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

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 $n = 3$, $X = {{\textbf {P}}^3}$.