  We study the 1-form diffeomorphism cohomologies within a local conformal
Lagrangian Field Theory model built on a two dimensional Riemann surface with
no boundary. We consider the case of scalar matter fields and the complex
structure is parametrized by Beltrami differential. The analysis is first
performed at the Classical level, and then we improve the quantum extension,
introducing the current in the Lagrangian dynamics, coupled to external source
fields. We show that the anomalies which spoil the current conservations take
origin from the holomorphy region of the external fields, and only the
differential spin 1 and 2 currents (as well their c.c) could be anomalous.
