Non-embeddable $1$-convex manifolds

Stevens, Jan

Annales de l'Institut Fourier, Tome 64 (2014) no. 5, pp. 2205-2222.

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 $c{D}_4$.

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 $c{D}_4$-singularities.

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

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

[3] Arnol’d, V. I.; Guseĭn-Zade, S. M.; Varchenko, A. N. Singularities of differentiable maps. Vol. I, Monographs in Mathematics, 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 | Zbl

[4] Bassanelli, Giovanni; Leoni, Marco Some examples of 1-convex non-embeddable threefolds, Rev. Roumaine Math. Pures Appl., Volume 52 (2007) no. 6, pp. 611-617 | MR | Zbl

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

[6] Colţoiu, Mihnea On $1$-convex manifolds with $1$-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 | Zbl

[7] Colţoiu, Mihnea 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 | DOI | MR | Zbl

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

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

[10] Kollár, János Flips, flops, minimal models, etc, Surveys in differential geometry (Cambridge, MA, 1990), Lehigh Univ., Bethlehem, PA, 1991, pp. 113-199 | MR | Zbl

[11] Laufer, Henry B. On $\mathbf{C}{P}^1$ 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 | Zbl

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

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

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

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

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

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

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

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

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

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

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