[Paper Review] Testing Lipschitz non normally embedded complex spaces

Maciej P. Denkowski, Mihai Tibăr|arXiv (Cornell University)|Mar 20, 2017

Advanced Topology and Set Theory5 citations

TL;DR

This paper introduces a sectional criterion to test whether complex analytic germs are Lipschitz non-normally embedded (non-NE). By slicing the germ with admissible linear subspaces, the method reduces the non-NE property to the existence of a non-NE curve in the slice. The key contribution is Theorem 1.3, which establishes that if any such slice contains a non-NE curve, then the original germ is non-NE, providing an effective and sharp test for non-NE singularities in complex spaces of pure dimension >1.

ABSTRACT

We introduce a sectional criterion for testing if complex analytic germs $(X,0) \subset (\bC^n, 0)$ are Lipschitz non normally embedded.

Motivation & Objective

• To develop an effective, geometric criterion for detecting Lipschitz non-normal embedding (non-NE) in complex analytic germs.
• To address the difficulty of verifying non-NE status, which typically requires comparing distances along pairs of arcs.
• To generalize existing results on non-NE singularities, particularly for Brieskorn-type hypersurfaces and tangent cones with multiple components.
• To establish a sharp condition under which slicing by linear subspaces preserves or detects non-NE behavior.
• To provide a systematic, algorithmic approach to testing non-NE via curve slices, applicable to higher-dimensional complex spaces.

Proposed method

• The method employs linear projections π: ℂⁿ → ℂᵏ that are general with respect to the germ (X,0), ensuring transversality to the tangent cone.
• It defines admissible (n−k+1)-planes P = π⁻¹(ℓ) where ℓ is a line in ℂᵏ not contained in the discriminant locus Δₚ of π|X.
• The core idea is that if any such slice P ∩ X contains a non-NE curve germ Γ at 0, then (X,0) is non-NE.
• The proof relies on Puiseux parametrizations of two distinct real curves in Γ, showing their outer distance is O(t^q) for q > 1.
• It uses the inner distance lower bound d_X(γ₁(t),γ₂(t)) ≥ d(tw, Δₚ), and the fact that d(tw, Δₚ) ≥ rt for small t when ℓ ∉ C₀(Δₚ).
• The ratio d_X / d outer grows as t^{1−q} → ∞ as t → 0⁺, proving non-NE via blow-up of the bi-Lipschitz constant.

Experimental results

Research questions

• RQ1 Can the non-NE property of a complex analytic germ (X,0) be tested via its intersections with linear subspaces?
• RQ2 Under what conditions does the non-NE property of a slice imply the non-NE property of the original germ?
• RQ3 Is there a sharp, geometric criterion for detecting non-NE singularities that avoids direct arc-based distance comparisons?
• RQ4 Does the presence of a multiple irreducible component in the tangent cone imply non-NE for the germ?
• RQ5 Can the sectional method be applied to classify non-NE behavior in Pham-Brieskorn hypersurfaces?

Key findings

• Theorem 1.3 establishes that if an admissible (n−k+1)-plane P intersects (X,0) in a non-NE curve germ Γ, then (X,0) is non-NE at 0.
• The admissibility condition (ℓ ∉ C₀(Δₚ)) is sharp: the method fails without it, as shown in Example 4.2 with the cusp x² + y² = z³.
• Corollary 1.4 proves that if the tangent cone C₀(X) has an irreducible component of multiplicity ≥2, then (X,0) is non-NE.
• For Pham-Brieskorn hypersurfaces X = {∑aᵢxᵢᵏⁱ = 0} with 1 < k₁ < k₂ ≤ … ≤ kₙ, the germ is non-NE at 0 due to the multiple component {x₁=0} in C₀(X) with multiplicity k₁ > 1.
• The method is effective and sharp: it detects non-NE status without requiring explicit arc parametrizations, offering a geometric alternative to direct distance comparisons.
• The result confirms that non-NE behavior is intrinsic and independent of embedding, as the NE property is preserved under bi-Lipschitz homeomorphisms.