Local polar varieties in the geometric study of singularities
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 27 (2018) no. 4, pp. 679-775.

Ce texte présente plusieurs aspects de la théorie de l’équisingularité des espaces analytiques complexes telle qu’elle est définie par les conditions de Whitney. Le but est de décrire des points de vue géométrique, topologique et algébrique une partition canonique localement finie d’un espace analytique complexe réduit X en strates non singulières telles que la géométrie locale de X soit constante le long de chaque strate. Les variétés polaires locales apparaissent dans le titre parce qu’elles jouent un rôle central dans l’unification des points de vue. Le point de vue géométrique conduit à l’étude des directions limites en un point donné de X n des hyperplans de n tangents à X en des points non singuliers. Ceci amène à réaliser que les conditions de Whitney, qui servent à définir la stratification, sont en fait de nature lagrangienne. Les variétés polaires locales sont utilisées pour analyser la structure de l’ensemble des positions limites d’hyperplans tangents. Cette structure aide à comprendre comment une singularité diffère de son cône tangent, supposé réduit. Les multiplicités des variétés polaires locales sont reliées à des invariants topologiques locaux, des caractéristiques d’Euler–Poincaré évanescentes, par une formule qui se révèle, dans le cas particulier où la singularité est le sommet du cône sur une variété projective réduite, donner une formule du type Plücker pour le calcul du degré de la variété duale d’une variété projective.

This text presents several aspects of the theory of equisingularity of complex analytic spaces from the standpoint of Whitney conditions. The goal is to describe from the geometrical, topological, and algebraic viewpoints a canonical locally finite partition of a reduced complex analytic space X into nonsingular strata with the property that the local geometry of X is constant on each stratum. Local polar varieties appear in the title because they play a central role in the unification of viewpoints. The geometrical viewpoint leads to the study of spaces of limit directions at a given point of X n of hyperplanes of n tangent to X at nonsingular points, which in turn leads to the realization that the Whitney conditions, which are used to define the stratification, are in fact of a Lagrangian nature. The local polar varieties are used to analyze the structure of the set of limit directions of tangent hyperplanes. This structure helps in particular to understand how a singularity differs from its tangent cone, assumed to be reduced. The multiplicities of local polar varieties are related to local topological invariants, local vanishing Euler–Poincaré characteristics, by a formula which turns out to contain, in the special case where the singularity is the vertex of the cone over a reduced projective variety, a Plücker-type formula for the degree of the dual of a projective variety.

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DOI : https://doi.org/10.5802/afst.1582
@article{AFST_2018_6_27_4_679_0,
     author = {Flores, Arturo Giles and Teissier, Bernard},
     title = {Local polar varieties in the geometric study of singularities},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {679--775},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 27},
     number = {4},
     year = {2018},
     doi = {10.5802/afst.1582},
     mrnumber = {3884609},
     zbl = {1409.14002},
     language = {en},
     url = {http://www.numdam.org/item/AFST_2018_6_27_4_679_0/}
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Flores, Arturo Giles; Teissier, Bernard. Local polar varieties in the geometric study of singularities. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 27 (2018) no. 4, pp. 679-775. doi : 10.5802/afst.1582. http://www.numdam.org/item/AFST_2018_6_27_4_679_0/

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