Prescribing the Preschwarzian in several complex variables

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We solve the several complex variables preSchwarzian operator equation [Df(z)]⁻¹D²f(z)=A(z), z∈ ⁿ, where A(z) is a bilinear operator and f is a ⁿ valued locally biholomorphic function on a domain in ⁿ. Then one can define a several variables f→ fα transform via the operator equation [Dfα(z)]⁻¹D²fα(z)=α[Df(z)]⁻¹D²f(z), and thereby, study properties of fα. This is a natural generalization of the one variable operator fα(z) in DSS66 and the study of its univalence properties, e.g., the work of Royster Ro65 and many others. Möbius invariance and the multivariables Schwarzian derivative operator of T. Oda O play a central role in this work.