logo AFST

Weak Whitney regularity implies equimultiplicity for families of complex hypersurfaces
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 25 (2016) no. 1, pp. 161-170.

Nous démontrons que la régularité faible de Whitney pour une famille d’hypersurfaces complexes à singularités isolées implique l’équimultiplicité.

We prove that weak Whitney regularity for a family of complex hypersurfaces with isolated singularities implies equimultiplicity.

@article{AFST_2016_6_25_1_161_0,
     author = {David Trotman and Duco van Straten},
     title = {Weak {Whitney} regularity implies equimultiplicity for families of complex hypersurfaces},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {161--170},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 25},
     number = {1},
     year = {2016},
     doi = {10.5802/afst.1490},
     mrnumber = {3485294},
     zbl = {1354.32019},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1490/}
}
David Trotman; Duco van Straten. Weak Whitney regularity implies equimultiplicity for families of complex hypersurfaces. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 25 (2016) no. 1, pp. 161-170. doi : 10.5802/afst.1490. https://afst.centre-mersenne.org/articles/10.5802/afst.1490/

[1] Bekka (K.).— Sur les propriétés topologiques et métriques des espaces stratifiés, Doctoral thesis, Université de Paris-Sud (1988).

[2] Bekka (K.).— C-régularité et trivialité topologique, Singularity theory and applications, Warwick 1989 (eds. D. M. Q. Mond and J. Montaldi), Springer Lecture Notes 1462, p. 42-62 (1991). | Article

[3] Bekka (K.) and Trotman (D.).— Propriétés métriques de familles Φ-radiales de sous-variétés différentiables, C. R. Acad. Sci. Paris Ser. A-B, 305, p. 389-392 (1987). | Zbl 0642.57016

[4] Bekka (K.) and Trotman (D.).— Weakly Whitney stratified sets, Real and complex singularities (Proceedings, Sao Carlos 1998, edited by J. W. Bruce and F. Tari), Chapman and Hall/CRC Res. Notes Math. 412, p. 1-15 (2000).

[5] Bekka (K.) and Trotman (D.).— On metric properties of stratified sets, Manuscripta Math., 111, p. 71-95 (2003). | Article | MR 1981597 | Zbl 1033.58005

[6] Bekka (K.) and Trotman (D.).— Briançon-Speder examples and the failure of weak Whitney regularity, Journal of Singularities 7, p. 88-107 (2013). | Article | Zbl 1293.32033

[7] Briançon (J.) and Speder (J.-P.).— La trivialité topologique n’implique pas les conditions de Whitney, C. R. Acad. Sci. Paris Ser. A-B, 280, p. 365-367 (1975). | Zbl 0331.32010

[8] Briançon (J.) and Speder (J.-P.).— Les conditions de Whitney impliquent μ * -constant, Annales de l’Institut Fourier, Grenoble, 26 (2), p. 153-163 (1976). | Article | MR 419825 | Zbl 0331.32012

[9] Ferrarotti (M.).— Volume on stratified sets, Annali di Matematica Pura e applicata, serie 4,144, p. 183-201 (1986). | Article | MR 870876

[10] Ferrarotti (M.).— Some results about integration on regular stratified sets, Annali di Matematica Pura e applicata, serie 4, 150, p. 263-279 (1988). | Article | MR 946036

[11] Fulton (W.), Hansen (J.).— A connectedness theorem for projective varieties, with applications to intersections and singularities of mappings, Ann. of Math. 110, p. 159-166 (1979). | Article | MR 541334 | Zbl 0389.14002

[12] Fulton (W.).— On the topology of algebraic varieties, Proc. Symp. Pure Math. 46, Amer. Math. Soc., Providence, RI, p. 15-46 (1987). | Article | Zbl 0703.14012

[13] Goresky (R. M.).— Triangulation of stratified objects, Proc. Amer. Math. Soc. 72, no. 1, p. 193-200 (1978). | Article | MR 500991 | Zbl 0392.57001

[14] Henry (J.-P.) and Merle (M.).— Sections planes, limites d’espaces tangents et transversalité de variétés polaires, C. R. Acad. Sci. Paris, Série A, 291, p. 291-294 (1980). | Zbl 0472.32005

[15] Hironaka (H.).— Normal cones of analytic Whitney stratifications, I. H. E. S. Publ. Math. 36, p. 127-138 (1969). | Article | Zbl 0219.57022

[16] (D. T.) and Saito (K.).— La constance du nombre de Milnor donne des bonnes stratifications, C. R. Acad. Sci. Paris 277, p. 793-795 (1973). | Zbl 0283.32007

[17] Mather (J.).— Notes on topological stability, Harvard University, 1970, Bull. Amer. Math. Soc 49, p. 475-506 (2012). | Article | MR 2958928 | Zbl 1260.57049

[18] Navarro Aznar (V.).— Conditions de Whitney et sections planes, Inventiones Math. 61, p. 199-225 (1980). | Article | MR 592691 | Zbl 0449.32013

[19] Navarro Aznar (V.) and Trotman (D.) J. A..— Whitney regularity and generic wings, Ann. Inst. Fourier, Grenoble 31(2), p. 87-111 (1981). | Article | Numdam | MR 617242 | Zbl 0442.58002

[20] Orro (P.) and Trotman (D.).— Cône normal à une stratification régulière, Seminari Geometria 1998-99, Università degli Studi Bologna 12, p. 169-175 (2000).

[21] Orro (P.) and Trotman (D.).— Transverse regular stratifications, Real and Complex Singularities, edited by M. Manoel, M. C. Romero Fuster and C. T. C. Wall, 10th international workshop, Sao Carlos, Brazil 2008, London Mathematical Society Lecture Note Series 380, Cambridge University Press, p. 298-304 (2010). | Article | Zbl 1218.58004

[22] Parusinski (A.).— Bi-Lipschitz trivialization of the distance function to a stratum of a stratification, Ann. Pol. Math., 87, p. 213-218 (2005). | Article | MR 2208547 | Zbl 1090.32016

[23] Pflaum (M.).— Analytic and Geometric study of Stratified Spaces, Springer Lecture Notes in Math. 1768 (2001). | Article | Zbl 0988.58003

[24] Schürmann (J.).— Topology of singular spaces and constructible sheaves, Mathematics Institute of the Polish Academy of Sciences. Mathematical Monographs (New Series) 63, Birkhauser Verlag, Basel (2003). | Zbl 1041.55001

[25] Teissier (B.).— Cycles évanescents, sections planes et conditions de Whitney, Singularités à Cargèse, Astérisque 7-8, Soc. Math. France, Paris, p. 285-362 (1973). | Numdam | Zbl 0295.14003

[26] Thom (R.).— Local topological properties of differentiable mappings, Differential Analysis, Bombay Colloq., Oxford Univ. Press, London, p. 191-202 (1964).

[27] Thom (R.).— Ensembles et morphismes stratifiés, Bull. A. M. S. 75, p. 240-284 (1969). | Article | Zbl 0197.20502

[28] Wall (C. T. C.).— Regular stratifications, Dynamical systems-Warwick 1974, Lecture Notes in Math. 468, Springer, Berlin, p. 332-344 (1975). | Article

[29] Whitney (H.).— Local properties of analytic varieties, Diff. and Comb. Topology (ed. S. Cairns), Princeton Univ. Press, Princeton, p. 205-244 (1965). | Article

[30] Whitney (H.).— Tangents to an analytic variety, Ann. of Math. 81, p. 496-549 (1965). | Article | MR 192520 | Zbl 0152.27701