no. 9
Algebraic geometry/Topology
Milnor and Tjurina numbers for a hypersurface germ with isolated singularity
[Nombres de Milnor et Tjurina pour les germes d'hypersurfaces à singularité isolée]

Présenté par : Claire Voisin

Liu, Yongqiang1
1 KU Leuven, Department of Mathematics, Celestijnenlaan 200B, 3001 Leuven, Belgium
Comptes Rendus. Mathématique, Tome 356 (2018) no. 9, pp. 963-966.

Soit f:(Cn,0)→(C,0) un germe de fonction analytique au voisinage de l'origine avec une seule singularité isolée. Soient μ et τ les nombres de Milnor et Tjurina correspondants. Nous montrons que μτ≤n. Comme application, nous donnons une minoration du nombre de Tjurina en fonction de n et de la multiplicité de f à l'origine.

Assume that f:(Cn,0)→(C,0) is an analytic function germ at the origin with only isolated singularity. Let μ and τ be the corresponding Milnor and Tjurina numbers. We show that μτ≤n. As an application, we give a lower bound for the Tjurina number in terms of n and the multiplicity of f at the origin.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2018.07.004
Liu, Yongqiang 1

1 KU Leuven, Department of Mathematics, Celestijnenlaan 200B, 3001 Leuven, Belgium 
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Liu, Yongqiang. Milnor and Tjurina numbers for a hypersurface germ with isolated singularity. Comptes Rendus. Mathématique, Tome 356 (2018) no. 9, pp. 963-966. doi : 10.1016/j.crma.2018.07.004. http://www.numdam.org/articles/10.1016/j.crma.2018.07.004/

[1] Briançon, J.; Skoda, H. Sur la clôture intégrale d'un idéal de germes de fonctions holomorphes en un point de Cn, C. R. Acad. Sci. Paris, Ser. A, Volume 278 (1974), pp. 949-951 (in French)

[2] Dimca, A. Differential forms and hypersurface singularities, Singularity theory and its applications, Part I, Coventry, 1988/1989 (Lecture Notes in Mathematics), Volume vol. 1462, Springer, Berlin (1991), pp. 122-153

[3] Dimca, A.; Greuel, G.-M. On 1-forms on isolated complete intersection curve singularities | arXiv

[4] Greuel, G.-M.; Lossen, C.; Shustin, E. Introduction to Singularities and Deformations, Springer Monographs in Mathematics, Springer, Berlin, 2007

[5] Huh, J. Milnor numbers of projective hypersurfaces with isolated singularities, Duke Math. J., Volume 163 (2014), pp. 1525-1548

[6] Kouchnirenko, A.G. Polyèdres de Newton et nombres de Milnor, Invent. Math., Volume 32 (1976) no. 1, pp. 1-31

[7] Malgrange, B. Letter to the editors, Invent. Math., Volume 20 (1973), pp. 171-172

[8] Saito, K. Quasihomogene isolierte Singularitäten von Hyperflächen, Invent. Math., Volume 14 (1971), pp. 123-142 (in German)

[9] Scherk, J. On the monodromy theorem for isolated hypersurface singularities, Invent. Math., Volume 58 (1980) no. 3, pp. 289-301

[10] Teissier, B. (Singularités à Cargèse. Asterisque), Volume vol. 7 et 8, Société mathématique de France, Paris (1973), pp. 285-362

[11] Teissier, B. Sur une inégalité à la Minkowski pour les multiplicités, Ann. of Math. (2), Volume 106 (1977) no. 1, pp. 38-44

[12] Wahl, J. A characterization of quasihomogeneous Gorenstein surface singularities, Compos. Math., Volume 55 (1985) no. 3, pp. 269-288

[13] Yau, S.S.-T.; Zuo, H. Complete characterization of isolated homogeneous hypersurface singularities, Pac. J. Math., Volume 273 (2015) no. 1, pp. 213-224

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