Interplay between the cosmological constant and "microscopic" properties of string vacua As far as I understand, string phenomenology is usually concerned with compactifications of string theory, M-theory or F-theory in which the uncompactified dimensions form a 4-dimensional Minkowski spacetime. However, we know our actual universe has a positive cosmological constant hence its asymptotics are that of a De Sitter spacetime. On the intuitive level it makes sense to me, since the microscopic physics should have little to do with spacetime asymptotics. However, from another point of view I see a problem.
It seems to me that a cosmological constant in the effective 4-dimensional field theory requires a non-vanishing Ricci tensor in the compactified dimensions. For example, the classical case study for anti-De Sitter string theory is AdS_4 x S_6. The compactified dimensions form the sphere, a manifold with positive curvature, compensating the negative curvature of AdS.
This non-vanishing Ricci tensor seems to require different topology from a vanishing Ricci tensor.  Hence all standard compactifications like Calabi-Yau manifolds, G2 manifolds etc. don't seem to be compatible with a non-vanishing cosmological constant.
What am I missing here?
 A: First of all, in the most recent decade, string phenomenology isn't talking about strictly Minkowski vacua. E.g. in the KKLT paper, you will see $AdS_4$ vacua uplifted to $dS_4$ by antibranes and no Minkowski space at any place in between.
The fact that a nonzero C.C. is generated for the large 3+1 dimensions doesn't mean that one can't find any shape of the hidden dimensions that exactly obey the equations of motion. Just like there exists a "tiny C.C." deformation of the flat Minkowski space, namely the $dS_4$ space with a small C.C. around us, there also exist solutions for the compact 6/7 dimensions that have a tiny (but nonzero) Ricci tensor proportional to the Ricci scalar. In the Calabi-Yau case, these deformed solutions will strictly no longer be $SU(3)$ holonomy manifolds; they will be $U(3)$ holonomy (Kähler) manifolds if we acknowledge that the Ricci curvature, while tiny, is nonzero.
In braneworld compactifications and compactifications on singular compactified manifolds, the energy density is typically concentrated at the loci of the branes or the singularities.
