3

I am really struggling with the notion of complex vector field on a $2n$-dimensional smooth manifold and I am hoping to work out a down-to-earth example. I am very confused so the questions might be a bit wacky.

Let $X=\mathbb{R}^2$ with global coordinates $(x_1, x_2)$.

What does a complex vector field on $\mathbb{R}^2$ look like?

For example, let $z=x_1 + i x_2$, then,

(1) Is a complex vector field on $\mathbb{R}^2$ generated by $\partial_z$ OR $\{\partial_z, \partial_{\overline{z}} \}$?

(2) Are the coefficients on $\partial_z$ (or, possibly, $\{\partial_z, \partial_{\overline{z}} \}$) smooth functions in $(x_1,x_2)$ of holomorphic functions in $z$, or just some function in $(z, \overline{z})$?

user7090
  • 5,453
  • 1
  • 22
  • 53
  • By complex vector field do you mean a smooth section of the complexified tangent bundle, a smooth section of the $(1, 0)$ part of the complexified tangent bundle, something else...? – Andrew D. Hwang Apr 11 '21 at 14:53
  • 1
    It probably means a smooth section of the $(1,0)$ part of the complexified tangent bundle. – user7090 Apr 11 '21 at 14:54

1 Answers1

4

$\newcommand{\Reals}{\mathbf{R}}\newcommand{\dd}{\partial}$I'll write $x_{1} = x$ and $x_{2} = y$ (so $z = x + iy$), since this tends to simplify notation in higher complex dimension.

Over the smooth, complex-valued functions, the complexified tangent bundle of $\Reals^{2}$ is spanned by $$ \frac{\dd}{\dd x},\quad \frac{\dd}{\dd y},\quad i \frac{\dd}{\dd x},\quad i \frac{\dd}{\dd y}. $$ The $(1, 0)$ and $(0, 1)$ parts are respectively spanned by $$ \frac{\dd}{\dd z} = \frac{1}{2}\biggl[\frac{\dd}{\dd x} - i \frac{\dd}{\dd y}\biggr],\qquad \frac{\dd}{\dd \bar{z}} = \frac{1}{2}\biggl[\frac{\dd}{\dd x} + i \frac{\dd}{\dd y}\biggr]. $$ A complex vector field, i.e., smooth section of the $(1, 0)$ part, may therefore be written $$ 2f(x, y)\dfrac{\dd}{\dd z} = f(x, y) \biggl[\frac{\dd}{\dd x} - i \frac{\dd}{\dd y}\biggr] $$ for some smooth, complex-valued function $f$.

Andrew D. Hwang
  • 78,195
  • 1
    Smooth sections of them are, yes. (E.g., a holomorphic vector field would be a holomorphic function times $\partial/\partial z$.) – Andrew D. Hwang Apr 11 '21 at 15:24