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We study supersymmetric AdS$_D$ backgrounds of eleven-dimensional or type IIsupergravity preserving $\mathcal{N}$ supersymmetries using generalisedgeometry. We show that a large class correspond precisely to spaces admitting ageneralised $G_{D,\mathcal{N}}$ structure with a weak integrability condition,which we call "weak generalised holonomy". This class contains all suchbackgrounds with odd $D$, but for even $D$ we must impose a technicalassumption concerning the inner products of Killing spinors. It is known thatall compact smooth solutions are included, though our analysis is purely local.We show that the Killing superalgebras of such backgrounds are isomorphic tothe standard $\mathcal{N}$-extended AdS$_D$ superalgebras, and give a geometricproof of the existence of isometries of the internal spaces which realise theAdS R-symmetry, as often assumed in explicit studies of such solutions.

Supersymmetric AdS backgrounds and weak generalised holonomy