The following are equivalent for a metric ${\|\hskip 4.30554pt\|}$ on the line bundle $L^{\rm an}$ over a proper scheme $X$:

• (a)

  ${\|\hskip 4.30554pt\|}$ is an algebraic metric;

• (b)

  ${\|\hskip 4.30554pt\|}$ is a formal metric;

• (c)

  ${\|\hskip 4.30554pt\|}$ is a piecewise linear metric.

The equivalence remains true if we replace “metric” by “semipositive metric” in every item.