  We study the general structure of the AdS_5/CFT_4 correspondence in type IIB
string theory from the perspective of generalized geometry. We begin by
defining a notion of "generalized Sasakian geometry," which consists of a
contact structure together with a differential system for three symplectic
forms on the four-dimensional transverse space to the Reeb vector field. A
generalized Sasakian manifold which satisfies an additional "Einstein"
condition provides a general supersymmetric AdS_5 solution of type IIB
supergravity with fluxes. We then show that the supergravity action restricted
to a space of generalized Sasakian structures is simply the contact volume, and
that its minimization determines the Reeb vector field for such a solution. We
conjecture that this contact volume is equal to the inverse of the trial
central charge whose maximization determines the R-symmetry of any
four-dimensional N=1 superconformal field theory. This variational procedure
allows us to compute the contact volumes for a predicted infinite family of
solutions, and we find perfect agreement with the central charges and R-charges
of BPS operators in the dual mass-deformed generalized conifold theories.
