[A topological introduction to Lipschitz geometry of complex singularities]
These are the notes of the three lectures I gave during the IXth Winterbraids School which took place in Reims from 4 to 7th March 2019. The aim of this course was to introduce researchers working in low-dimensional topology to Lipschitz geometry of complex singularities. In these lectures, I focussed on topological points of view on the objects, avoiding as much as possible technical material from algebraic geometry and singularity theory such as resolution of singularities, Nash transform, generic projections of curves and surfaces, etc.
It starts with an introduction to Lipschitz geometry of singular spaces. It then gives the complete classification of Lipschitz geometry of complex curves and covers the result of [17]. The last part is an introduction to Lipschitz geometry of complex surfaces and states the thick-thin decomposition Theorem of a normal complex surface proved in [6].
@article{WBLN_2019__6__A3_0,
author = {Pichon, Anne},
title = {A topological introduction to {Lipschitz} geometry of complex singularities},
booktitle = {Winter Braids IX (Reims, 2019)},
series = {Winter Braids Lecture Notes},
note = {talk:3},
pages = {1--17},
publisher = {Winter Braids School},
year = {2019},
doi = {10.5802/wbln.29},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/wbln.29/}
}
TY - JOUR AU - Pichon, Anne TI - A topological introduction to Lipschitz geometry of complex singularities BT - Winter Braids IX (Reims, 2019) AU - Collectif T3 - Winter Braids Lecture Notes N1 - talk:3 PY - 2019 SP - 1 EP - 17 PB - Winter Braids School UR - http://archive.numdam.org/articles/10.5802/wbln.29/ DO - 10.5802/wbln.29 LA - en ID - WBLN_2019__6__A3_0 ER -
%0 Journal Article %A Pichon, Anne %T A topological introduction to Lipschitz geometry of complex singularities %B Winter Braids IX (Reims, 2019) %A Collectif %S Winter Braids Lecture Notes %Z talk:3 %D 2019 %P 1-17 %I Winter Braids School %U http://archive.numdam.org/articles/10.5802/wbln.29/ %R 10.5802/wbln.29 %G en %F WBLN_2019__6__A3_0
Pichon, Anne. A topological introduction to Lipschitz geometry of complex singularities, dans Winter Braids IX (Reims, 2019), Winter Braids Lecture Notes (2019), Exposé no. 3, 17 p. doi : 10.5802/wbln.29. http://archive.numdam.org/articles/10.5802/wbln.29/
[1] Lev Birbrair, Alexandre Fernandes, Inner metric geometry of complex algebraic surfaces with isolated singularities, Comm. Pure Appl. Math. 61 (2008), 1483–1494. | DOI | MR | Zbl
[2] Lev Birbrair, Alexandre Fernandez, Walter D. Neumann, Bi-Lipschitz geometry of weighted homogeneous surface singularities. Math. Ann. 342 (2008), 139–144. | DOI | MR | Zbl
[3] Lev Birbrair, Alexandre Fernandez, Walter D. Neumann, Bi-Lipschitz geometry of complex surface singularities, Geometriae Dedicata 129 (2009), 259–267. | DOI | MR | Zbl
[4] Lev Birbrair, A. Fernandez, Walter D. Neumann, Separating sets, metric tangent cone and applications for complex algebraic germs, Selecta Math. 16 (2010), 377-391. | DOI | MR | Zbl
[5] Lev Birbrair, Alexandre Fernandes, Jose Edson Sampaio, Misha Verbitsky, Multiplicity of singularities is not a bi-Lipschitz invariant, Math. Ann. 377 (2020), no. 1-2, 115-121. | DOI | MR | Zbl
[6] Lev Birbrair, Walter D Neumann and Anne Pichon, The thick-thin decomposition and the bilipschitz classification of normal surface singularities, Acta Math 212 (2014), 199–254. | DOI | MR | Zbl
[7] Alan Durfee, Neighborhoods of algebraic sets, Trans. Amer. Math. Soc. 276 (1983), 517–530. | DOI | MR | Zbl
[8] Alexandre Fernandes, Topological equivalence of complex curves and bi-Lipschitz maps, Michigan Math. J. 51 (2003), 593–606. | DOI | Zbl
[9] Alexandre Fernandes, Javier Fernandes de Bobadilla and Jose Edson Sampaio, Multiplicity and degree as bi-Lipschitz invariants for complex sets, (2017) arXiv:1706.06614.
[10] Alexandre Fernandes, Jose Edson Sampaio, Multiplicity of analytic hypersurface singularities under bi-Lipschitz homeomorphisms. J. Topol. 9 (2016), no. 3, 927–933. | DOI | MR | Zbl
[11] Javier Fernandez de Bobadilla, Sonja Heinze, Maria Pe Pereira, Jose Edson Sampaio, Moderately Discontinuous Homology, (2020), arXiv:1910.12552
[12] Javier Fernandez de Bobadilla, Sonja Heinze, Maria Pe Pereira, Moderately Discontinuous Homotopy, (2020), arXiv:2007.01538
[13] Tzee-Char Kuo and Yung-Chen Lu, On analytic function germs of two complex variables, Topology 16 (1977), 299–301. | DOI | MR | Zbl
[14] Lê Dũng Tráng, Françoise Michel, Claude Weber, Sur le comportement des polaires associées aux germes de courbes planes. Compositio Mathematica 72 (1989), 87-113. | Zbl
[15] Lê Dũng Tráng, Bernard Teissier, Sur la géométrie des surfaces complexes I. Tangentes exceptionnelles, in Amer. J. Math. 101, (1979), 420-452. | DOI | Zbl
[16] Tadeusz Mostowski, Lipschitz equisingularity, Dissertationes Math. (Rozprawy Mat.) 243 (1985), 46pp.
[17] Walter D Neumann and Anne Pichon, Lipschitz geometry of complex curves, Journal of Singularities 10 (2014), 225-234. | DOI | MR | Zbl
[18] Walter D Neumann and Anne Pichon, Lipschitz geometry of complex surfaces: analytic invariants and equisingularity, arXiv:1211.4897v1.
[19] Adam Parusiński, Lipschitz properties of semi-analytic sets, Université de Grenoble, Annales de l’Institut Fourier, 38 (1988) 189–213. | DOI | MR | Zbl
[20] Adam Parusiński, Lipschitz stratification of subanalytic sets, Ann. Sci. Ec. Norm. Sup. (4) 27 (1994), 661–696. | DOI | MR | Zbl
[21] Frédéric Pham and Bernard Teissier, Fractions Lipschitziennes d’une algèbre analytique complexe et saturation de Zariski. Prépublications Ecole Polytechnique No. M17.0669 (1969). Available at http://hal.archives-ouvertes.fr/hal-00384928/fr/.
[22] Pham F., Teissier B. (2020) Lipschitz Fractions of a Complex Analytic Algebra and Zariski Saturation. In: Neumann W., Pichon A. (eds) Introduction to Lipschitz Geometry of Singularities. Lecture Notes in Mathematics, vol 2280. Springer (2020). | DOI | Zbl
[23] Bernard Teissier, Variétés polaires, II. Multiplicités polaires, sections planes, et conditions de Whitney, Algebraic geometry (La Ràbida, 1981) 314–491, Lecture Notes in Math. 961 (Springer, Berlin, 1982). | DOI
[24] C.T.C. Wall, Singular points of plane curves, Cambridge University Press (2004). | DOI | Zbl
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