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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the contact between complex manifolds and real hypersurfaces in $ {\bf C}\sp{3}$


Author: Thomas Bloom
Journal: Trans. Amer. Math. Soc. 263 (1981), 515-529
MSC: Primary 32F30; Secondary 53B35
MathSciNet review: 594423
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Abstract: Let $ m$ be a real $ {\mathcal{C}^\infty }$ hypersurface of an open subset of $ {{\mathbf{C}}^3}$ and let $ p \in M$. Let $ {a^1}(M,p)$ denote the maximal order of contact of a one-dimensional complex submanifold of a neighborhood of $ p$ in $ {{\mathbf{C}}^3}$ with $ M$ at $ p$. Let $ {c^1}(M,p)$ denote the $ \sup \{ m \in {\mathbf{Z}}\vert$ for all tangential holomorphic vector fields $ L$ with $ L(p) \ne 0$ then $ {L^{{i_0}}}{\bar L^{{j_0}}} \ldots {L^{{i_n}}}{\bar L^{{j_n}}}({\mathfrak{L}_M}(L))(p) = 0\} $ where $ {i_0}, \ldots ,{i_n};{j_0}, \ldots ,{j_n}$ are positive integers such that $ \sum\nolimits_{t = 0}^n {{i_t} + {j_t} = m - 3} $ and $ {\mathfrak{L}_M}(L)$ denotes the Levi form of $ M$ evaluated on the vector field $ L$.

Theorem. If $ M$ is pseudoconvex near $ p \in M$ then $ {a^1}(M,p) = {c^1}(M,p)$.


References [Enhancements On Off] (What's this?)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1981-0594423-0
PII: S 0002-9947(1981)0594423-0
Keywords: $ \bar \partial $-Neumann problem, pseudoconvex hypersurface, tangential holomorphic vector field, type of a point
Article copyright: © Copyright 1981 American Mathematical Society