Supersymmetry of AdS and flat IIB backgrounds

We present a systematic description of all warped $AdS_n____times_w M^{10-n}$ and ${____mathbb{R}}^{n-1,1}____times_w M^{10-n}$ 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 $N=2^{[{n____over2}]} k$ for $n____leq 4$ and $N=2^{[{n____over2}]+1} k$ for $4<n____leq 6$ supersymmetries and for $k____in {____mathbb{N}}_{+}$ 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 $M^{10-n}$ establishing a new class of Lichnerowicz type theorems. Furthermore, we adapt some of these results to ${____mathbb{R}}^{n-1,1}____times_w M^{10-n}$ backgrounds with fluxes by taking the AdS radius to infinity. We find that these backgrounds preserve $N=2^{[{n____over2}]} k$ for $2<n____leq 4$ and $N=2^{[{n+1____over2}]} k$ for $4