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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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