We explore all warped AdS 4 × w M D−4 backgrounds with the most general allowed fluxes that preserve more than 16 supersymmetries in D = 10-and 11-dimensional supergravities. After imposing the assumption that either the internal space M D−4 is compact without boundary or the isometry algebra of the background decomposes into that of AdS 4 and that of M D−4 , we find that there are no such backgrounds in IIB supergravity. Similarly in IIA supergravity, there is a unique such background with 24 supersymmetries locally isometric to AdS 4 × CP 3 , and in D = 11 supergravity all such backgrounds are locally isometric to the maximally supersymmetric AdS 4 × S 7 solution.

We show that there are no smooth warped AdS 3 solutions in 10-and 11-dimensional supergravities which preserve strictly more than 16 supersymmetries and have internal space a compact without boundary manifold.

We demonstrate that warped Minkowski space backgrounds, R n−1,1 × w M d−n , n ≥ 3, that preserve strictly more than 16 supersymmetries in d = 11 and type II d = 10 supergravities and with fields which may not be smooth everywhere are locally isometric to the R d−1,1 Minkowski vacuum. In particular, all such flux compactification vacua of these theories have the same local geometry as the maximally supersymmetric vacuum R n−1,1 × T d−n .