Étude des jets de Demailly-Semple en dimension 3

Rousseau, Erwan

Annales de l'Institut Fourier, Tome 56 (2006) no. 2, pp. 397-421.

Dans cet article nous faisons l’étude algébrique des jets de Demailly-Semple en dimension 3 en utilisant la théorie des invariants des groupes non réductifs. Cette étude fournit la caractérisation géométrique du fibré des jets d’ordre 3 sur une variété de dimension 3 et permet d’effectuer, par Riemann-Roch, un calcul de caractéristique d’Euler.

In this article, the algebraic characterization of Demailly-Semple jets in dimension 3 is given using the invariant theory of non reductive groups. This work provides the geometric characterization of the 3-jets bundle on a manifold of dimension 3 and, by Riemann-Roch, the computation of the Euler characteristic.

Classification : 32Q45, 13A50, 06B15
Mots clés : hyperbolicité des variétés complexes, huperbolicité au sens de Kobayashi, fibrés des jets de différentielles, représentations des groupes linéaires, théorie des invariants des groupes non réductifs

[1] Demailly, J.-P. Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials, Proc. Sympos. Pure Math., Volume 62, Amer. Math.Soc., Providence, RI (1997), pp. 285-360 | MR | Zbl

[2] Demailly, J.-P.; El Goul, J. Hyperbolicity of generic surfaces of high degree in projective 3-space, Amer. J. Math, Volume 122 (2000), pp. 515-546 | DOI | MR | Zbl

[3] Dethloff, G.; Lu, S. Logarithmic jet bundles and applications, Osaka J. of Math., Volume 38 (2001), pp. 185-237 | MR | Zbl

[4] Dethloff, G.; Schumacher, G.; Wong, P.M. Hyperbolicity of the complement of plane algebraic curves, Amer.J. Math, Volume 117 (1995), pp. 573-599 | DOI | MR | Zbl

[5] Dethloff, G.; Schumacher, G.; Wong, P.M. On the hyperbolicity of the complements of curves in algebraic surfaces : the three component case, Duke. Math. J., Volume 78 (1995), pp. 193-212 | DOI | MR | Zbl

[6] Fulton, W. Young Tableaux, London Mathematical Society Student Texts 35, Cambrige University Press, 1997 | MR | Zbl

[7] Green, M.; Griffiths, P. Two applications of algebraic geometry to entire holomorphic mappings, The Chern Symposium 1979, Proc. Inter. Sympos. Berkeley, CA, Springer-Verlag, New-York (1980), pp. 41-74 | MR | Zbl

[8] Iitaka, S. On the logarithmic Kodaira dimension of algebraic varieties, Complex Anal. and Alg. Geom., Ianami Shoten (1977), pp. 175-189 | MR | Zbl

[9] Iitaka, S. Geometry on complements of lines in ${\mathbb{P}}^2$, Tokyo J. Math., Volume 1 (1978), pp. 1-19 | DOI | MR | Zbl

[10] Kobayashi, S.; Ochiai, T. On complex manifolds with positive tangent bundles, Journal of the Mathematical Society of Japan, Volume 22 (1970), pp. 499-525 | DOI | MR | Zbl

[11] Pommerening, K. Invariant theory, LNM, Volume 1278 (1987)

[12] Popov, V.L. Invariant theory, 4, EMS, Springer-Verlag, 1989

[13] Procesi, C. Classical invariant theory, Brandeis Lect. Notes, Volume 1 (1982) | MR

[14] Rousseau, E. Sur la conjecture de Kobayashi et l’hyperbolicité des hypersurfaces projectives en dimension 2 et 3 (Université de Bretagne Occidentale)

[15] Rousseau, E. Hyperbolicité du complémentaire d’une courbe : le cas de deux composantes, CRAS, Volume Ser. I 336 (2003), pp. 635-640 | MR | Zbl

[16] Sakai, F. Symmetric powers of the cotangent bundle and classification of algebraic varieties, Lect. Notes in Math., Volume 732, Berlin, Heidelberg, New York, Springer (1979) | MR | Zbl

[17] Siu, Y.-T.; Yeung, S.K. Hyperbolicity of the complement of a generic smooth curve of high degree in the complex projective plane, Invent. Math., Volume 124 (1996), pp. 573-618 | DOI | MR | Zbl

[18] Tan, L. On the Popov-Pommerening conjecture for groups of type $A_n$, Proc. AMS, Volume 106 (1989), pp. 611-616 | MR | Zbl