Uniform weak RC-positivity of TX on a compact Kähler manifold X implies X is projective and rationally connected; the same condition on any holomorphic vector bundle E yields a Hermitian metric with positive mean curvature.
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Uniform weak RC-positivity and rational connectedness
Uniform weak RC-positivity of TX on a compact Kähler manifold X implies X is projective and rationally connected; the same condition on any holomorphic vector bundle E yields a Hermitian metric with positive mean curvature.