Introdução à Obstrução de Euler Local: dos sólidos platônicos às variedades determinantais

Raphael de Omena Marinho

doi:10.14244/lajm.v4i1.37

Autores

Raphael de Omena Marinho UFC

DOI:

https://doi.org/10.14244/lajm.v4i1.37

Palavras-chave:

teoria de obstrução, variedade singular, campo de vetor, índice

Resumo

A obstrução local de Euler foi inicialmente definida por MacPherson para responder à conjectura de Deligne e Grothendieck sobre a existência e unicidade das classes de Chern para variedades algébricas singulares. No contexto de índices de campos vetoriais, Brasselet e Schwartz caracterizaram esse invariante, abordagem que seguiremos neste texto. Com base nessa definição, também apresentamos a obstrução de Euler de uma função e a obstrução de Euler de uma aplicação.

Referências

[1] MacPherson R. Chern Classes for Singular Algebraic Varieties. Annals of Mathematics. 1974;100(2):423-32. DOI: https://doi.org/10.2307/1971080

[2] Schwartz MH. Classes Caractéristiques Définies par une Stratification d’une Variété Analytique Complexe. CR Acad Sci Paris. 1965;260:3262-4.

[3] Brasselet JP, Schwartz MH. Sur les classes de Chern d’un ensemble analytique complexe, caractéristique d’Euler–Poincaré. Astérisque. 1981;82:93-147.

[4] Grulha NG Jr. Obstrução de Euler de Aplicações Analíticas [Tese (Doutorado em Matemática)]. São Paulo: Universidade de São Paulo; 2007.

[5] Brasselet JP. An Introduction to Characteristic Classes. Rio de Janeiro: IMPA; 2011.

[6] Martins EBC. Obstruction theory, characteristic classes and applications [Dissertação (Mestrado em Matemática)]. São Paulo: Universidade de São Paulo; 2022.

[7] Munkres JR. Elements of algebraic topology. CRC press; 2018. DOI: https://doi.org/10.1201/9780429493911

[8] Alexander JW. A proof of the invariance of certain constants of analysis situs. Transactions of the American Mathematical Society. 1915;16(2):148-54. DOI: https://doi.org/10.1090/S0002-9947-1915-1501007-5

[9] Hatcher A. Algebraic Topology. Cambridge: Cambridge University Press; 2002.

[10] Brasselet JP, Thuy NTB. Teorema de Poincaré-Hopf. CQD-Revista Eletrônica Paulista de Matemática. 2019. DOI: https://doi.org/10.21167/cqdvol16201923169664jpbntbt134162

[11] Sullivan D. Combinatorial invariants of analytic spaces. In: Proceedings of Liverpool Singularities—Symposium I. Springer; 2006. p. 165-77. DOI: https://doi.org/10.1007/BFb0066822

[12] Brasselet JP. Characteristic Classes and Singular Varieties. School on Singularities in Geometry, Topology, Foliations and Dynamics; 2002.

[13] Goresky M, MacPherson R. Stratified Morse Theory. In: Stratified Morse Theory. Springer; 1988. p. 3-22. DOI: https://doi.org/10.1007/978-3-642-71714-7_1

[14] Whitney H. Local Properties of Analytic Varieties. Differential and Combinatorial Topology: A Symposium in Honor of Marston Morse. 1965:496-549. DOI: https://doi.org/10.1515/9781400874842-014

[15] Whitney H. Tangents to an Analytic Variety. Annals of Mathematics. 1965:496-549. DOI: https://doi.org/10.2307/1970400

[16] Brasselet JP, Lê DT, Seade J. Euler obstruction and indices of vector fields. Topology. 2000;39(6):1193-208. DOI: https://doi.org/10.1016/S0040-9383(99)00009-9

[17] Brasselet JP, Grulha Jr NG, Nguyen TTB. Local Euler Obstruction, Old and New III. Journal of Singularities. 2022;25:90-122. DOI: https://doi.org/10.5427/jsing.2022.25e

[18] Brasselet JP, Massey D, Parameswaran A, Seade J. Euler obstruction and defects of functions on singular varieties. Journal of the London Mathematical Society. 2004;70(1):59-76. DOI: https://doi.org/10.1112/S0024610704005447

[19] Seade J, Tibar M, Verjovsky A. Milnor numbers and Euler obstruction. Bulletin of the Brazilian Mathematical Society. 2005;36:275-83. DOI: https://doi.org/10.1007/s00574-005-0039-x

[20] Ament D, Nuño-Ballesteros J, Oréfice-Okamoto B, Tomazella J. The Euler obstruction of a function on a determinantal variety and on a curve. Bulletin of the Brazilian Mathematical Society, New Series. 2016;47(3):955-70. DOI: https://doi.org/10.1007/s00574-016-0198-y

[21] Decker W, Greuel GM, Pfister G, Schönemann H. SINGULAR 4-4-0 — A computer algebra system for polynomial computations; 2024. http://www.singular.uni-kl.de.

[22] Dutertre N, Grulha Jr NG. Lê–Greuel type formula for the Euler obstruction and applications. Advances in Mathematics. 2014;251:127-46. DOI: https://doi.org/10.1016/j.aim.2013.10.023

[23] Brasselet JP, Seade J, Suwa T. Vector Fields on Singular Varieties. Springer Science & Business Media; 2009. DOI: https://doi.org/10.1007/978-3-642-05205-7

[24] Ebeling W, Gusein-Zade SM. Chern obstructions for collections of 1-forms on singular varieties. In: Singularity Theory: Dedicated to Jean-Paul Brasselet on His 60th Birthday. World Scientific; 2007. p. 557-64. DOI: https://doi.org/10.1142/9789812707499_0021

[25] Brasselet JP, Grulha Jr NG, Ruas MAS. The Euler obstruction and the Chern obstruction. Bulletin of the London Mathematical Society. 2010;42(6):1035-43. DOI: https://doi.org/10.1112/blms/bdq063

[26] Grulha Jr NG, Ruiz CM, Santana H. The geometrical information encoded by the Euler obstruction of a map. International Journal of Mathematics. 2022;33(04):2250029. DOI: https://doi.org/10.1142/S0129167X2250029X

[27] Arbarello E, Cornalba M, Griffiths PA, Harris J. Geometry of Algebraic Curves. New York: Springer-Verlag; 2011. DOI: https://doi.org/10.1007/978-3-540-69392-5

[28] Bonnard B, Faugère JC, Jacquemard A, Safey El Din M, Verron T. Determinantal sets, singularities and application to optimal control in medical imagery. In: Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation; 2016. p. DOI: https://doi.org/10.1145/2930889.2930916

103-10.

[29] Jockers H, Kumar V, Lapan JM, Morrison DR, Romo M. Nonabelian 2D Gauge theories for determinantal Calabi-Yau varieties. Journal of High Energy Physics. 2012;2012(11):1-47. DOI: https://doi.org/10.1007/JHEP11(2012)166

[30] Gusein-Zade SM, Ebeling W. On indices of 1-forms on determinantal singularities. Proceedings of the Steklov Institute of Mathematics. 2009;267(1):113-24. DOI: https://doi.org/10.1134/S0081543809040099

[31] Pereira MdS. Variedades determinantais e singularidades de matrizes [Tese (Doutorado em Matemática)]. São Paulo: Universidade de São Paulo; 2010.

[32] Gaffney T, Grulha Jr NG, Ruas MAS. The Local Euler Obstruction and Topology of the Stabilization of Associated Determinantal Varieties. Mathematische Zeitschrift. 2019;291(3):905-30. DOI: https://doi.org/10.1007/s00209-018-2141-y

[33] Damon J, Pike B. Solvable Groups, Free Divisors and Nonisolated Matrix Singularities II: Vanishing Topology. Geometry & Topology. 2014;18(2):911-62. DOI: https://doi.org/10.2140/gt.2014.18.911

[34] de Omena R, Grulha Jr NG, Pereira M. From Milnor Number to the Euler Obstruction of a Map on Isolated Determinantal Singularities. Manuscripta Mathematica. 2024:1-17. DOI: https://doi.org/10.1007/s00229-023-01528-w

[35] Gaffney T. Polar multiplicities and equisingularity of map germs. Topology. 1993;32(1):185-223. DOI: https://doi.org/10.1016/0040-9383(93)90045-W

[36] de Omena R. Do Número de Milnor à Obstrução de Euler de uma Aplicação em Singularidades Determinantais [Tese (Doutorado em Matemática)]. São Paulo: Universidade de São Paulo; 2023.