## Compactifications of $\textbf {C}^{n}$

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- by L. Brenton and J. Morrow PDF
- Trans. Amer. Math. Soc.
**246**(1978), 139-153 Request permission

## 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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## Additional Information

- © Copyright 1978 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**246**(1978), 139-153 - MSC: Primary 32J10; Secondary 32C40
- DOI: https://doi.org/10.1090/S0002-9947-1978-0515533-X
- MathSciNet review: 515533