Étude des jets de Demailly-Semple en dimension 3  [ Demailly-Semple jets in dimension 3 ]
Annales de l'Institut Fourier, Volume 56 (2006) no. 2, p. 397-421

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.

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.

@article{AIF_2006__56_2_397_0,
     author = {Rousseau, Erwan},
     title = {\'Etude des jets de Demailly-Semple en dimension 3},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {56},
     number = {2},
     year = {2006},
     pages = {397-421},
     doi = {10.5802/aif.2187},
     mrnumber = {2226021},
     zbl = {1092.58003},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_2006__56_2_397_0}
}
Rousseau, Erwan. Étude des jets de Demailly-Semple en dimension 3. Annales de l'Institut Fourier, Volume 56 (2006) no. 2, pp. 397-421. doi : 10.5802/aif.2187. http://www.numdam.org/item/AIF_2006__56_2_397_0/

[1] Demailly, J.-P. Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials, Proc. Sympos. Pure Math., Amer. Math.Soc., Providence, RI, Tome 62 (1997), pp. 285-360 | MR 1492539 | Zbl 0919.32014

[2] Demailly, J.-P.; El Goul, J. Hyperbolicity of generic surfaces of high degree in projective 3-space, Amer. J. Math, Tome 122 (2000), pp. 515-546 | Article | MR 1759887 | Zbl 0966.32014

[3] Dethloff, G.; Lu, S. Logarithmic jet bundles and applications, Osaka J. of Math., Tome 38 (2001), pp. 185-237 | MR 1824906 | Zbl 0982.32022

[4] Dethloff, G.; Schumacher, G.; Wong, P.M. Hyperbolicity of the complement of plane algebraic curves, Amer.J. Math, Tome 117 (1995), pp. 573-599 | Article | MR 1333937 | Zbl 0842.32021

[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., Tome 78 (1995), pp. 193-212 | Article | MR 1328756 | Zbl 0847.32028

[6] Fulton, W. Young Tableaux, London Mathematical Society Student Texts 35, Cambrige University Press (1997) | MR 1464693 | Zbl 0878.14034

[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 609557 | Zbl 0508.32010

[8] Iitaka, S. On the logarithmic Kodaira dimension of algebraic varieties, Complex Anal. and Alg. Geom., Ianami Shoten (1977), pp. 175-189 | MR 569688 | Zbl 0351.14016

[9] Iitaka, S. Geometry on complements of lines in 2 , Tokyo J. Math., Tome 1 (1978), pp. 1-19 | Article | MR 502810 | Zbl 0391.14004

[10] Kobayashi, S.; Ochiai, T. On complex manifolds with positive tangent bundles, Journal of the Mathematical Society of Japan, Tome 22 (1970), pp. 499-525 | Article | MR 275477 | Zbl 0197.36003

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

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

[13] Procesi, C. Classical invariant theory, Brandeis Lect. Notes, Tome 1 (1982) | MR 743262

[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, Tome Ser. I 336 (2003), pp. 635-640 | MR 1988123 | Zbl 1034.32017

[16] Sakai, F. Symmetric powers of the cotangent bundle and classification of algebraic varieties, Lect. Notes in Math., Berlin, Heidelberg, New York, Springer, Tome 732 (1979) | MR 555717 | Zbl 0415.14020

[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., Tome 124 (1996), pp. 573-618 | Article | MR 1369429 | Zbl 0856.32017

[18] Tan, L. On the Popov-Pommerening conjecture for groups of type A n , Proc. AMS, Tome 106 (1989), pp. 611-616 | MR 969528 | Zbl 0682.14005