. Noncommutative coordinates on Riemann surfaces

The non commutative coordinates arise from the composition of local symplectic diffeomorphism algebra quantum anomalies compensations We show that the definition of a projective coordinate frame within a Laguerre-Forsyth scheme, leads to the extension of the factorized diffeomorphism algebra. The quantum improvement of this symmetry can be performed only if these coordinates switch, at the quantum level, into a non-commutative regime.