Burgos Gil, José I.Sombra, Martín2020-07-142020-07-142019-10-010373-0956https://hdl.handle.net/2445/168550Let L be an ample line bundle on a smooth projective variety $X$ over a non-archimedean field $K$. For a continuous metric on $L^{\text {an }},$ we show In the following two cases that the semipositive envelope is a continuous semipositive metric on $L^{\text {an }}$ and that the non-archimedean Monge-Ampère equation has a solution. First, we prove it for curves using results of Thuillier. Second, we show it under the assumption that $X$ is a surface defined geometrically over the function field of a curve over a perfect field $k$ of positive characteristic. The second case holds in higher dimensions if we assume resolution of singularities over $k .$ The proof follows a strategy from Boucksom, Favre and Jonsson, replacing multiplier ideals by test ideals. Finally, the appendix by Burgos and Sombra provides an example of a semipositive metric whose retraction is not semipositive. The example is based on the construction of a toric variety which has two SNC-models which induce the same skeleton but different retraction maps.9 p.application/pdfeng(c) Association des Annales de l'Institut Fourier, 2019Funcions de diverses variables complexesÀlgebra commutativaGeometria algebraicaFunctions of several complex variablesCommutative algebraAlgebraic geometryinfo:eu-repo/semantics/article7026702020-07-14info:eu-repo/semantics/openAccess