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

 

Compactifications of $ {\bf C}\sp{n}$


Authors: L. Brenton and J. Morrow
Journal: Trans. Amer. Math. Soc. 246 (1978), 139-153
MSC: Primary 32J10; Secondary 32C40
MathSciNet review: 515533
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Abstract: Let X be a compactification of $ {{\text{C}}^n}$. We assume that X is a compact complex manifold and that $ A\, = \,X\, - \,{{\text{C}}^n}$ is a proper subvariety of X. If we suppose that A is a Kähler manifold, then we prove that X is projective algebraic, $ {H^{\ast}}\left( {A,\,{\textbf{Z}}} \right)\, \cong \,{H^{\ast}}\left( {{{\textbf{P}}^{n\, - \,1}},\,{\textbf{Z}}} \right)$, and $ {H^{\ast}}\left( {X,\,{\textbf{Z}}} \right)\, \cong \,{H^{\ast}}\left( {{{\textbf{P}}^n},\,{\textbf{Z}}} \right)$. Various additional conditions are shown to imply that $ X\, = \,{{\textbf{P}}^n}$. It is known that no additional conditions are needed to imply $ X\, = \,{{\textbf{P}}^n}$ in the cases $ n\, = \,1,\,2$. In this paper we prove that if $ n\, = \,3$, $ X\, = \,{{\textbf{P}}^3}$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1978-0515533-X
PII: S 0002-9947(1978)0515533-X
Article copyright: © Copyright 1978 American Mathematical Society