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

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?

score 1 · Accepted Answer · answered Jun 02 '20 at 02:56

1

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

answered Jun 02 '20 at 02:56

Invariance

1,700

What exactly is the standard definition of holomorphic vector fields? – Mathgrad Jun 02 '20 at 03:07
You can see this: https://math.stackexchange.com/questions/198410/what-is-a-holomorphic-vector-field?rq=1 – Invariance Jun 02 '20 at 03:20
Okay, thank you. – Mathgrad Jun 02 '20 at 03:27
Sure thing. Just please remember to accept the answer when you are satisfied that you accept it. :) – Invariance Jun 02 '20 at 03:32
No problem. This is good for me. – Mathgrad Jun 02 '20 at 03:52

What is the coordinate definition of holomorphic vector fields?

1 Answers1