All AdS4 and AdS3 supergravity backgrounds with N > 16 supersymmetries: uniqueness and (non-)existence

We investigate all warped AdS₄×M^D-4 and AdS₃×M^D-3 backgrounds with the most general allowed fluxes that preserve more than 16 supersymmetries in D=10- and 11-dimensional supergravities. Assuming either that the internal manifold is compact without boundary or that the isometry algebra of the background decomposes into that of AdS₄/AdS₃ and that of the transverse space, we find that there are no AdS₄ backgrounds in IIB supergravity. Similarly, in IIA supergravity, we find a unique such background with 24 supersymmetries, locally isometric to AdS₄×CP³, and in D=11 all such backgrounds are locally isometric to the maximally supersymmetric AdS₄×S⁷ solution. Finally, we establish a non-existence theorem for AdS₃ solutions preserving strictly more than 16 supersymmetries.