  We consider nilmanifolds with left-invariant complex structure and prove that
small deformations of such structures are again left invariant if the
Dolbeault-cohomology of the nilmanifold can be calculated using left-invariant
forms. By a result of Console and Fino this is generically the case. Our main
tool is an analog of Dolbeault-cohomology for Lie-algebras with complex
structure.
