The Dolbeault resolution of the sheaf of holomorphic vector fields $Lie$ on a
complex manifold $M$ relates $Lie$ to a sheaf of differential graded Lie
algebras, known as the Fr\"olicher-Nijenhuis algebra $g$. We establish -
following B. L. Feigin - an isomorphism between the differential graded
cohomology of the space of global sections of $g$ and the hypercohomology of
the sheaf of continuous cochain complexes of $Lie$. We calculate this
cohomology up to the singular cohomology of some mapping space. We use and
generalize results of N. Kawazumi on complex Gelfand-Fuks cohomology.
Applications are - again following B. L. Feigin - in conformal field theory,
and in the theory of deformations of complex structures.
In an erratum to this paper, we admit that the sheaf of continuous cochains
of a sheaf of vector fields with values in the ground fields does not make much
sense. The most important cochains (like evaluations in a point or integrations
over the manifold) do not come from sheaf homomorphisms. The main result of the
above article (theorem 7) remains true.