  In complete analogy with the Beltrami parametrization of complex structures
on a (compact) Riemann surface, we use in this paper the Kodaira-Spencer
deformation theory of complex structures on a (compact) complex manifold of
higher dimension. According to the Newlander-Nirenberg theorem, a smooth change
of local complex coordinates can be implemented with respect to an integrable
complex structure parametrized by a Beltrami differential. The question of
constructing a local field theory on a complex compact manifold is addressed
and the action of smooth diffeomorphisms is studied in the BRS algebraic
approach. The BRS cohomology for the diffeomorphisms gives rise to generalized
Gel'fand-Fuchs cocycles provided that the Kodaira-Spencer integrability
condition is satisfied. The diffeomorphism anomaly is computed and turns out to
be holomorphically split as in the bidimensional Lagrangian conformal models.
Moreover, its algebraic structure is much more complicated than the one
proposed in a quite recent paper hep-th/9606082 (Nucl. Phys. B484 (1997) 196).
