Nous montrons que chaque petite résolution d’une singularité de hypersurface 3-dimensionnelle terminale peut se produire sur une variété 1-convexe non plongeable.
Nous donnons un exemple explicite d’une variété non plongeable contenant une courbe exceptionnelle rationnelle irréductible avec fibré normal du type . À cette fin, nous étudions de petites résolutions des singularités .
We show that every small resolution of a 3-dimensional terminal hypersurface singularity can occur on a non-embeddable -convex manifold.
We give an explicit example of a non-embeddable manifold containing an irreducible exceptional rational curve with normal bundle of type . To this end we study small resolutions of -singularities.
Classification : 32S45, 32F10, 32Q15, 32T15, 13C20, 14E30
Mots clés : variétés 1-convexes, petites résolutions
@article{AIF_2014__64_5_2205_0, author = {Stevens, Jan}, title = {Non-embeddable $1$-convex manifolds}, journal = {Annales de l'Institut Fourier}, pages = {2205--2222}, publisher = {Association des Annales de l'institut Fourier}, volume = {64}, number = {5}, year = {2014}, doi = {10.5802/aif.2909}, mrnumber = {3330936}, zbl = {06387336}, language = {en}, url = {archive.numdam.org/item/AIF_2014__64_5_2205_0/} }
Stevens, Jan. Non-embeddable $1$-convex manifolds. Annales de l'Institut Fourier, Tome 64 (2014) no. 5, pp. 2205-2222. doi : 10.5802/aif.2909. http://archive.numdam.org/item/AIF_2014__64_5_2205_0/
[1] On the embedding of 1-convex manifolds with 1-dimensional exceptional set, Ann. Inst. Fourier (Grenoble), Volume 51 (2001) no. 1, pp. 99-108 | Article | Numdam | MR 1821070 | Zbl 0966.32008
[2] Transforms of currents by modifications and 1-convex manifolds, Osaka J. Math., Volume 40 (2003) no. 3, pp. 717-740 | MR 2003745 | Zbl 1034.32009
[3] Singularities of differentiable maps. Vol. I, Monographs in Mathematics, Volume 82, Birkhäuser Boston, Inc., Boston, MA, 1985, pp. xi+382 (The classification of critical points, caustics and wave fronts, Translated from the Russian by Ian Porteous and Mark Reynolds) | MR 777682 | Zbl 0554.58001
[4] Some examples of 1-convex non-embeddable threefolds, Rev. Roumaine Math. Pures Appl., Volume 52 (2007) no. 6, pp. 611-617 | MR 2387599 | Zbl 1174.32014
[5] Higher-dimensional complex geometry (Astérisque 166, (1988), 144 pp.) | MR 1004926 | Zbl 0689.14016
[6] On -convex manifolds with -dimensional exceptional set, Rev. Roumaine Math. Pures Appl., Volume 43 (1998) no. 1-2, pp. 97-104 (Collection of papers in memory of Martin Jurchescu) | MR 1655264 | Zbl 0932.32018
[7] Some remarks about 1-convex manifolds on which all holomorphic line bundles are trivial, Bull. Sci. Math., Volume 130 (2006) no. 4, pp. 337-340 | Article | MR 2237448 | Zbl 1111.32007
[8] Gorenstein threefold singularities with small resolutions via invariant theory for Weyl groups, J. Algebraic Geom., Volume 1 (1992) no. 3, pp. 449-530 | MR 1158626 | Zbl 0788.14036
[9] General hyperplane sections of nonsingular flops in dimension , Math. Res. Lett., Volume 1 (1994) no. 1, pp. 49-52 | Article | MR 1258489 | Zbl 0834.32007
[10] Flips, flops, minimal models, etc, Surveys in differential geometry (Cambridge, MA, 1990), Lehigh Univ., Bethlehem, PA, 1991, pp. 113-199 | MR 1144527 | Zbl 0755.14003
[11] On as an exceptional set, Recent developments in several complex variables (Proc. Conf., Princeton Univ., Princeton, N. J., 1979) (Ann. of Math. Stud.) Volume 100, Princeton Univ. Press, Princeton, N.J., 1981, pp. 261-275 | MR 627762 | Zbl 0523.32007
[12] Irreducible exceptional submanifolds, of the first kind, of three-dimensional complex-analytic manifolds, Soviet Math. Dokl., Volume 6 (1965), p. 402-403 | MR 222916 | Zbl 0158.33103
[13] Factorization of birational maps in dimension , Singularities, Part 2 (Arcata, Calif., 1981) (Proc. Sympos. Pure Math.) Volume 40, Amer. Math. Soc., Providence, RI, 1983, pp. 343-371 | MR 713260 | Zbl 0544.14005
[14] The Grothendieck-Lefschetz theorem for normal projective varieties, J. Algebraic Geom., Volume 15 (2006) no. 3, pp. 563-590 | Article | MR 2219849 | Zbl 1123.14004
[15] The Noether-Lefschetz theorem for the divisor class group, J. Algebra, Volume 322 (2009) no. 9, pp. 3373-3391 | Article | MR 2567426 | Zbl 1189.14010
[16] Minimal models of canonical -folds, Algebraic varieties and analytic varieties (Tokyo, 1981) (Adv. Stud. Pure Math.) Volume 1, North-Holland, Amsterdam, 1983, pp. 131-180 | MR 715649 | Zbl 0558.14028
[17] Familien negativer Vektorraumbündel und -konvexe Abbildungen, Abh. Math. Sem. Univ. Hamburg, Volume 47 (1978), pp. 150-170 (Special issue dedicated to the seventieth birthday of Erich Kähler) | Article | MR 492393 | Zbl 0391.32011
[18] On certain non-Kählerian strongly pseudoconvex manifolds, J. Geom. Anal., Volume 4 (1994) no. 2, pp. 233-245 | Article | MR 1277508 | Zbl 0807.32018
[19] On the Kählerian geometry of -convex threefolds, Forum Math., Volume 7 (1995) no. 2, pp. 131-146 | MR 1316945 | Zbl 0839.32003
[20] Resolution of singularities of flat deformations of double rational points, Funkcional. Anal. i Priložen., Volume 4 (1970) no. 1, pp. 77-83 | MR 267129 | Zbl 0221.32008
[21] On embeddable 1-convex spaces, Osaka J. Math., Volume 38 (2001) no. 2, pp. 287-294 | MR 1833621 | Zbl 0982.32010
[22] On the quasi-projectivity of compactifiable strongly pseudoconvex manifolds, Bull. Sci. Math., Volume 129 (2005) no. 6, pp. 501-522 | Article | MR 2142895 | Zbl 1083.32010