A singular Hermitian metric on a holomorphic line bundle assigns, in a local frame, a squared norm
 |
where the weight 
is locally integrable
and may be singular. It is semipositive when its curvature current 
is nonnegative,
equivalently when these local weights are plurisubharmonic. A semipositive metric
has minimal singularities if each other semipositive weight 
on the same line bundle satisfies
for a constant 
.
For rational line bundles these notions are defined
by clearing denominators and rescaling the weights.
A metric with locally bounded weights has bounds both above and below on relatively compact coordinate neighborhoods; its weights need not be continuous.
