Non-embeddable $1$-convex manifolds

[Variétés 1-convexes non plongeables]

Stevens, Jan

Annales de l'Institut Fourier, Tome 64 (2014), p. 2205-2222 / Harvested from Numdam

Résumé

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 $ (1, -3) $. À cette fin, nous étudions de petites résolutions des singularités $cD_4$.

Abstract

We show that every small resolution of a 3-dimensional terminal hypersurface singularity can occur on a non-embeddable $1$-convex manifold.

We give an explicit example of a non-embeddable manifold containing an irreducible exceptional rational curve with normal bundle of type $(1,-3)$. To this end we study small resolutions of $cD_4$-singularities.

Détail

Publié le : 2014-01-01

MR 3330936 | Zbl 06387336

DOI : https://doi.org/10.5802/aif.2909
Classification:  32S45,  32F10,  32Q15,  32T15,  13C20,  14E30
Mots clés: variétés 1-convexes, petites résolutions

Bibliographie

[1] Alessandrini, L.; Bassanelli, G. On the embedding of 1-convex manifolds with 1-dimensional exceptional set, Ann. Inst. Fourier (Grenoble), Tome 51 (2001) no. 1, pp. 99-108 | Article | Numdam | MR 1821070 | Zbl 0966.32008

[2] Alessandrini, Lucia; Bassanelli, Giovanni Transforms of currents by modifications and 1-convex manifolds, Osaka J. Math., Tome 40 (2003) no. 3, pp. 717-740 | MR 2003745 | Zbl 1034.32009

[3] Arnol’D, V. I.; Guseĭn-Zade, S. M.; Varchenko, A. N. Singularities of differentiable maps. Vol. I, Birkhäuser Boston, Inc., Boston, MA, Monographs in Mathematics, Tome 82 (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] Bassanelli, Giovanni; Leoni, Marco Some examples of 1-convex non-embeddable threefolds, Rev. Roumaine Math. Pures Appl., Tome 52 (2007) no. 6, pp. 611-617 | MR 2387599 | Zbl 1174.32014

[5] Clemens, Herbert; Kollár, János; Mori, Shigefumi Higher-dimensional complex geometry (Astérisque 166, (1988), 144 pp.) | MR 1004926 | Zbl 0689.14016

[6] Colţoiu, Mihnea On $1$-convex manifolds with $1$-dimensional exceptional set, Rev. Roumaine Math. Pures Appl., Tome 43 (1998) no. 1-2, pp. 97-104 (Collection of papers in memory of Martin Jurchescu) | MR 1655264 | Zbl 0932.32018

[7] Colţoiu, Mihnea Some remarks about 1-convex manifolds on which all holomorphic line bundles are trivial, Bull. Sci. Math., Tome 130 (2006) no. 4, pp. 337-340 | Article | MR 2237448 | Zbl 1111.32007

[8] Katz, Sheldon; Morrison, David R. Gorenstein threefold singularities with small resolutions via invariant theory for Weyl groups, J. Algebraic Geom., Tome 1 (1992) no. 3, pp. 449-530 | MR 1158626 | Zbl 0788.14036

[9] Kawamata, Yujiro General hyperplane sections of nonsingular flops in dimension $3$, Math. Res. Lett., Tome 1 (1994) no. 1, pp. 49-52 | Article | MR 1258489 | Zbl 0834.32007

[10] Kollár, János 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] Laufer, Henry B. On ${\bf C}P^{1}$ as an exceptional set, Recent developments in several complex variables (Proc. Conf., Princeton Univ., Princeton, N. J., 1979), Princeton Univ. Press, Princeton, N.J. (Ann. of Math. Stud.) Tome 100 (1981), pp. 261-275 | MR 627762 | Zbl 0523.32007

[12] Moĭšezon, B. G. Irreducible exceptional submanifolds, of the first kind, of three-dimensional complex-analytic manifolds, Soviet Math. Dokl., Tome 6 (1965), p. 402-403 | MR 222916 | Zbl 0158.33103

[13] Pinkham, Henry C. Factorization of birational maps in dimension $3$, Singularities, Part 2 (Arcata, Calif., 1981), Amer. Math. Soc., Providence, RI (Proc. Sympos. Pure Math.) Tome 40 (1983), pp. 343-371 | MR 713260 | Zbl 0544.14005

[14] Ravindra, G. V.; Srinivas, V. The Grothendieck-Lefschetz theorem for normal projective varieties, J. Algebraic Geom., Tome 15 (2006) no. 3, pp. 563-590 | Article | MR 2219849 | Zbl 1123.14004

[15] Ravindra, G. V.; Srinivas, V. The Noether-Lefschetz theorem for the divisor class group, J. Algebra, Tome 322 (2009) no. 9, pp. 3373-3391 | Article | MR 2567426 | Zbl 1189.14010

[16] Reid, Miles Minimal models of canonical $3$-folds, Algebraic varieties and analytic varieties (Tokyo, 1981), North-Holland, Amsterdam (Adv. Stud. Pure Math.) Tome 1 (1983), pp. 131-180 | MR 715649 | Zbl 0558.14028

[17] Schneider, Michael Familien negativer Vektorraumbündel und $1$-konvexe Abbildungen, Abh. Math. Sem. Univ. Hamburg, Tome 47 (1978), pp. 150-170 (Special issue dedicated to the seventieth birthday of Erich Kähler) | Article | MR 492393 | Zbl 0391.32011

[18] Tan, Vo Van On certain non-Kählerian strongly pseudoconvex manifolds, J. Geom. Anal., Tome 4 (1994) no. 2, pp. 233-245 | Article | MR 1277508 | Zbl 0807.32018

[19] Tan, Vo Van On the Kählerian geometry of $1$-convex threefolds, Forum Math., Tome 7 (1995) no. 2, pp. 131-146 | MR 1316945 | Zbl 0839.32003

[20] Tjurina, G. N. Resolution of singularities of flat deformations of double rational points, Funkcional. Anal. i Priložen., Tome 4 (1970) no. 1, pp. 77-83 | MR 267129 | Zbl 0221.32008

[21] Vâjâitu, Viorel On embeddable 1-convex spaces, Osaka J. Math., Tome 38 (2001) no. 2, pp. 287-294 | MR 1833621 | Zbl 0982.32010

[22] Vo Van, Tan On the quasi-projectivity of compactifiable strongly pseudoconvex manifolds, Bull. Sci. Math., Tome 129 (2005) no. 6, pp. 501-522 | Article | MR 2142895 | Zbl 1083.32010