AdS5 backgrounds with 24 supersymmetries

We prove a non-existence theorem for smooth AdS(5) solutions with connected, compact without boundary internal space that preserve strictly 24 supersymmetries. In particular, we show that D = 11 supergravity does not admit such solutions, and that all such solutions of IIB supergravity are locally isometric to the AdS(5) x S-5 maximally supersymmetric background. Furthermore, we prove that (massive) IIA supergravity also does not admit such solutions, provided that the homogeneity conjecture for massive IIA supergravity is valid. In the context of AdS/CFT these results imply that if gravitational duals for strictly N = 3 superconformal theories in 4-dimensions exist, they are either singular or their internal spaces are not compact.