Lipschitz stratification of complex hypersurfaces in codimension 2. (English) Zbl 1541.14004

Parusiński, Adam; Păunescu, Laurenţiu

J. Eur. Math. Soc. (JEMS) 25, No. 5, 1743-1781 (2023).

Zariski equisingularity is known to imply bi-Lipschitz triviality in the case of plane curve singularities, i.e., such families are topologically simple in a bi-Lipschitz sense. The goal of the article is to demonstrate that Zariski equisingular families of surface singularities in \(\mathbb{C}^3\) also display bi-Lipschitz triviality by extending to such singularities.
The paper explores the conjecture by J.-P. Henry and Mostowski that Zariski equisingular families of surface singularities in \(\mathbb{C}^3\) could naturally lead to Lipschitz stratifications. The main theorem says that generically linearly Zariski equisingular families of surface singularities in \(\mathbb{C}^3\) are bi-Lipschitz trivial. This theorem is proved using local parameterizations of geometric constructs.

Reviewer: Meral Tosun (İstanbul)

MSC:

14B05	Singularities in algebraic geometry
32S60	Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)
32S15	Equisingularity (topological and analytic)
32B15	Analytic subsets of affine space

Keywords:

Zariski equisingularity; polar curves, Lipschitz stratifications

References:

[1]	S /. Then it is enough to extend vj S to a Lipschitz vector field on S [ 0 , since then such an extension defines a Lipschitz vector field on S [ .s/ [ 0 .s/ for every s sufficiently small, with the Lipschitz constant independent of s. Case 2: dist. .s/; 0 .s// dist. 0 .s/;
[2]	S /. Then it suffices to extend v from to a Lip-schitz vector field on [ 0 . Note that we may suppose that on both arcs , 0 we have y D O.x/, z D O.x/, that is, they are in the form (32). Indeed, by the Transversality Assumptions the variable z restricted to an arc in X cannot dominate x and y, that is, x D o.z/;
[3]	small. This is a change of coordinates in the target of the projection .x; y; z; t / 7 ! .x; y; t / and affects neither the discriminant locus nor Zariski’s equisingularity. To make the proof more precise we will use the constant “ of Definition 4.1 and denote the resulting union of polar wedges and the singular set by P W ” . If both .s/; 0 .s/ belong to P W ” then the claim follows from the first part of the proof (Section 6). In Case 1, given a stratified Lipschitz vector field v onto S we extend it on 0 .
[4]	By Proposition 9.1 we may suppose that dist. .s/; C j / & s m j for every j , and there-fore, for b small, say b Ä ”, dist. .s/; C j / dist. .s/;
[5]	Ä ”=2. Thus there exists a quasi-wing QW containing 0 and moreover dist. 0 .s/;
[6]	S / D dist. b 0 . 0 .s//; b 0 / s l , where l D max ¹max l i ; max r k º and b 0 denotes the discrim-inant of b 0 . Then there is a Lipschitz extension of v to QW by Proposition 9.4. Similarly, in Case 2 we may suppose dist. .s/; C j / dist. 0 .s/; C j / & s m j for every j , since otherwise, by Proposition 9.1, both .s/; 0 .s/ belong to P W ” . Then, choosing b appropriately, we may suppose that dist. b . .s//; b . 0 .s/// dist. .s/; 0 .s// s l :
[7]	Let QW be a quasi-wing containing . It always exists by Corollary 7.8, and 0 is contained either in QW or in another quasi-wing QW 0 such that QW and QW 0 are nicely-
[8]	Birbrair, L., Neumann, W. D., Pichon, A.: The thick-thin decomposition and the bilipschitz classification of normal surface singularities. Acta Math. 212, 199-256 (2014) Zbl 1303.14016 MR 3207758 · Zbl 1303.14016
[9]	Briançon, J., Henry, J. P. G.: Équisingularité générique des familles de surfaces à singularité isolée. Bull. Soc. Math. France 108, 259-281 (1980) Zbl 0482.14004 MR 606093 · Zbl 0482.14004
[10]	Cubides Kovacsisc, P., Halupczok, I.: Arc-wise analytic t-stratifications. arXiv:2106.11289 (2021)
[11]	Federer, H.: Geometric Measure Theory. Grundlehren Math. Wiss. 153, Springer, New York (1969) Zbl 0176.00801 MR 0257325 · Zbl 0176.00801
[12]	Fernández de Bobadilla, J., Fernandes, A., Sampaio, J. E.: Multiplicity and degree as bi-Lipschitz invariants for complex sets. J. Topol. 11, 958-966 (2018) Zbl 1409.32024 MR 3989434 · Zbl 1409.32024
[13]	Halupczok, I., Yin, Y.: Lipschitz stratifications in power-bounded o-minimal fields. J. Eur. Math. Soc. 20, 2717-2767 (2018) Zbl 06965903 MR 3861807 · Zbl 1522.03140
[14]	Luengo, I.: An example concerning a question of Zariski. Bull. Soc. Math. France 113, 379-386 (1985) Zbl 0605.14013 MR 850774 · Zbl 0605.14013
[15]	McShane, E. J.: Extension of range of functions. Bull. Amer. Math. Soc. 40, 837-842 (1934) Zbl 0010.34606 MR 1562984 · JFM 60.0973.02
[16]	Mostowski, T.: Lipschitz equisingularity. Dissertationes Math. (Rozprawy Mat.) 243, 46 pp. (1985) Zbl 0578.32020 MR 808226
[17]	Mostowski, T.: Tangent cones and Lipschitz stratifications. In: Singularities (Warszawa, 1985), Banach Center Publ. 20, PWN, Warszawa, 303-322 (1988) Zbl 0662.32012 MR 1101848 · Zbl 0662.32012
[18]	Neumann, W. D., Pichon, A.: Lipschitz geometry of complex surfaces: analytic invariants and equisingularity. arXiv:1211.4897 (2012)
[19]	Nguyen, N., Valette, G.: Lipschitz stratifications in o-minimal structures. Ann. Sci. École Norm. Sup. (4) 49, 399-421 (2016) Zbl 1353.32011 MR 3481354 · Zbl 1353.32011
[20]	Parusiński, A.: Lipschitz properties of semi-analytic sets. Ann. Inst. Fourier (Grenoble) 38, 189-213 (1988) Zbl 0631.32006 MR 978246 · Zbl 0631.32006
[21]	Parusiński, A.: Lipschitz stratification. In: Global Analysis in Modern Mathematics (Orono, ME, 1991; Waltham, MA, 1992), Publish or Perish, Houston, TX, 73-89 (1993) Zbl 0932.32010 MR 1278751 · Zbl 0932.32010
[22]	Parusiński, A.: Integrability of some functions on semi-analytic sets. In: Singularités Franco-Japonaises, Sémin. Congr. 10, Soc. Math. France, Paris, 243-254 (2005) Zbl 1084.32007 MR 2145958 · Zbl 1084.32007
[23]	Parusiński, A.: Algebro-geometric equisingularity of Zariski. In: Handbook of Geometry and Topology of Singularities II, J. L. Cisneros-Molina et al. (eds.), Springer, 177-222 (2021) Zbl 07464608 MR 4261550 · Zbl 1486.32011
[24]	Parusiński, A., Pȃunescu, L.: Arc-wise analytic stratification, Whitney fibering conjecture and Zariski equisingularity. Adv. Math. 309, 254-305 (2017) Zbl 1375.32048 MR 3607278 · Zbl 1375.32048
[25]	Pham, F.: Fractions lipschitziennes et saturation de Zariski des algèbres analytiques com-plexes. In: Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, 649-654 (1971) Zbl 0245.32003 MR 0590058 · Zbl 0245.32003
[26]	Siebenmann, L., Sullivan, D.: On complexes that are Lipschitz manifolds. In: Geometric Topo-logy (Athens, GA, 1977), Academic Press, New York, 503-525 (1979) Zbl 0478.57008 MR 537747 · Zbl 0478.57008
[27]	Speder, J. P.: Équisingularité et conditions de Whitney. Amer. J. Math. 97, 571-588 (1975) Zbl 0323.14006 MR 444665 · Zbl 0323.14006
[28]	Teissier, B.: The hunting of invariants in the geometry of discriminants. In: Real and Complex Singularities (Oslo, 1976), 565-678 (1977) Zbl 0388.32010 MR 0568901 · Zbl 0388.32010
[29]	Teissier, B.: Variétés polaires. II. Multiplicités polaires, sections planes, et conditions de Whit-ney. In: Algebraic Geometry (La Rábida, 1981), Lecture Notes in Math. 961, Springer, Berlin, 314-491 (1982) Zbl 0585.14008 MR 708342 · Zbl 0585.14008
[30]	Valette, G.: Lipschitz triangulations. Illinois J. Math. 49, 953-979 (2005) Zbl 1154.14323 MR 2210270 · Zbl 1154.14323
[31]	Whitney, H.: Analytic extensions of differentiable functions defined in closed sets. Trans. Amer. Math. Soc. 36, 63-89 (1934) Zbl 60.0217.01 MR 1501735 · JFM 60.0217.01
[32]	Whitney, H.: Tangents to an analytic variety. Ann. of Math. (2) 81, 496-549 (1965) Zbl 0152.27701 MR 192520 · Zbl 0152.27701
[33]	Zariski, O.: Studies in equisingularity. I. Equivalent singularities of plane algebroid curves. Amer. J. Math. 87, 507-536 (1965) Zbl 0132.41601 MR 177985 · Zbl 0132.41601
[34]	Zariski, O.: Studies in equisingularity. II. Equisingularity in codimension 1 (and characteristic zero). Amer. J. Math. 87, 972-1006 (1965) Zbl 0146.42502 MR 191898 · Zbl 0146.42502
[35]	Zariski, O.: On equimultiple subvarieties of algebroid hypersurfaces. Proc. Nat. Acad. Sci. U.S.A. 72, 1425-1426 (1975) Zbl 0304.14008 MR 389894 · Zbl 0304.14008
[36]	Zariski, O.: Foundations of a general theory of equisingularity on r-dimensional algebroid and algebraic varieties, of embedding dimension r C 1. Amer. J. Math. 101, 453-514 (1979) Zbl 0417.14008 MR 528001 · Zbl 0417.14008