Équisingularité réelle II : invariants locaux et conditions de régularité
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 2, pp. 221-269.

On définit, pour un germe d'ensemble sous-analytique, deux nouvelles suites finies d'invariants numériques. La première a pour termes les localisations des courbures de Lipschitz-Killing classiques, la seconde est l'équivalent réel des caractéristiques évanescentes complexes introduites par M. Kashiwara. On montre que chaque terme d'une de ces suites est combinaison linéaire des termes de l'autre, puis on relie ces invariants à la géométrie des discriminants des projections du germe sur des plans de toutes les dimensions. Il apparaît alors que ces invariants sont continus le long de strates de Verdier d'une stratification sous-analytique d'un fermé.

For germs of subanalytic sets, we define two finite sequences of new numerical invariants. The first one is obtained by localizing the classical Lipschitz-Killing curvatures, the second one is the real analogue of the vanishing Euler characteristics introduced by M. Kashiwara. We show that each invariant of one sequence is a linear combination of the invariants of the other sequence. We then connect our invariants to the geometry of the discriminants of all dimension. Finally we prove that these invariants are continuous along Verdier strata of a closed subanalytic set.

@article{ASENS_2008_4_41_2_221_0,
     author = {Comte, Georges and Merle, Michel},
     title = {\'Equisingularit\'e r\'eelle {II} : invariants locaux et conditions de r\'egularit\'e},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {221--269},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {4e s{\'e}rie, 41},
     number = {2},
     year = {2008},
     doi = {10.24033/asens.2067},
     mrnumber = {2468482},
     zbl = {1163.32012},
     language = {fr},
     url = {http://www.numdam.org/articles/10.24033/asens.2067/}
}
TY  - JOUR
AU  - Comte, Georges
AU  - Merle, Michel
TI  - Équisingularité réelle II : invariants locaux et conditions de régularité
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2008
SP  - 221
EP  - 269
VL  - 41
IS  - 2
PB  - Société mathématique de France
UR  - http://www.numdam.org/articles/10.24033/asens.2067/
DO  - 10.24033/asens.2067
LA  - fr
ID  - ASENS_2008_4_41_2_221_0
ER  - 
%0 Journal Article
%A Comte, Georges
%A Merle, Michel
%T Équisingularité réelle II : invariants locaux et conditions de régularité
%J Annales scientifiques de l'École Normale Supérieure
%D 2008
%P 221-269
%V 41
%N 2
%I Société mathématique de France
%U http://www.numdam.org/articles/10.24033/asens.2067/
%R 10.24033/asens.2067
%G fr
%F ASENS_2008_4_41_2_221_0
Comte, Georges; Merle, Michel. Équisingularité réelle II : invariants locaux et conditions de régularité. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 2, pp. 221-269. doi : 10.24033/asens.2067. http://www.numdam.org/articles/10.24033/asens.2067/

[1] A. Bernig & L. Bröcker, Lipschitz-Killing invariants, Math. Nachr. 245 (2002), 5-25. | Zbl

[2] A. Bernig & L. Bröcker, Courbures intrinsèques dans les catégories analytico-géométriques, Ann. Inst. Fourier (Grenoble) 53 (2003), 1897-1924. | Numdam | Zbl

[3] W. Blaschke, Vorlesungen über Integralgeometrie, 3e éd., Deutscher Verlag der Wissenschaften, 1955. | JFM | MR | Zbl

[4] V. G. Boltianskiĭ, Hilbert's third problem, John Wiley & Sons, 1978. | Zbl

[5] J. Briançon & J.-P. Speder, Les conditions de Whitney impliquent (μ * ) constant, Ann. Inst. Fourier (Grenoble) 26 (1976), 153-163. | Numdam | Zbl

[6] L. Bröcker & M. Kuppe, Integral geometry of tame sets, Geom. Dedicata 82 (2000), 285-323. | Zbl

[7] H. Brodersen & D. Trotman, Whitney (b)-regularity is weaker than Kuo’s ratio test for real algebraic stratifications, Math. Scand. 45 (1979), 27-34. | Zbl

[8] J.-L. Brylinski, A. S. Dubson & M. Kashiwara, Formule de l'indice pour modules holonomes et obstruction d'Euler locale, C. R. Acad. Sci. Paris Sér. I Math. 293 (1981), 573-576. | Zbl

[9] J. Cheeger, W. Müller & R. Schrader, Kinematic and tube formulas for piecewise linear spaces, Indiana Univ. Math. J. 35 (1986), 737-754. | Zbl

[10] G. Comte, Formule de Cauchy-Crofton pour la densité des ensembles sous-analytiques, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), 505-508. | MR | Zbl

[11] G. Comte, Équisingularité réelle : nombres de Lelong et images polaires, Ann. Sci. École Norm. Sup. 33 (2000), 757-788. | Numdam | MR | Zbl

[12] G. Comte, J.-M. Lion & J.-P. Rolin, Nature log-analytique du volume des sous-analytiques, Illinois J. Math. 44 (2000), 884-888. | Zbl

[13] R. N. Draper, Intersection theory in analytic geometry, Math. Ann. 180 (1969), 175-204. | MR | Zbl

[14] L. v. d. Dries, Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series 248, Cambridge University Press, 1998. | MR | Zbl

[15] L. v. d. Dries & C. Miller, Geometric categories and o-minimal structures, Duke Math. J. 84 (1996), 497-540. | Zbl

[16] A. S. Dubson, Classes caractéristiques des variétés singulières, C. R. Acad. Sci. Paris Sér. A-B 287 (1978), A237-A240. | MR | Zbl

[17] A. S. Dubson, Calcul des invariants numériques des singularités et des applications, Thèse, Unversität Bonn, 1981.

[18] H. Federer, The (ϕ,k) rectifiable subsets of n-space, Trans. Amer. Soc. 62 (1947), 114-192. | MR | Zbl

[19] H. Federer, Curvature measures, Trans. Amer. Math. Soc. 93 (1959), 418-491. | MR | Zbl

[20] H. Federer, Geometric measure theory, Die Grund. Math. Wiss., Band 153, Springer New York Inc., New York, 1969. | MR | Zbl

[21] J. H. G. Fu, Tubular neighborhoods in Euclidean spaces, Duke Math. J. 52 (1985), 1025-1046. | MR | Zbl

[22] J. H. G. Fu, Curvature measures and generalized Morse theory, J. Differential Geom. 30 (1989), 619-642. | MR | Zbl

[23] J. H. G. Fu, Kinematic formulas in integral geometry, Indiana Univ. Math. J. 39 (1990), 1115-1154. | MR | Zbl

[24] J. H. G. Fu, Curvature of singular spaces via the normal cycle, in Differential geometry : geometry in mathematical physics and related topics (Los Angeles, CA, 1990), Proc. Sympos. Pure Math. 54, Amer. Math. Soc., 1993, 211-221. | MR | Zbl

[25] J. H. G. Fu, Curvature measures of subanalytic sets, Amer. J. Math. 116 (1994), 819-880. | MR | Zbl

[26] M. Goresky & R. Macpherson, Stratified Morse theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 14, Springer, 1988. | Zbl

[27] P. M. Gruber & R. Schneider, Problems in geometric convexity, in Contributions to geometry (Proc. Geom. Sympos., Siegen, 1978), Birkhäuser, 1979, 255-278. | Zbl

[28] H. Hadwiger, Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer, 1957. | MR | Zbl

[29] R. M. Hardt, Stratification of real analytic mappings and images, Invent. Math. 28 (1975), 193-208. | MR | Zbl

[30] R. M. Hardt, Semi-algebraic local-triviality in semi-algebraic mappings, Amer. J. Math. 102 (1980), 291-302. | MR | Zbl

[31] J.-P. Henry & M. Merle, Limites de normales, conditions de Whitney et éclatement d'Hironaka, in Singularities, Part 1 (Arcata, Calif., 1981), Proc. Sympos. Pure Math. 40, Amer. Math. Soc., 1983, 575-584. | Zbl

[32] J.-P. Henry & M. Merle, Conditions de régularité et éclatements, Ann. Inst. Fourier (Grenoble) 37 (1987), 159-190. | Numdam | Zbl

[33] J.-P. Henry, M. Merle & C. Sabbah, Sur la condition de Thom stricte pour un morphisme analytique complexe, Ann. Sci. École Norm. Sup. 17 (1984), 227-268. | Numdam | Zbl

[34] H. Hironaka, Normal cones in analytic Whitney stratifications, Publ. Math. I.H.É.S. 36 (1969), 127-138. | Numdam | MR | Zbl

[35] H. Hironaka, Subanalytic sets, in Number theory, algebraic geometry and commutative algebra, in honor of Yasuo Akizuki, Kinokuniya, 1973, 453-493. | MR | Zbl

[36] H. Hironaka, Stratification and flatness, in Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976), Sijthoff and Noordhoff, 1977, 199-265. | MR | Zbl

[37] M. Kashiwara, Index theorem for a maximally overdetermined system of linear differential equations, Proc. Japan Acad. 49 (1973), 803-804. | MR | Zbl

[38] M. Kashiwara, B-functions and holonomic systems. Rationality of roots of B-functions, Invent. Math. 38 (1976/77), 33-53. | MR | Zbl

[39] D. A. Klain, A short proof of Hadwiger's characterization theorem, Mathematika 42 (1995), 329-339. | MR | Zbl

[40] J. F. Knight, A. Pillay & C. Steinhorn, Definable sets in ordered structures II, Trans. Amer. Math. Soc. 295 (1986), 593-605. | Zbl

[41] T.-C. Kuo, The ratio test for analytic Whitney stratifications, in Proceedings of Liverpool Singularities-Symposium, I (1969/70), Lecture Notes in Math., Vol. 192, Springer, 1971, 141-149. | MR | Zbl

[42] M. Kuppe, Integralgeometrie Whitney-stratifizierbarer Mengen, Thèse, Universität Münster, 1999. | Zbl

[43] K. Kurdyka, J.-B. Poly & G. Raby, Moyennes des fonctions sous-analytiques, densité, cône tangent et tranches, in Real analytic and algebraic geometry (Trento, 1988), Lecture Notes in Math. 1420, Springer, 1990, 170-177. | Zbl

[44] K. Kurdyka & G. Raby, Densité des ensembles sous-analytiques, Ann. Inst. Fourier (Grenoble) 39 (1989), 753-771. | Numdam | Zbl

[45] J. Lafontaine, Mesures de courbure des variétés lisses et des polyèdres (d'après Cheeger-Müller et Schröder), Séminaire Bourbaki, Vol. 1985/86, Exp. no 664, Astérisque 145-146 (1987), 241-256. | Numdam | MR | Zbl

[46] R. Langevin, Introduction to integral geometry, 21o Colóquio Brasileiro de Matemática, Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 1997. | MR | Zbl

[47] R. Langevin, La petite musique de la géométrie intégrale, in La recherche de la vérité, Écrit. Math., ACL-Éd. Kangourou, Paris, 1999, 117-143. | MR | Zbl

[48] R. Langevin & T. Shifrin, Polar varieties and integral geometry, Amer. J. Math. 104 (1982), 553-605. | Zbl

[49] P. Lelong, Intégration sur un ensemble analytique complexe, Bull. Soc. Math. France 85 (1957), 239-262. | Numdam | MR | Zbl

[50] J.-M. Lion, Densité des ensembles semi-pfaffiens, Ann. Fac. Sci. Toulouse Math. 7 (1998), 87-92. | Numdam | MR | Zbl

[51] J.-M. Lion & J.-P. Rolin, Intégration des fonctions sous-analytiques et volumes des sous-ensembles sous-analytiques, Ann. Inst. Fourier (Grenoble) 48 (1998), 755-767. | Numdam | Zbl

[52] T. L. Loi, Verdier and strict Thom stratifications in o-minimal structures, Illinois J. Math. 42 (1998), 347-356. | MR | Zbl

[53] R. Macpherson, Chern classes for singular algebraic varieties, Ann. of Math. 100 (1974), 423-432. | MR | Zbl

[54] J. Mather, Notes on topological stability, Harvard University, 1970. | Zbl

[55] P. Mcmullen & R. Schneider, Valuations on convex bodies, in Convexity and its applications, Birkhäuser, 1983, 170-247. | Zbl

[56] M. Merle, Variétés polaires, stratifications de Whitney et classes de Chern des espaces analytiques complexes (d'après Lê-Teissier), Séminaire Bourbaki, Vol. 1982/83, Exp. no 600, Astérisque 105 (1983), 65-78. | Numdam | MR | Zbl

[57] V. Navarro Aznar, Conditions de Whitney et sections planes, Invent. Math. 61 (1980), 199-225. | MR | Zbl

[58] V. Navarro Aznar, Stratifications régulières et variétés polaires locales, manuscrit, 1981.

[59] V. Navarro Aznar & D. Trotman, Whitney regularity and generic wings, Ann. Inst. Fourier (Grenoble) 31 (1981), 87-111. | Numdam | Zbl

[60] P. Orro, Conditions de régularité, espaces tangents et fonctions de Morse, Thèse, Université d'Orsay, 1984.

[61] P. Orro & D. Trotman, On the regular stratifications and conormal structure of subanalytic sets, Bull. London Math. Soc. 18 (1986), 185-191. | Zbl

[62] A. Pillay & C. Steinhorn, Definable sets in ordered structures. I, Trans. Amer. Math. Soc. 295 (1986), 565-592. | Zbl

[63] C. H. Sah, Hilbert's third problem : scissors congruence, Research Notes in Math. 33, Pitman (Advanced Publishing Program), 1979. | MR | Zbl

[64] L. A. Santaló, Integral geometry and geometric probability, Encyclopedia of Mathematics and its Applications, Vol. 1, Addison-Wesley, 1976. | MR | Zbl

[65] W. Schickhoff, Whitneysche Tangentenkegel, Multiplizitätsverhalten, Normal-Pseudoflachheit und Äquisingularitätstheorie für Ramissche Räume, Schr. Math. Inst. Univ. Münster Heft 12 (1977).

[66] R. Schneider, Curvature measures of convex bodies, Ann. Mat. Pura Appl. 116 (1978), 101-134. | MR | Zbl

[67] R. Schneider, A uniqueness theorem for finitely additive invariant measures on a compact homogeneous space, Rend. Circ. Mat. Palermo 30 (1981), 341-344. | MR | Zbl

[68] R. Schneider, Integral geometry - Measure theoretic approach and stochastic applications, Advanced course on integral geometry, CRM, 1984. | Zbl

[69] R. Schneider, Convex bodies : the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications 44, Cambridge University Press, 1993. | MR | Zbl

[70] M. Shiota, Geometry of subanalytic and semialgebraic sets, Progress in Mathematics 150, Birkhäuser, 1997. | MR | Zbl

[71] J. Steiner, Über parallele Flächen, Monatsber. Preuß. Akad. Wissen. Berlin (1840), 114-118, Ges. Werke, vol. 2 (1882), Reimer, Berlin, 171-176.

[72] J. Steiner, Von dem Krümmungsschwerpunkte ebener Curven, J. reine angew. Mathematik 21 (1840), 33-63, Ges. Werke, vol. 2 (1882), Reimer, Berlin, 99-159. | Zbl

[73] B. Teissier, Cycles évanescents, sections planes et conditions de Whitney, in Singularités à Cargèse (Rencontre Singularités Géom. Anal., Inst. Études Sci., Cargèse, 1972), Astérisque, 7-8, Soc. Math. France, 1973, 285-362. | MR | Zbl

[74] B. Teissier, Variétés polaires II. Multiplicités polaires, sections planes, et conditions de Whitney, in Algebraic geometry (La Rábida, 1981), Lecture Notes in Math. 961, Springer, 1982, 314-491. | MR | Zbl

[75] R. Thom, Ensembles et morphismes stratifiés, Bull. Amer. Math. Soc. 75 (1969), 240-284. | MR | Zbl

[76] L. D. Tráng & B. Teissier, Variétés polaires locales et classes de Chern des variétés singulières, Ann. of Math. 114 (1981), 457-491. | Zbl

[77] L. D. Tráng & B. Teissier, Errata : “Local polar varieties and Chern classes of singular varieties”, Ann. of Math. 115 (1982), 668. | Zbl

[78] L. D. Tráng & B. Teissier, Cycles evanescents, sections planes et conditions de Whitney II, in Singularities, Part 2 (Arcata, Calif., 1981), Proc. Sympos. Pure Math. 40, Amer. Math. Soc., 1983, 65-103. | Zbl

[79] D. Trotman, Counterexamples in stratification theory : two discordant horns, in Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976), Sijthoff and Noordhoff, 1977, 679-686. | MR | Zbl

[80] D. Trotman, Comparing regularity conditions on stratifications, in Singularities, Part 2 (Arcata, Calif., 1981), Proc. Sympos. Pure Math. 40, Amer. Math. Soc., 1983, 575-586. | MR | Zbl

[81] G. Valette, Détermination et stabilité du type métrique des singularités, Thèse, Université de Provence, 2003.

[82] G. Valette, Volume, density and Whitney conditions, à paraître dans Ann. Pol. Math..

[83] J.-L. Verdier, Stratifications de Whitney et théorème de Bertini-Sard, Invent. Math. 36 (1976), 295-312. | MR | Zbl

[84] H. Weyl, On the Volume of Tubes, Amer. J. Math. 61 (1939), 461-472. | JFM | MR

[85] O. Zariski, Studies in equisingularity. I. Equivalent singularities of plane algebroid curves, Amer. J. Math. 87 (1965), 507-536. | MR | Zbl

[86] O. Zariski, Studies in equisingularity II 87 (1965), 972-1006. | MR | Zbl

[87] O. Zariski, Studies in equisingularity III. Saturation of local rings and equisingularity, Amer. J. Math. 90 (1968), 961-1023. | MR | Zbl

[88] O. Zariski, Some open questions in the theory of singularities, Bull. Amer. Math. Soc. 77 (1971), 481-491. | MR | Zbl

[89] O. Zariski, On equimultiple subvarieties of algebroid hypersurfaces, Proc. Nat. Acad. Sci. U.S.A. 72 (1975), 1425-1426, Correction : Proc. Nat. Acad. Sci. U.S.A. 72 (1975), 3260. | MR | Zbl

[90] O. Zariski, Foundations of a general theory of equisingularity on r-dimensional algebroid and algebraic varieties, of embedding dimension r+1, Amer. J. Math. 101 (1979), 453-514. | MR | Zbl

Cité par Sources :