What is the coordinate definition of holomorphic vector fields? We can write a holomorphic vector field in local co-ordinates as $X=X^i\dfrac{\partial}{\partial z_i}$, where $\dfrac{\partial}{\partial z_i}$ forms a local frame for $T^{(1,0)}M$. My questions are


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*Does $X^i$ have to be holomorphic functions?

*Can we say that $\dfrac{\partial X^i}{\partial \bar{z}^j}=0$ for all $i,j$?

*If 2 is not true, then what is the right co-ordinate condition on $X^i$s?

*

*is right due to the definition of the holomorphic vector field.

*is also right.

