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Lifting cycles to deformations of two-dimensional pseudoconvex manifolds


Author: Henry B. Laufer
Journal: Trans. Amer. Math. Soc. 266 (1981), 183-202
MSC: Primary 32G05; Secondary 14J15, 32G10
DOI: https://doi.org/10.1090/S0002-9947-1981-0613791-4
MathSciNet review: 613791
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Abstract: Let $ M$ be a strictly pseudoconvex manifold with exceptional set $ A$. Let $ D \geqslant 0$ be a cycle on $ A$. Let $ \omega :\mathfrak{M} \to Q$ be a deformation of $ M$. Kodaira's theory for deforming submanifolds of $ \mathfrak{M}$ is extended to the subspace $ D$. Let $ \mathfrak{J}$ be the sheaf of germs of infinitesimal deformations of $ D$. Suppose that $ {H^1}(D,\mathfrak{J}) = 0$. If $ \omega $ is the versal deformation, then $ D$ lifts to above a submanifold of $ Q$. This lifting is a complete deformation of $ D$ with a smooth generic fiber.

If all of the fibers of $ \mathfrak{M}$ are isomorphic, then $ \omega $ is the trivial deformation. If $ M$ has no exceptional curves of the first kind, then there exists $ \omega $ such that only any given irreducible component of $ A$ disappears as part of the exceptional set.


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  • [1] M. Artin, Algebraic construction of Brieskorn's resolutions, J. Algebra 29 (1974), 330-348. MR 0354665 (50:7143)
  • [2] M. Artin and M. Schlessinger, Algebraic construction of Brieskorn's resolutions, preprint.
  • [3] A. Douady, Le problème des modules locaux pour les espaces $ C$-analytiques compacts, Ann. Sci. École Norm. Sup. (4) 7 (1974), 569-602 (1975). MR 0382729 (52:3611)
  • [4] R. Elkik, Algébrisation du module formel d'une singularité isolé, Astérisque 16 (1974), 133-144. MR 0352093 (50:4580)
  • [5] W. Fischer and H. Grauert, Lokal-triviale Familien kompakter komplexer Mannigfaltigkeiten, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1965, 89-94. MR 0184258 (32:1731)
  • [6] H. Grauert, Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann. 146 (1962), 331-368. MR 0137127 (25:583)
  • [7] -, Über die Deformation isolierter Singularitäten analytischer Mengen, Invent. Math. 15 (1972), 171-199. MR 0293127 (45:2206)
  • [8] -, Der Satz von Kuranishi für kompakte komplexe Räume, Invent. Math. 25 (1974), 107-142. MR 0346194 (49:10920)
  • [9] R. Gunning and R. Narasimhan, Immersion of open Riemann surfaces, Math. Ann. 174 (1967), 103-108. MR 0223560 (36:6608)
  • [10] R. Gunning and H. Rossi, Analytic functions of several complex variables, Prentice-Hall, Englewood Cliffs, N. J., 1965. MR 0180696 (31:4927)
  • [11] A. Kas and M. Schlessinger, On the versal deformation of a complex space with an isolated singularity, Math. Ann. 196 (1972), 23-29. MR 0294701 (45:3769)
  • [12] K. Kodaira, A theorem of completeness of characteristic systems for analytic families of compact submanifolds of complex manifolds, Ann. of Math. 75 (1962), 146-162. MR 0133841 (24:A3665b)
  • [13] K. Kodaira and D. Spencer, A theorem of completeness of characteristic systems of complete continuous systems, Amer. J. Math. 81 (1959), 577-500. MR 0112156 (22:3011)
  • [14] H. Laufer, On rational singularities, Amer. J. Math. 94 (1972), 597-608. MR 0330500 (48:8837)
  • [15] -, Deformations of resolutions of two-dimensional singularities, Rice Univ. Studies 59 (1973), 53-96. MR 0367277 (51:3519)
  • [16] -, Taut two-dimensional singularities, Math. Ann. 205 (1973), 131-164. MR 0333238 (48:11563)
  • [17] -, Ambient deformations for one-dimensional exceptional sets, 1978, preprint.
  • [18] -, Ambient deformations for exceptional sets in two-manifolds, Invent. Math. 55 (1979), 1-36. MR 553993 (81f:32027)
  • [19] -, Versal deformations for two-dimensional pseudoconvex manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1980), 511-521. MR 597550 (82g:32024)
  • [20] -, On $ C{{\mathbf{P}}^1}$ as an exceptional set, Recent Developments in Several Complex Variables (J. Fornaess, Ed.), Ann. of Math. Studies No. 100, Princeton Univ. Press, Princeton, N. J., 1981, pp. 261-275. MR 627762 (82m:32012)
  • [21] D. Lê and C. Ramanujan, The invariance of Milnor's number implies the invariance of the topological type, Amer. J. Math. 98 (1976), 67-78. MR 0399088 (53:2939)
  • [22] G. Pourcin, Déformation de singularités isolées, Quelques Problèmes de Modules (Sem. Géométrie Analytique, Ecole Norm. Sup., Paris, 1971-1972), Astérisque, no. 16, Soc. Math. France, Paris, 1974, pp. 161-173. MR 0369742 (51:5974)
  • [23] O. Riemenschneider, Familien komplexer Räume mit streng pseudokonvexer spezieller Faser, Comment. Math. Helv. 51 (1976), 547-565. MR 0437808 (55:10730)
  • [24] Y.-T. Siu, Analytic sheaf cohomology groups of dimension $ n$ of $ n$-dimensional complex spaces, Trans. Amer. Math. Soc. 143 (1969), 77-94. MR 0252684 (40:5902)
  • [25] B. Teissier, Deformations à type topologique constant. I, Astérisque 16 (1974), 215-227. MR 0414931 (54:3023)
  • [26] J. Wahl, Equisingular deformations of normal surface singularities. I, Ann. of Math. 104 (1976), 325-356. MR 0422270 (54:10261)
  • [27] -, The discriminant locus of a rational singularity, informal research announcement, 1976.
  • [28] -, Simultaneous resolution and discriminantal loci, Duke Math. J. 46 (1979), 341-374. MR 534056 (80g:14008)
  • [29] H. Whitney, Tangents to an analytic variety, Ann. of Math. 81 (1965), 496-549. MR 0192520 (33:745)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1981-0613791-4
Keywords: Deformation, pseudoconvex, exceptional set, obstruction
Article copyright: © Copyright 1981 American Mathematical Society

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