We determine the necessary and sufficient conditions for warped product $dS_n$ solutions, $5 \leq n \leq 10$, to preserve supersymmetry in $D=11$ supergravity, without assuming factorization of the Killing spinors. We prove that for $7 \leq n \leq 10$, all such solutions are flat, with vanishing 4-form. We also show that the only warped product $dS_6$ solutions are either the maximally supersymmetric $AdS_7 \times S^4$ solution, or $\mathbb{R}^{1,6} \times N_4$ where $N_4$ is hyperK\"ahler, with vanishing 4-form. Supersymmetric warped product $dS_5$ 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 hyperK\"ahler manifold. If the 4-form is covariantly constant, then $N_4$ admits a hyperK\"ahler potential.

We consider black holes in five-dimensional N = 2 U(1)-gauged supergravity coupled to vector multiplets, with horizons that are homogeneous but not isotropic. We write down the equations of motion for electric and magnetic ansätze, and solve them explicitely for the case of pure gauged supergravity with magnetic U(1) field strength and Sol horizon. The thermodynamics of the resulting solution, which exhibits anisotropic scaling, is discussed. If the horizon is compactified, the geometry approaches asymptotically a torus bundle over AdS 3. Furthermore, we prove a no-go theorem that states the nonexistence of supersymmetric, static, Sol-invariant, electrically or magnetically charged solutions with spatial cross-sections modelled on solvegeometry. Finally, we study the attractor mechanism for extremal static non-BPS black holes with nil-or solvegeometry horizons. It turns out that there are no such attractors for purely electric field strengths, while in the magnetic case there are attractor geometries, where the values of the scalar fields on the horizon are computed by extremization of an effective potential V eff , which contains the charges as well as the scalar potential of the gauged supergravity theory. The entropy density of the extremal black hole is then given by the value of V eff in the extremum.

Extreme near-horizon geometries in D = 11 supergravity preserving four supersymmetries are classified. It is shown that the Killing spinors fall into three possible orbits, corresponding to pairs of spinors defined on the spatial cross-sections of the horizon which have isotropy groups SU(3), G2, or SU(4). In each case, the conditions on the geometry and the 4-form flux are determined. The integrability conditions obtained from the Killing spinor equations are also investigated.

We classify all warped dS5 backgrounds in D=11 supergravity with enhanced supersymmetry. We show that backgrounds preserving N=16 supersymmetries consist of either a stack of M5 branes with transverse space R5, or a generalized M5-brane configuration with transverse space R × N4, where N4 is a hyper-Kähler manifold and the M5-brane harmonic function is determined by a hyper-Kähler potential on N4. Moreover, we find that there are no backgrounds preserving exactly N=24 supersymmetries. Backgrounds preserving N=32 supersymmetries correspond to either R1,10 or AdS7 × S4