Any complete non-compact Kähler surface with positive sectional curvature is biholomorphic to ℂ².
Mok:An embedding theorem of complete Ka ¨hler manifolds of positive Ricci cur- vature onto quasi–projective varieties,Math
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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.