ScalingStacks

Definition 4.17 . [05B0]

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Definition 4.17.

Let KK be a complete, non-archimedean, non-trivially valued field, VV a strictly KK-analytic space and LL a line bundle on VV. A continuous metric ∥⋅∥\|\cdot\| on LL is called locally semipositive if for any x∈Vx\in V there is an open neighbourhood UU of xx such that ∥⋅∥|U\|\cdot\|\Big|_{U} is a uniform limit of semipositive piecewise ℚ\mathbb{Q}-linear metrics on L|UL\Big|_{U}. It is called locally potentially semipositive if its base change to the completion of an algebraic closure of KK is locally semipositive. If VV is an open subset of XanX^{\textup{an}} for a separated scheme XX of finite type over KK then using the Remarks 4.14 and 4.16 we define the Monge-Ampère measure c1​(L¯1)∧…∧c1​(L¯n)c_{1}(\overline{L}_{1})\wedge...\wedge c_{1}(\overline{L}_{n}) for locally potentially semipositive metrized line bundles L¯1,…,L¯n\overline{L}_{1},...,\overline{L}_{n} on VV.

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