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Compactifications of 𝐶ⁿ

1978; American Mathematical Society; Volume: 246; Linguagem: Inglês

10.1090/s0002-9947-1978-0515533-x

1088-6850

Lawrence Brenton, James Morrow,

Homotopy and Cohomology in Algebraic Topology

Let X be a compactification of C n {{\text {C}}^n} . We assume that X is a compact complex manifold and that A = X − C n 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 ∗ ( A , Z ) ≅ H ∗ ( P n − 1 , Z ) {H^{\ast }}\left ( {A,\,{\textbf {Z}}} \right )\, \cong \,{H^{\ast }}\left ( {{{\textbf {P}}^{n\, - \,1}},\,{\textbf {Z}}} \right ) , and H ∗ ( X , Z ) ≅ H ∗ ( P n , Z ) {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 = P n X\, = \,{{\textbf {P}}^n} . It is known that no additional conditions are needed to imply X = P n X\, = \,{{\textbf {P}}^n} in the cases n = 1 , 2 n\, = \,1,\,2 . In this paper we prove that if n = 3 n\, = \,3 , X = P 3 X\, = \,{{\textbf {P}}^3} .