LOCAL EULER OBSTRUCTION, OLD AND NEW, III
Autor(a) principal: | |
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Data de Publicação: | 2022 |
Outros Autores: | , |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Institucional da UNESP |
Texto Completo: | http://dx.doi.org/10.5427/jsing.2022.25e http://hdl.handle.net/11449/240683 |
Resumo: | The first part of the article is a survey of papers originating from a joint course given by the first and third named authors in São José do Rio Preto. That is an historical journey from Athens to São Carlos, going from the discovery of the Plato polyhedra to characteristic classes of a singular variety, by M.-H. Schwartz and R. MacPherson, from the Euler formula and Poincaré-Hopf Theorem to the study of local Euler obstruction. In 1965, Marie-Hélène Schwartz defined characteristic classes for singular complex varieties, as cohomology classes of an ambient manifold, with support on the singular varieties. In 1974, Robert MacPherson showed existence of homology characteristic classes for singular varieties, proving a Deligne and Grothendieck conjecture. One of the main ingredients of his definition is the local Euler obstruction, defined by differential forms. An equivalent definition of the local Euler obstruction, using vector fields, has been given by Jean-Paul Brasselet and MarieHélène Schwartz in their proof of the coincidence of two previous definitions of characteristic classes via Alexander isomorphism. In 1998, the first author published a survey, Local Euler obstruction, old and new followed in 2010 by a survey by the two first authors Local Euler obstruction, old and new, II. The notion of local Euler obstruction was revealed to be very useful to describe the local complexity of stratified singular varieties and developed in many areas, study of foliations, determinantal varieties. Nowadays, a full book would be necessary to write a complete survey on the subject. Many São Carlense researchers published various papers related to local Euler obstruction. Celebrating 30 years of International Workshops on Real and Complex Singularities in São Carlos is the occasion to “take stock” of the successes they achieved in this area alone or with coauthors. That is the second part of the article. |
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LOCAL EULER OBSTRUCTION, OLD AND NEW, IIIThe first part of the article is a survey of papers originating from a joint course given by the first and third named authors in São José do Rio Preto. That is an historical journey from Athens to São Carlos, going from the discovery of the Plato polyhedra to characteristic classes of a singular variety, by M.-H. Schwartz and R. MacPherson, from the Euler formula and Poincaré-Hopf Theorem to the study of local Euler obstruction. In 1965, Marie-Hélène Schwartz defined characteristic classes for singular complex varieties, as cohomology classes of an ambient manifold, with support on the singular varieties. In 1974, Robert MacPherson showed existence of homology characteristic classes for singular varieties, proving a Deligne and Grothendieck conjecture. One of the main ingredients of his definition is the local Euler obstruction, defined by differential forms. An equivalent definition of the local Euler obstruction, using vector fields, has been given by Jean-Paul Brasselet and MarieHélène Schwartz in their proof of the coincidence of two previous definitions of characteristic classes via Alexander isomorphism. In 1998, the first author published a survey, Local Euler obstruction, old and new followed in 2010 by a survey by the two first authors Local Euler obstruction, old and new, II. The notion of local Euler obstruction was revealed to be very useful to describe the local complexity of stratified singular varieties and developed in many areas, study of foliations, determinantal varieties. Nowadays, a full book would be necessary to write a complete survey on the subject. Many São Carlense researchers published various papers related to local Euler obstruction. Celebrating 30 years of International Workshops on Real and Complex Singularities in São Carlos is the occasion to “take stock” of the successes they achieved in this area alone or with coauthors. That is the second part of the article.Aix-Marseille UniversitéFundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)I2M-CNRS-Aix-Marseille UniversityICMC-USPIBILCE-UNESPIBILCE-UNESPFAPESP: 2019/21181-0I2M-CNRS-Aix-Marseille UniversityUniversidade de São Paulo (USP)Universidade Estadual Paulista (UNESP)Brasselet, Jean-PaulGrulha, Nivaldo G.Bích, Thủy Nguyễn Thị [UNESP]2023-03-01T20:28:08Z2023-03-01T20:28:08Z2022-01-01info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/article90-122http://dx.doi.org/10.5427/jsing.2022.25eJournal of Singularities, v. 25, p. 90-122.1949-2006http://hdl.handle.net/11449/24068310.5427/jsing.2022.25e2-s2.0-85136283146Scopusreponame:Repositório Institucional da UNESPinstname:Universidade Estadual Paulista (UNESP)instacron:UNESPengJournal of Singularitiesinfo:eu-repo/semantics/openAccess2023-03-01T20:28:08Zoai:repositorio.unesp.br:11449/240683Repositório InstitucionalPUBhttp://repositorio.unesp.br/oai/requestopendoar:29462023-03-01T20:28:08Repositório Institucional da UNESP - Universidade Estadual Paulista (UNESP)false |
dc.title.none.fl_str_mv |
LOCAL EULER OBSTRUCTION, OLD AND NEW, III |
title |
LOCAL EULER OBSTRUCTION, OLD AND NEW, III |
spellingShingle |
LOCAL EULER OBSTRUCTION, OLD AND NEW, III Brasselet, Jean-Paul |
title_short |
LOCAL EULER OBSTRUCTION, OLD AND NEW, III |
title_full |
LOCAL EULER OBSTRUCTION, OLD AND NEW, III |
title_fullStr |
LOCAL EULER OBSTRUCTION, OLD AND NEW, III |
title_full_unstemmed |
LOCAL EULER OBSTRUCTION, OLD AND NEW, III |
title_sort |
LOCAL EULER OBSTRUCTION, OLD AND NEW, III |
author |
Brasselet, Jean-Paul |
author_facet |
Brasselet, Jean-Paul Grulha, Nivaldo G. Bích, Thủy Nguyễn Thị [UNESP] |
author_role |
author |
author2 |
Grulha, Nivaldo G. Bích, Thủy Nguyễn Thị [UNESP] |
author2_role |
author author |
dc.contributor.none.fl_str_mv |
I2M-CNRS-Aix-Marseille University Universidade de São Paulo (USP) Universidade Estadual Paulista (UNESP) |
dc.contributor.author.fl_str_mv |
Brasselet, Jean-Paul Grulha, Nivaldo G. Bích, Thủy Nguyễn Thị [UNESP] |
description |
The first part of the article is a survey of papers originating from a joint course given by the first and third named authors in São José do Rio Preto. That is an historical journey from Athens to São Carlos, going from the discovery of the Plato polyhedra to characteristic classes of a singular variety, by M.-H. Schwartz and R. MacPherson, from the Euler formula and Poincaré-Hopf Theorem to the study of local Euler obstruction. In 1965, Marie-Hélène Schwartz defined characteristic classes for singular complex varieties, as cohomology classes of an ambient manifold, with support on the singular varieties. In 1974, Robert MacPherson showed existence of homology characteristic classes for singular varieties, proving a Deligne and Grothendieck conjecture. One of the main ingredients of his definition is the local Euler obstruction, defined by differential forms. An equivalent definition of the local Euler obstruction, using vector fields, has been given by Jean-Paul Brasselet and MarieHélène Schwartz in their proof of the coincidence of two previous definitions of characteristic classes via Alexander isomorphism. In 1998, the first author published a survey, Local Euler obstruction, old and new followed in 2010 by a survey by the two first authors Local Euler obstruction, old and new, II. The notion of local Euler obstruction was revealed to be very useful to describe the local complexity of stratified singular varieties and developed in many areas, study of foliations, determinantal varieties. Nowadays, a full book would be necessary to write a complete survey on the subject. Many São Carlense researchers published various papers related to local Euler obstruction. Celebrating 30 years of International Workshops on Real and Complex Singularities in São Carlos is the occasion to “take stock” of the successes they achieved in this area alone or with coauthors. That is the second part of the article. |
publishDate |
2022 |
dc.date.none.fl_str_mv |
2022-01-01 2023-03-01T20:28:08Z 2023-03-01T20:28:08Z |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://dx.doi.org/10.5427/jsing.2022.25e Journal of Singularities, v. 25, p. 90-122. 1949-2006 http://hdl.handle.net/11449/240683 10.5427/jsing.2022.25e 2-s2.0-85136283146 |
url |
http://dx.doi.org/10.5427/jsing.2022.25e http://hdl.handle.net/11449/240683 |
identifier_str_mv |
Journal of Singularities, v. 25, p. 90-122. 1949-2006 10.5427/jsing.2022.25e 2-s2.0-85136283146 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
Journal of Singularities |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
90-122 |
dc.source.none.fl_str_mv |
Scopus reponame:Repositório Institucional da UNESP instname:Universidade Estadual Paulista (UNESP) instacron:UNESP |
instname_str |
Universidade Estadual Paulista (UNESP) |
instacron_str |
UNESP |
institution |
UNESP |
reponame_str |
Repositório Institucional da UNESP |
collection |
Repositório Institucional da UNESP |
repository.name.fl_str_mv |
Repositório Institucional da UNESP - Universidade Estadual Paulista (UNESP) |
repository.mail.fl_str_mv |
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