Singular semipositive metrics in non-Archimedean geometry

Let X X be a smooth projective Berkovich space over a complete discrete valuation field K K of residue characteristic zero, endowed with an ample line bundle L L . We introduce a general notion of (possibly singular) semipositive (or plurisubharmonic) metrics on L L and prove the analogue of the following two basic results in the complex case: the set of semipositive metrics is compact modulo scaling, and each semipositive metric is a decreasing limit of smooth semipositive ones. In particular, for continuous metrics, our definition agrees with the one by S.-W. Zhang. The proofs use multiplier ideals and the construction of suitable models of X X over the valuation ring of K K , using toroidal techniques.