We study the space of supersymmetric
AdS
5
solutions of type IIB supergravity corresponding to the conformal manifold of the dual
N
=
1
conformal field theory. We show that the background geometry naturally encodes a generalized holomorphic structure, dual to the superpotential of the field theory, with the existence of the full solution following from a continuity argument. In particular, this work allows us to address the long-standing problem of finding the gravity dual of the generic
N
=
1
deformations of
N
=
4
conformal field theory: even if we are not able to give it in a fully explicit form, we provide a proof-of-existence of the supergravity solution. Using this formalism, we derive a new result for the Hilbert series of the deformed field theories.