Varietes polaires II Multiplicites polaires, sections planes, et conditions de whitney

Abstract

To each reduced equidimensional analytic algebra O _X,x, one can associate a sequence of d integers, where O _X,x:

$$M_{X,x}^* = \{ m_x (X),m_x (P_1 (X)),...,m_x (P_{d - 1} (X))\}$$

where for 0≤k≤d−1, P_k (X) is a general local polar variety of codimension k of X, as defined by Lê D.T. and myself, and m_x denotes the multiplicity at x.

One can visualize P_k(X) as follows : Pick an embedding X⊂ℂ^N of a representative of (X, x) and take a general linear projection p : ℂ^N→ℂ^d-k+1. The closure in X of the critical locus of the restriction p|X^o of p to the nonsingular part X^o of X is purely of codimension k or empty, its multiplicity at x is independent of the choice of the general linear projection p and of the embedding. It is denoted by m_X(p_k(X)). Note that P_o(X)=X. I prove here the Theorem : Let X be a reduced purely d-dimensional complex-analytic space, and Y a non-singular subspace of X. Given a point 0∈Y, the following conditions are equivalent :

• i) The pair (X^o, Y) satisfies the Whitney conditions at 0.

• ii) The map from Y to ℕ^dgiven by y ↦ M ^*_X,y is constant in a neighbourhood of 0 in Y.

Equivalently, (X^o,Y) does not satisfy the Whitney conditions at 0 if and only if one of the general local polar varieties P_k(X) is not equimultiple along Y at 0.

So the following picture already shows the general phenomenon : here X is the surface in ℂ³ defined by y²−x³−t²x²=0, Y is the t-axis, and p is the projection onto the (x,t)-plane : the curve defined by x+t²=0, y=0 is a general polar curve for X, it is not equimultiple along Y so (X^o,Y) does not satisfy the Whitney conditions at 0, which is obvious from the definition (see Chap. III)

An important feature of the sequence M ^*_X,x is that it is an analytic invariant of the germ (X,x) which can be computed topologically, just like the multiplicity can be computed by counting the number of points of intersection with X of a general (N-d)-plane in ℂ^N near x (see Chap. IV and VI).