Kodaira-Spencer deformation of complex structures and Lagrangian field theory

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).