What is the coordinate definition of holomorphic vector fields?

Question

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

1. Does $X^i$ have to be holomorphic functions?

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

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

Answer 1

1. is right due to the definition of the holomorphic vector field.
2. is also right.