complex differential geometry

I recently asked a question in my mathematical physics class: In complex manifolds, why don't we combine the real part of one variable with the imaginary part of the other variable to obtain more general CR-equations? My lectures mentioned something about conformal manifolds. But I didn't understand quite what he was talking about. Could you please refer me to literature which deals with the question that I put? I haven't read any literature on Functions of Several Complex Variables, but I could imagine that they the CR equations could look something like this: $$\frac{\partial u}{\partial x_i}=\frac{\partial v}{\partial y_j}$$ $$\frac{\partial u}{\partial y_i}=-\frac{\partial v}{\partial x_j}$$ where i and j are natural numbers and run from 1 to dimension of the complex space.

thank you

• What exactly are you proposing (write the equations you have in mind)? – Moishe Kohan Apr 29 '17 at 10:17
• @MoisheCohen I edited the question. – eeqesri Apr 30 '17 at 7:57
• Your equation is more or less the general CR equation (where you should have $i=j$ and $z_i = x_i + \sqrt{-1} y_i$ are the complex coordinates). – user99914 Apr 30 '17 at 8:01
• I know that usually i=j. but the whole point is that why should we consider only the case where i=j? Why not combine the real and imaginary parts of different complex variables? – eeqesri Apr 30 '17 at 16:33
• It is not exactly clear what you are proposing: Are you asking what happens if you have these equations for all $i, j$ or only for some $i, j$? If you mean, for all $i, j$ then your function simply becomes a holomorphic function of the form $f(z, z + a_2, z+ a_3,..., z+ a_n)$ for some complex numbers $a_2,..., a_n$. – Moishe Kohan May 1 '17 at 2:35