  The category of exploded manifolds is an extension of the category of smooth
manifolds related to tropical geometry in which some adiabatic limits appear as
smooth families. This paper studies the dbar equation on variations of a given
family of curves in an exploded manifold. Roughly, we prove that the dbar
equation on variations of an exploded family of curves behaves as nicely as the
dbar equation on variations of a smooth family of smooth curves, even though
exploded families of curves allow the development of normal crossing or log
smooth singularities. The resulting regularity results are used in a series of
separate papers to construct Gromov Witten invariants for exploded manifolds.
