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Variétés polaires. II: Multiplicités polaires, sections planes et conditions de Whitney. (Polar varieties. II: Polar multiplicities, plane sections and the Whitney conditions). (French) Zbl 0585.14008
Algebraic geometry, Proc. int. Conf., La Rábida/Spain 1981, Lect. Notes Math. 961, 314-491 (1982).
[For the entire collection see Zbl 0487.00004. For part I see Inst. Élie Cartan, Univ. Nancy I 3, 33-55 (1981; Zbl 0572.14002).]
Let (X,x) be a germ of a reduced analytic space of (pure) dimension d. A collection of d natural numbers (polar multiplicities) $M^*_{X,x}=\{m_ x(X),\quad m_ x(P_ 1(X)),...,m_ x(P_{d- 1}(X))\}$ corresponds to it. Here $$P_ k(X)$$ is the (local) polar variety in general position of codimension k for the germ X, $$m_ x$$ the multiplicity in the point x. The local polar variety $$P_ k(X)$$ of codimension k can be defined in the following way. Let $$(X,x)\to ({\mathbb{C}}^ N,0)$$ be an imbedding of the germ (X,x) into a complex linear space, p: ($${\mathbb{C}}^ N,0)\to ({\mathbb{C}}^{d-k+1},0)$$ be the projection along a subspace L in general position of dimension $$N-d+k-1$$. The polar variety $$P_ k(x)$$ is the closure in X for the set of critical points of the restriction of the projection p to the set $$X^ 0$$ of non-singular points of the germ X. It is either empty or an analytic supspace of pure codimension k in X. Its multiplicity in the point x is denoted by $$m_ x(P_ k(X))$$. It does not depend on the space L along which the projection is realized if L is chosen in general position. We have $$X=P_ 0(X)$$, i.e., $$m_ x(X)=m_ x(P_ 0(x))$$. - Main result of the paper under review:
Theorem: Let X be a reduced, complex-analytic space of pure dimension d, Y be a non-singular analytic subspace of the space X, $$0\in Y$$. The following conditions are equivalent: (1) the pair $$(X^ 0,Y)$$ satisfies the Whitney conditions (a),(b) in 0 $$(X^ 0$$ is the set of non-singular points of the space X); (2) the collection of polar multiplicities $$M^*_{X,y} (y\in Y)$$ is constant for all $$y\in Y$$ of a neighbourhood of the point 0.
Thus, the pair $$(X^ 0,Y)$$ does not satisfy the Whitney conditions in the point 0 if and only if the multiplicity of one of the local polar varieties $$P_ k(X)$$ of general form is not constant in a neighbourhood of 0 on Y. The fact, that the collections of polar multiplicities $$M^*_{X,x}$$ can be (in a certain sense) computed by topological methods while the immediate checking of fulfilment of the Whitney conditions requires analytical computations, accounts for the significance of this result.

##### MSC:
 14B05 Singularities in algebraic geometry 13H15 Multiplicity theory and related topics 32Sxx Complex singularities