Let M=G/Γ be a compact nilmanifold endowed with an invariant complex structure. We prove that, on an open set of any connected component of the moduli space C(g) of invariant complex structures on M, the Dolbeault cohomology of M is isomorphic to the one of the differential bigraded algebra associated to the complexification
Nearby in the stack
^
of the Lie algebra of
G
. To obtain this result, we first prove the above isomorphism for compact nilmanifolds endowed with a rational invariant complex structure. This is done using a descending series associated to the complex structure and the Borel spectral sequences for the corresponding set of holomorphic fibrations. Then we apply the theory of Kodaira-Spencer for deformations of complex structures.