  Cohesive modules give a dg-enhancement of the bounded derived category of
coherent sheaves on a complex manifold via superconnections. In this paper we
discuss the deformation theory of cohesive modules on compact complex
manifolds. This generalizes the deformation theory of holomorphic vector
bundles and coherent sheaves. We also develop the theory of Kuranishi maps and
obstructions of deformations of cohesive modules and give some examples of
unobstructed deformations.
