Global pluripotential theory on hybrid spaces

Let (X,L) be a polarized scheme over Banach ring A. We define and study class PSH(X,L) of plurisubharmonic metrics on the Berkovich analytification X an . focus mainly case where A is hybrid power series, so that space associated to degeneration complex manifolds X. then prove any metric with logarithmic growth at zero admits canonical extension hyb also discuss continuity family Monge-Ampère measures continuous metric.