Any complete non-compact Kähler surface with positive sectional curvature is biholomorphic to ℂ².
Demailly:A numerical criterion for very ample line bundles,J.Diff.Geom.,37 (1993), no
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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.