Supersymmetry of AdS and flat 11B backgrounds

We present a systematic description of all warped AdS(n) x(w) M10-n and R-n-1,R-1 x(w) M10-n IIB backgrounds and identify the a priori number of supersymmetries N preserved by these solutions. In particular, we find that the AdS(n), backgrounds preserve N = 2([n\2]k) for n <= 4 and N = 2([n\2]+) (1)k for 4 < n <= 6 supersymmetries and for k ENF suitably restricted. In addition under some assumptions required for the applicability of the maximum principle, we demonstrate that the Killing spinors of AdS(n) backgrounds can be identified with the zero modes of Dirac-like operators on M10-n establishing a new class of Lichnerowicz type theorems. Furthermore, we adapt some of these results to R-n-1,R-1 x(w) M10-n backgrounds with fluxes by taking the AdS radius to infinity. We find that these backgrounds preserve N = 2([n\2]k) for 2 < n <= 4 and N = 2([n+1/2])k for 4 < n <= 7 supersymmetries. We also demonstrate that the Killing spinors of AdS(n) x(w) M10-n do not factorize into Killing spinors on AdS, and Killing spinors on