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Surface charges of a $p$-form theory on the boundary of an AdS$_{d+1}$spacetime are computed. Counter-terms on the boundary produce divergentcorner-terms which holographically renormalize the symplectic form. Differentchoices of boundary conditions lead to various expressions for the charges andthe associated fluxes. With the usual standard AdS boundary conditions, thereare conserved zero-mode charges. Moreover, we explore two leaky boundaryconditions which admit an infinite number of charges forming an Abelian algebraand non-vanishing flux. Finally, we discuss magnetic $p$-form charges andelectric/magnetic duality.

$p$-Form Surface Charges on AdS: Renormalization and Conservation