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Local Polar Varieties in the Geometric Study of Singularities Arturo Giles Flores, Bernard Teissier

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Local Polar Varieties in the Geometric Study of Singularities Arturo Giles Flores, Bernard Teissier Local polar varieties in the geometric study of singularities Arturo Giles Flores, Bernard Teissier To cite this version: Arturo Giles Flores, Bernard Teissier. Local polar varieties in the geometric study of singularities. 2016. hal-01348838v3 HAL Id: hal-01348838 https://hal.archives-ouvertes.fr/hal-01348838v3 Preprint submitted on 10 Dec 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution - NoDerivatives| 4.0 International License Local polar varieties in the geometric study of singularities Arturo Giles Flores and Bernard Teissier December 11, 2017 This paper is dedicated to those administrators of research who realize how much damage is done by the evaluation of mathematical research solely by the rankings of the journals in which it is published, or more generally by bibliomet- ric indices. We hope that their lucidity will become widespread in all countries. Abstract This text presents several aspects of the theory of equisingularity of com- plex 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 ⊂ Cn of hyperplanes of Cn 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 lo- cal topological invariants, local vanishing Euler-Poincar´echaracteristics, 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 Pl¨ucker-type formula for the degree of the dual of a projective variety. R´esum´e Ce texte pr´esente plusieurs aspects de la th´eoriede l'´equisingularit´edes espaces analytiques complexes telle qu'elle est d´efiniepar les conditions de Whitney. Le but est de d´ecriredes points de vue g´eom´etrique,topologique et alg´ebriqueune partition canonique localement finie d'un espace analy- tique complexe r´eduit X en strates non singuli`erestelles que la g´eom´etrie locale de X soit constante le long de chaque strate. Les vari´et´espolaires lo- cales apparaissent dans le titre parce qu'elles jouent un r^olecentral dans 1 Local polar varieties 2 l’unification des points de vue. Le point de vue g´eom´etriqueconduit `a l'´etudedes directions limites en un point donn´ede X ⊂ Cn des hyper- plans de Cn tangents `a X en des points non singuliers. Ceci am`ene`a r´ealiserque les conditions de Whitney, qui servent `ad´efinirla stratifi- cation, sont en fait de nature lagrangienne. Les vari´et´espolaires locales sont utilis´eespour analyser la structure de l'ensemble des positions limites d'hyperplans tangents. Cette structure aide `acomprendre comment une singularit´ediff`erede son c^onetangent, suppos´er´eduit. Les multiplicit´es des vari´et´espolaires locales sont reli´ees`ades invariants topologiques lo- caux, des caract´eristiquesd'Euler-Poincar´e´evanescentes, par une formule qui se r´ev`ele, dans cas particulier o`ula singularit´eest le sommet du c^one sur une vari´et´eprojective r´eduite,donner une formule du type Pl¨ucker pour le calcul du degr´ede la vari´et´eduale d'une vari´et´eprojective. Contents 1 Introduction 2 2 Limits of tangent spaces, the conormal space and the tangent cone. 8 2.1 Some symplectic Geometry . .9 2.2 Conormal space. 11 2.3 Conormal spaces and projective duality . 12 2.4 Tangent cone . 17 2.5 Multiplicity . 24 3 Normal Cone and Polar Varieties: the normal/conormal diagram 25 3.1 Specialization to the normal cone of κ−1(x) ⊂ C(X)....... 32 3.2 Local Polar Varieties . 32 3.3 Limits of tangent spaces . 38 4 Whitney Stratifications 45 4.1 Whitney's conditions . 46 4.2 Stratifications . 48 4.3 Whitney stratifications and polar varieties . 52 4.4 Relative Duality . 54 5 Whitney stratifications and tubular neighborhoods 59 6 Whitney stratifications and the local total topological type 62 7 Specialization to the Tangent Cone and Whitney equisingular- ity 68 8 Polar varieties, Whitney stratifications, and projective duality 74 Local polar varieties 3 9 APPENDIX: SOME COMPLEMENTS 82 9.1 Whitney's conditions and the condition (w) . 82 9.2 Other applications . 83 1 Introduction The origin of these notes is a course imparted by the second author in the \2ndo Congreso Latinoamericano de Matem´aticos" celebrated in Cancun, Mexico on July 20 -26, 2004. The first redaction was subsequently elaborated by the au- thors. The theme of the course was the local study of analytic subsets of Cn, which is the local study of reduced complex analytic spaces. That is, we will consider subsets defined in a neighbourhood of a point 0 2 Cn by equations: f1(z1; : : : ; zn) = ··· = fk(z1; : : : ; zn) = 0 fi 2 Cfz1; : : : ; zng; fi(0) = 0; i = 1; : : : ; k: Meaning that the subset X ⊂ Cn is thus defined in a neighbourhood U of 0, where all the series fi converge. Throughout this text, the word \local" means that we work with “sufficiently small" representatives of a germ (X; x). For simplicity we assume throughout that the spaces we study are equidimensional; all their irreducible components have the same dimension. The reader who needs the general case of reduced spaces should have no substantial difficulty in making the extension. The purpose of the course was to show how to stratify X. In other words, par- 1 tition X into finitely many nonsingular complex analytic manifolds fXαgα2A, which will be called strata, such that: i) The closure Xα is a closed complex analytic subspace of X, for all α 2 A. ii) The frontier Xβ n Xβ is a union of strata Xα, for all β 2 A. iii) Given any x 2 Xα, the \geometry" of all the closures Xβ containing Xα is locally constant along Xα in a neighbourhood of x. The stratification process being such that at each step one can take connected components, one usually assumes that the strata are connected and then the closures Xα are equidimensional. Recall that two closed subspaces X ⊂ U ⊂ Cn and X0 ⊂ U 0 ⊂ Cn, where U and U 0 are open, have the same embedded topological type if there exists a homeomorphism φ: U ! U 0 such that φ(X) = X0. If X and X0 are representa- tives of germs (X; x) and (X0; x0) we require that φ(x) = x0 and we say the two germs have the same embedded topological type. If by \geometry" we mean the embedded local topological type at x 2 Xα of n n Xβ ⊂ C and of its sections by affine subspaces of C of general directions 1We would prefer to use regular to emphasize the absence of singularities, but this term has too many meanings. Local polar varieties 4 passing near x or through x, which we call the total local topological type, there is a minimal such partition, in the sense that any other partition with the same properties must be a sub-partition of it. Characterized by differential-geometric conditions, called Whitney conditions, bearing on limits of tangent spaces and of secants, it plays an important role in generalizing to singular spaces geometric concepts such as Chern classes and integrals of curvature. The existence of such partitions, or stratifications, without proof of the existence of a minimal one, and indeed the very concept of stratification2, are originally due to Whitney in [Whi1], [Whi2]. In these papers Whitney also initated the study in complex analytic geometry of limits of tangent spaces at nonsingular points. In algebraic geometry the first general aproach to such limits is due to Semple in [Se]. In addition to topological and differential-geometric characterizations, the partition can also be described algebraically by means of Polar Varieties, and this is one of the main points of these lectures. Apart from the characterization of Whitney conditions by equimultiplicity of polar varieties, one of the main results appearing in these lectures is therefore the S equivalence of Whitney conditions for a stratification X = α Xα of a complex n analytic space X ⊂ C with the local topological triviality of the closures Xβ of strata along each Xα which they contain, to which one must add the local topological triviality along Xα of the intersections of the Xβ with (germs of) n general nonsingular subspaces of C containing Xα. Other facts concerning Whitney conditions also appear in these notes, for example that the Whitney conditions are in fact of a Lagrangian nature, related to a condition of relative projective duality between the irreducible components of the normal cones of the Xβ along the Xα and of some of their subcones, on the one hand, and the irreducible components of the space of limits of tangents hyperplanes to Xβ at nonsingular points approaching Xα, on the other.
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