Differential operators associated with holomorphic mappings

Two differential operators which act on holomorphic mappings to complex projective space are studied. One operator is of second order and characterizes projective linear mappings. The other operator is of third order and may be viewed as a curvature. The two operators together play a role analogous to the Schwarzian derivative.A canonical approximation to a holomorphic mapping is defined, and a relationship between the approximation and the operators is derived. In the one variable case, this reduces to a classical result relating the Schwarzian derivative and the best Möbius approximation to a holomorphic function.