  We associate to the resolved conifold an affine special K\"{a}hler (ASK)
manifold of complex dimension 1, and an instanton corrected hyperk\"{a}hler
(HK) manifold of complex dimension 2. We describe these geometries explicitly,
and show that the instanton corrected HK geometry realizes an Ooguri-Vafa-like
smoothing of the semi-flat HK metric associated to the ASK geometry. On the
other hand, the instanton corrected HK geometry associated to the resolved
conifold can be described in terms of a twistor family of two holomorphic
Darboux coordinates. We study a certain conformal limit of the twistor
coordinates, and conjecture a relation to a solution of a Riemann-Hilbert
problem previously considered by T. Bridgeland.
