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Non-embeddable 1-convex manifolds. (Variétés 1-convexes non plongeables.) (English. French summary) Zbl 1310.32033
Author’s 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.

MSC:
32S45 Modifications; resolution of singularities (complex-analytic aspects)
32F10 \(q\)-convexity, \(q\)-concavity
32T15 Strongly pseudoconvex domains
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[1] Alessandrini, L.; Bassanelli, G., On the embedding of 1-convex manifolds with 1-dimensional exceptional set, Ann. Inst. Fourier (Grenoble), 51, 1, 99-108, (2001) · Zbl 0966.32008
[2] Alessandrini, Lucia; Bassanelli, Giovanni, Transforms of currents by modifications and 1-convex manifolds, Osaka J. Math., 40, 3, 717-740, (2003) · Zbl 1034.32009
[3] Arnol’d, V. I.; Guseĭn-Zade, S. M.; Varchenko, A. N., Singularities of differentiable maps. Vol. I, 82, xi+382 pp., (1985), Birkhäuser Boston, Inc., Boston, MA · Zbl 0554.58001
[4] Bassanelli, Giovanni; Leoni, Marco, Some examples of 1-convex non-embeddable threefolds, Rev. Roumaine Math. Pures Appl., 52, 6, 611-617, (2007) · Zbl 1174.32014
[5] Clemens, Herbert; Kollár, János; Mori, Shigefumi, Higher-dimensional complex geometry · Zbl 0689.14016
[6] Colţoiu, Mihnea, On \(1\)-convex manifolds with \(1\)-dimensional exceptional set, Rev. Roumaine Math. Pures Appl., 43, 1-2, 97-104, (1998) · Zbl 0932.32018
[7] Colţoiu, Mihnea, Some remarks about 1-convex manifolds on which all holomorphic line bundles are trivial, Bull. Sci. Math., 130, 4, 337-340, (2006) · Zbl 1111.32007
[8] Katz, Sheldon; Morrison, David R., Gorenstein threefold singularities with small resolutions via invariant theory for Weyl groups, J. Algebraic Geom., 1, 3, 449-530, (1992) · Zbl 0788.14036
[9] Kawamata, Yujiro, General hyperplane sections of nonsingular flops in dimension \(3,\) Math. Res. Lett., 1, 1, 49-52, (1994) · Zbl 0834.32007
[10] Kollár, János, Surveys in differential geometry (Cambridge, MA, 1990), Flips, flops, minimal models, etc, 113-199, (1991), Lehigh Univ., Bethlehem, PA · Zbl 0755.14003
[11] Laufer, Henry B., Recent developments in several complex variables (Proc. Conf., Princeton Univ., Princeton, N. J., 1979), 100, On \(\textbf{C}P^1\) as an exceptional set, 261-275, (1981), Princeton Univ. Press, Princeton, N.J. · Zbl 0523.32007
[12] Moĭšezon, B. G., Irreducible exceptional submanifolds, of the first kind, of three-dimensional complex-analytic manifolds, Soviet Math. Dokl., 6, 402-403, (1965) · Zbl 0158.33103
[13] Pinkham, Henry C., Singularities, Part 2 (Arcata, Calif., 1981), 40, Factorization of birational maps in dimension \(3, 343-371, (1983),\) Amer. Math. Soc., Providence, RI · Zbl 0544.14005
[14] Ravindra, G. V.; Srinivas, V., The Grothendieck-Lefschetz theorem for normal projective varieties, J. Algebraic Geom., 15, 3, 563-590, (2006) · Zbl 1123.14004
[15] Ravindra, G. V.; Srinivas, V., The Noether-Lefschetz theorem for the divisor class group, J. Algebra, 322, 9, 3373-3391, (2009) · Zbl 1189.14010
[16] Reid, Miles, Algebraic varieties and analytic varieties (Tokyo, 1981), 1, Minimal models of canonical \(3\)-folds, 131-180, (1983), North-Holland, Amsterdam · Zbl 0558.14028
[17] Schneider, Michael, Familien negativer vektorraumbündel und \(1\)-konvexe abbildungen, Abh. Math. Sem. Univ. Hamburg, 47, 150-170, (1978) · Zbl 0391.32011
[18] Tan, Vo Van, On certain non-Kählerian strongly pseudoconvex manifolds, J. Geom. Anal., 4, 2, 233-245, (1994) · Zbl 0807.32018
[19] Tan, Vo Van, On the Kählerian geometry of \(1\)-convex threefolds, Forum Math., 7, 2, 131-146, (1995) · Zbl 0839.32003
[20] Tjurina, G. N., Resolution of singularities of flat deformations of double rational points, Funkcional. Anal. i Priložen., 4, 1, 77-83, (1970) · Zbl 0221.32008
[21] Vâjâitu, Viorel, On embeddable 1-convex spaces, Osaka J. Math., 38, 2, 287-294, (2001) · Zbl 0982.32010
[22] Vo Van, Tan, On the quasi-projectivity of compactifiable strongly pseudoconvex manifolds, Bull. Sci. Math., 129, 6, 501-522, (2005) · Zbl 1083.32010
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