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.


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