# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.