LOCAL EULER OBSTRUCTION, OLD AND NEW, III

Abstract

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.