New Constraints on Classical de Sitter: Flirting with the Swampland

We study the existence and stability of classical de Sitter solutions of type II supergravities with parallel Dp-branes and orientifold Op-planes. Together with the dilaton and volume scalar fields, we consider a third one that distinguishes between parallel and transverse directions to the Dp/Op. We derive the complete scalar potential for these three fields. This formalism allows us to reproduce known constraints obtained in 10d, and to derive new ones. Specifying to group manifolds with constant fluxes, we exclude a large region of parameter space, forbidding de Sitter solutions on nilmanifolds, semi-simple group manifolds, and some solvmanifolds (at least in some standard algebra basis). In the small remaining region, we identify a stability island, where the three scalars could be stabilized in any de Sitter solution. We discuss these results in the swampland context.