On Moduli Spaces in $\bf{AdS_4}$ Supergravity

We study the structure of the supersymmetric moduli spaces of ${\cal N}=1$ and ${\cal N}=2$ supergravity theories in $AdS_4$ backgrounds. In the ${\cal N}=1$ case, the moduli space cannot be a complex submanifold of the K\"ahler field space, but is instead real with respect to the inherited complex structure. In ${\cal N}=2$ supergravity the same result holds for the vector multiplet moduli space, while the hypermultiplet moduli space is a K\"ahler submanifold of the quaternionic-K\"ahler field space. These findings are in agreement with AdS/CFT considerations.