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on an elliptic curve C, the line bundle L dual to the tautological line bundle of the projectivized bundle M = P(E) does not admit any smooth Hermitian metric with nonnegative curvature. In fact, they showed that any singular Hermitian metric on L with nonnegative curvature must have logarithmic singularity, so the metric cannot even be continuous. Clearly, L is a nef line bundle on the ruled surface M , since E is nef on C.

A Degenerate Monge–Ampère equation and the Boundary Classes of Kähler Cones