Lipschitz Fractions of a Complex Analytic Algebra and Zariski Saturation

Abstract

This text is about the algebra of germs of Lipschitz meromorphic functions on a germ of reduced complex analytic space (X, 0). It is shown to be an analytic algebra, the Lipschitz saturation of the algebra of (X, 0), which in some important cases coincides with Zariski’s algebraic saturation. In the case of reduced germs of plane curves, the results in Sect. 10.6 imply that two such germs are topologically equivalent if and only if their Lipschitz saturations are analytically isomorphic. Applications to bi-Lipschitz equisingularity are given.

Appendix: Stratification, Whitney’s (a)-property and Transversality

A stratification ¹³ of an analytic (reduced) space X is a locally finite partition of X in smooth varieties called strata, such that:

• 1. the closure $\overline {W}$ of every stratum W is an (irreducible) analytic space;

• 2. the boundary $\partial W= \overline {W}\setminus W$ of every stratum W is a union of strata.

We call star of a stratum W the set of strata which have W in their boundary.

Let (W ₀, W) be an ordered pair of strata, with W ₀ ⊂ ∂W. We say that this ordered pair satisfies the property (a) of Whitney at a point x ₀ ∈ W ₀ if for every sequence of points x _i ∈ W tending to x ₀ in such a way that the tangent space $T_{x_i}(W)$ admits a limit, this limit contains the tangent space $T_{x_0}(W)$ (we suppose that X is locally embedded in a Euclidean space, in such a way that the tangent spaces are realized as subspaces of the same vector space; the property (a) of Whitney is independent of the chosen embedding). For every ordered pair of strata (W ₀, W) of a stratification, there exists a Zariski dense open set of points of W ₀ where the property (a) of Whitney is satisfied [Whi65]. We can then refine every stratification into a stratification such that the property (a) of Whitney is satisfied at every point for every ordered pair of strata.

Proposition 10.8.7

Let (X, x ₀) be a stratified germ of complex analytic space such that the ordered pairs of strata (W ₀, W) satisfy the property (a) of Whitney, where W ₀ denotes the stratum which contains x ₀ and where W is any stratum of the star of W ₀ . Let φ : X →C ^m be a morphism germ such that φ|W ₀ is effectively transversal to the value 0 at the point x ₀ . Then, for every stratum W, φ|W is effectively transversal to the value 0 at (at least) one point of W arbitrarily close to x ₀.

Proof

The transversality of φ|W at every point close to x ₀ is an obvious consequence of the property (a) of Whitney for the ordered pair (W ₀, W). It remains to prove the effective transversality i.e., to prove that (φ|W)⁻¹(0) is not empty. But $(\varphi {| \overline {W}} )^{-1}(0)$ is a closed analytic subset of $\overline {W}$, non empty (because it contains the point x ₀) and defined by m equations. Therefore its codimension is at most m. If (φ|W)⁻¹(0) were empty, then ∂W would contain at least one stratum W′ such that (φ|W′)⁻¹(0) is non empty and of dimension $\geqslant \dim W - m$. But on the other hand, the transversality of φ|W′ implies that (φ|W′)⁻¹(0) is a smooth variety of dimension $< \dim W' - m$, and then of dimension $< \dim W - m$. We then get a contradiction. □

Remark 10.8.8

Of course, in the statement of Proposition 10.8.7, we could replace the transversality relative to the value of 0 by the transversality relative to a smooth variety of C ^m.