In this paper we provide a definition of the D-map, namely of the mathematical construction implicitly utilized by supergravity that associates an axial symmetric Kähler surface to every positive definite potential function V (ϕ). The properties of the D-map are discussed in general. Then the D-map is applied to the list of integrable cosmological potentials classified by us in a previous publication with A. Sagnotti. Several interesting geometrical and analytical properties of the manifolds in the image of this D-map are discovered and illustrated. As a by-product of our analysis we demonstrate the existence of (integrable) Starobinsky–like potentials that can be embedded into supergravity. Some of them follow from constant curvature Kähler manifolds. In the quest for a microscopic interpretation of inflaton dynamics we present the Ariadne's thread provided by a new mathematical concept that we introduce under the name of axial symmetric descendants of one dimensional special Kähler manifolds. By means of this token we define a clearcut algorithm that to each potential function V (φ) associates a unique 4th order Picard-Fuchs equation of restricted type. Such an equation encodes information on the chiral ring of a superconformal field theory to be sought for, unveiling in this way a microscopic interpretation of the inflaton potential. We conjecture that the physical mechanism at the basis of the transition from a special manifold to its axial symmetric descendant is probably the construction of an Open String descendant of a Closed String model.