Supersymmetric $d{S}_n$ solutions for $n\ge 5$ in $D=11$ supergravity

We determine the necessary and sufficient conditions for warped product $d{S}_n$ solutions, $5\le n\le 10$, to preserve supersymmetry in $D=11$ supergravity, without assuming factorization of the Killing spinors. We prove that for $7\le n\le 10$, all such solutions are flat, with vanishing 4-form. We also show that the only warped product $d{S}_6$ solutions are either the maximally supersymmetric $Ad{S}_7\times S^4$ solution, or $$\mathbb{R}$^{1,6} \times N_4$where$N_4$ is hyperKahler, with vanishing 4-form. Supersymmetric warped product $d{S}_5$ solutions are then classified; it is shown that all such solutions are generalized M5-brane configurations, for which the transverse space is $\mathbb{R}\times N_4$, and $N_4$ is a hyperKahler manifold. If the 4-form is covariantly constant, then $N_4$ admits a hyperKahler potential.