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Relative Lipschitz saturation of complex algebraic varieties

  • François Bernard
Veröffentlicht/Copyright: 19. Juli 2025
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Abstract

This paper is devoted to the study of the relative Lipschitz saturation of complex algebraic varieties. More precisely, we investigate the concept of Lipschitz saturation of a variety in another, and we focus on the case where the dominant morphism between the two varieties is not necessarily finite. In particular, we answer, in the case of algebraic varieties, an open question of Pham and Teissier concerning the finiteness of the Lipschitz saturation of general algebras. As an application, we provide algebraic criteria for two algebraic varieties to be linked by an algebraic morphism, which is a locally biLipschitz homeomorphism on the closed points of the variety.

MSC 2010: 14M05; 14B05; 13B22

Funding statement: This research was partially supported by Plan d’investissements France 2030, IDEX UP ANR-18-IDEX-0001.

Acknowledgements

The author is deeply grateful to Goulwen Fichou and Jean-Philippe Monnier for very useful discussions, to Antoni Rangachev and Bernard Teissier for helping improve this paper, and to the reviewer’s careful reading.

  1. Communicated by: D. Plaumann

References

[1] W. A. Adkins, Weak normality and Lipschitz saturation for ordinary singularities. Compositio Math. 51 (1984), 149–157. MR739729 Zbl 0591.14009Suche in Google Scholar

[2] A. Andreotti, E. Bombieri, Sugli omeomorfismi delle varietà algebriche. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 23 (1969), 431–450. MR266923 Zbl 0184.24503Suche in Google Scholar

[3] A. Andreotti, F. Norguet, La convexité holomorphe dans l’espace analytique des cycles d’une variété algébrique. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 21 (1967), 31–82. MR239118 Zbl 0176.04001Suche in Google Scholar

[4] F. Bernard, Seminormalization and rational continuous functions on complex algebraic varieties. Preprint 2022, to appear in Annales de l’Institut Fourier, arXiv 2109.06542v2Suche in Google Scholar

[5] F. Bernard, G. Fichou, J.-P. Monnier, R. Quarez, Algebraic characterizations of homeomorphisms between algebraic varieties. Math. Z. 308 (2024), Paper No. 12, 28 pages. MR4780623 Zbl 0789964610.1007/s00209-024-03533-5Suche in Google Scholar

[6] T. da Silva, M. Ribeiro, Universally injective and integral contractions on relative Lipschitz saturation of algebras. J. Algebra 662 (2025), 902–922. MR4798552 Zbl 0794582610.1016/j.jalgebra.2024.08.024Suche in Google Scholar

[7] T. da Silva, G. Schultz-Netto, A survey on relative Lipschitz saturation of algebras and its relation with radicial algebras. Latin American Journal of Mathematics 3 (2024), 1–18.10.14244/lajm.v3i1.31Suche in Google Scholar

[8] D. Duarte, A. E. Flores, On the Lipschitz saturation of toric singularities. Preprint 2024, arXiv 2310.03216v2Suche in Google Scholar

[9] D. Eisenbud, Commutative algebra. Springer 1995. MR1322960 Zbl 0819.1300110.1007/978-1-4612-5350-1Suche in Google Scholar

[10] A. Fernandes, Topological equivalence of complex curves and bi-Lipschitz homeomorphisms. Michigan Math. J. 51 (2003), 593–606. MR2021010 Zbl 1055.1402810.1307/mmj/1070919562Suche in Google Scholar

[11] A. Fernandes, Z. Jelonek, Bi-Lipschitz characterization of space curves. Bull. Braz. Math. Soc. (N.S.) 54 (2023), Paper No. 29, 6 pages. MR4600223 Zbl 1541.1408410.1007/s00574-023-00348-6Suche in Google Scholar

[12] A. G. Flores, O. N. da Silva, B. Teissier, The biLipschitz geometry of complex curves: an algebraic approach. In: Introduction to Lipschitz geometry of singularities, volume 2280 of Lecture Notes in Math., 217–271, Springer 2020. MR4200100 Zbl 1457.3206910.1007/978-3-030-61807-0_8Suche in Google Scholar

[13] T. Gaffney, The genericity of the infinitesimal Lipschitz condition for hypersurfaces. J. Singul. 10 (2014), 108–123. MR3300289 Zbl 1513.32043Suche in Google Scholar

[14] T. Gaffney, T. da Silva, Infinitesimal Lipschitz conditions on a family of analytic varieties: genericity and necessity. São Paulo J. Math. Sci. 18 (2024), 1207–1238. MR4837312 Zbl 0796117410.1007/s40863-024-00452-5Suche in Google Scholar

[15] T. J. Gaffney, T. F. da Silva, The generic equivalence among the Lipschitz saturations of a sheaf of modules. Res. Math. Sci. 11 (2024), Paper No. 32, 16 pages. MR4732448 Zbl 1537.3209910.1007/s40687-024-00442-1Suche in Google Scholar

[16] S. Greco, C. Traverso, On seminormal schemes. Compositio Math. 40 (1980), 325–365. MR571055 Zbl 0412.14024Suche in Google Scholar

[17] A. Grothendieck, Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes. Inst. Hautes Études Sci. Publ. Math. no. 8 (1961), 222 pages. MR217084 Zbl 0118.36206Suche in Google Scholar

[18] R. Hartshorne, Ample subvarieties of algebraic varieties. Springer 1970. MR282977 Zbl 0208.4890110.1007/BFb0067839Suche in Google Scholar

[19] M. Hochster, Lectures on integral closure, the Briançon–Skoda theorem and related topics in commutative algebra. Lecture notes 2014, https://dept.math.lsa.umich.edu/∼hochster/615W14/615.pdf.Suche in Google Scholar

[20] C. Huneke, I. Swanson, Integral closure of ideals, rings, and modules. Cambridge Univ. Press 2006. MR2266432 Zbl 1117.13001Suche in Google Scholar

[21] Z. Jelonek, On algebraic bi-Lipschitz homeomorphisms. Preprint 2021, to appear in Proc. Amer. Math. Soc., arXiv 2104.0689410.1090/proc/16320Suche in Google Scholar

[22] J. V. Leahy, M. A. Vitulli, Weakly normal varieties: the multicross singularity and some vanishing theorems on local cohomology. Nagoya Math. J. 83 (1981), 137–152. MR632650 Zbl 0509.1400510.1017/S0027763000019450Suche in Google Scholar

[23] M. Lejeune-Jalabert, B. Teissier, Clôture intégrale des idéaux et équisingularité. Ann. Fac. Sci. Toulouse Math. (6) 17 (2008), 781–859. MR2499856 Zbl 1171.1300510.5802/afst.1203Suche in Google Scholar

[24] J. Lipman, Relative Lipschitz-saturation. Amer. J. Math. 97 (1975), 791–813. MR417169 Zbl 0403.1400110.2307/2373777Suche in Google Scholar

[25] S. Łojasiewicz, Introduction to complex analytic geometry. Birkhäuser Verlag, Basel 1991. MR1131081 Zbl 0747.3200110.1007/978-3-0348-7617-9Suche in Google Scholar

[26] M. Manaresi, Some properties of weakly normal varieties. Nagoya Math. J. 77 (1980), 61–74. MR556308 Zbl 0403.1400110.1017/S0027763000018663Suche in Google Scholar

[27] W. D. Neumann, A. Pichon, Lipschitz geometry of complex curves. J. Singul. 10 (2014), 225–234. MR3300297 Zbl 1457.32072Suche in Google Scholar

[28] F. Pham, B. Teissier, Lipschitz fractions of a complex analytic algebra and Zariski saturation. In: Introduction to Lipschitz geometry of singularities, volume 2280 of Lecture Notes in Math., 309–337, Springer 2020. MR4200102 Zbl 1457.3207210.1007/978-3-030-61807-0_10Suche in Google Scholar

[29] A. Pichon, An introduction to Lipschitz geometry of complex singularities. In: Introduction to Lipschitz geometry of singularities, volume 2280 of Lecture Notes in Math., 167–216, Springer 2020. MR4200099 Zbl 1457.3207310.1007/978-3-030-61807-0_7Suche in Google Scholar

[30] The Stacks Project Authors, Stacks Project. 2023, https://stacks.math.columbia.edu/Suche in Google Scholar

[31] C. Traverso, Seminormality and Picard group. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 24 (1970), 585–595. MR277542 Zbl 0205.50501Suche in Google Scholar

[32] M. A. Vitulli, Weak normality and seminormality. In: Commutative algebra—Noetherian and non-Noetherian perspectives, 441–480, Springer 2011. MR2762521 Zbl 1256.1300910.1007/978-1-4419-6990-3_17Suche in Google Scholar

Received: 2024-11-06
Revised: 2025-02-11
Published Online: 2025-07-19
Published in Print: 2025-07-28

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