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.

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 |