  We give a construction of NC-smooth thickenings (a notion defined by Kapranov
in math/9802041) of a smooth variety equipped with a torsion free connection.
We show that a twisted version of this construction realizes all NC-smooth
thickenings as 0th cohomology of a differential graded sheaf of algebras,
similarly to Fedosov's construction in \cite{Fed}. We use this dg resolution to
construct and study sheaves on NC-smooth thickenings. In particular, we
construct an NC version of the Fourier-Mukai transform from coherent sheaves on
a (commutative) curve to perfect complexes on the canonical NC-smooth
thickening of its Jacobian. We also define and study analytic NC-manifolds. We
prove NC-versions of some of GAGA theorems, and give a $C^\infty$-construction
of analytic NC-thickenings that can be used in particular for Kahler manifolds
with constant holomorphic sectional curvature. Finally, we describe an analytic
NC-thickening of the Poincare line bundle for the Jacobian of a curve, and the
corresponding Fourier-Mukai functor, in terms of A-infinity structures.
