Abstract
Let X be a smooth projective toric scheme over Spec(Z). Let D¯ be an equivariant line bundle on X, endowed with a continuous Hermitian metric which is invariant by the action of the compact torus of X(C). We show that its arithmetic volume is given in terms of the Legendre–Fenchel transform associated to the metric. When D¯ is supposed admissible, we characterize when it is arithmetically ample, nef, or big in terms of combinatorial date.
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