Any complete non-compact Kähler surface with positive sectional curvature is biholomorphic to ℂ².
Milnor,Singular points of complex hypersurfaces,Annals of Mathematics Stud- ies,61, Princeton University Press, Princeton, N.J (1968)
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Uniformisation of complete K\"ahler surfaces with positive sectional curvature
Any complete non-compact Kähler surface with positive sectional curvature is biholomorphic to ℂ².