c-map, very special quaternionic geometry and dual Kaehler spaces

We show that for all very special quaternionic manifolds a different N=1 reduction exists, defining a Kaehler geometry which is 'dual' to the original very special Kaehler geometry with metric Gab-bar=-∂a∂blnV (V=(1/6)dabcλaλbλc). The dual metric gab=V-2(G-1)ab is Kaehler and it also defines a flat potential as the original metric. Such geometries and some of their extensions find applications in type IIB compactifications on Calabi-Yau orientifolds