly, by the reviewer and the third author (cf. Semin. Bologna 1986). It differs from the one discussed above in the demonstration of the key lemma.

Reviewer: Z.Denkowska

##### MSC:

32B20 | Semi-analytic sets, subanalytic sets, and generalizations |