$\partial_{z}$ and $\partial_{\bar{z}}$: what are these vector fields from a geometrical point of view?

Question

In complex analysis, we are taught that instead of coordinates $x$, $y$ on the complex plane, one can use $z$, $\bar{z}$, then, for instance, the Cauchy-Riemann conditions become $\frac{\partial }{\partial \bar{z}}f(z, \bar{z})=0$, and $\Delta = \frac{1}{4}\frac{\partial^2}{\partial z \partial \bar{z}}$. This was explained to me in the following way: we simply perform a change of coordinates $z = x + y$, $\bar{z}=x-y$ on $\mathbb{R}^2$. However, if this was the case, functions with $\frac{\partial }{\partial \bar{z}}=0$ would be constant along lines $y=-x+C$, which is clearly not true (for instance, this would mean isolated zeros or isolated singularities are impossible). In general, this means that $\partial_{\bar{z}}$ can't be understood as a directional derivative in the complex plane, i.e. as a vector field on the complex plane with real coefficients, because holomorphic functions would then be constant along the integral lines of this vector field. So, I guess, either the derivative or the change of coordinates should be understood in some other sense. Reading old MSE answers to similar questions, I understood that

a) This confusion about the meaning of $\partial_{\bar{z}}$ is very common among people studying complex analysis, including, apparently, those who answer questions about Wirtinger derivatives (because often the answers boil down to treating $\partial_{z}$ and $\partial_{\bar{z}}$ as real vector fields).

b) The correct answer (as opposed to hand-waving the issue) has something to do with the structure of complex manifold on $\mathbb{C}$.

So my question is: where can I find an elementary introduction to the theory of complex manifolds that would carefully explain how $\partial_{z}$ and $\partial_{\bar{z}}$ fields work?

Edit: Perhaps, I should explain that my naive understanding of complex manifolds leads to further confusions. For instance, $\mathbb{C}$ is a one-dimensional complex manifold. The experience teaches me it should have one linearly independent vector at each tangent space, not two, which raises more questions about what $\partial_{z}$ and $\partial_{\bar{z}}$ really are.

Answer 1

$\newcommand{\dd}{\partial}\newcommand{\Cpx}{\mathbf{C}}$tl; dr: Holomorphic vector fields don't literally have a geometric interpretation the way real vector fields do. Instead, they're complex-valued differential operators, or complex linear combinations of real vector fields (their real and imaginary parts).

That said, we can view a holomorphic vector field geometrically as twice its real part.

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Let's carefully examine the complex line equipped with its standard holomorphic coordinate $z$, its complex structure $J$ given by "multiplication by $i$ at each point," and its holomorphic coordinate vector vield $\dd_{z}$.

One might imagine a world where these are primitive notions, where we do calculus without decomposing a complex number into real and imaginary parts, where we're blissfully unaware of complex conjugation (!), where differentiable means holomorphic. In this world, the "tangent bundle" of $\Cpx$ may be identified with $\Cpx \times \Cpx$. A point $(z, a)$ in the Cartesian product is viewed as the vector $a\, \dd_{z}$ at the point $z$. A holomorphic function $f$ may be identified with a holomorphic vector field $\bigl(z, f(z)\bigr)$. A flow line of this field is a holomorphic mapping $\Phi$ satisfying $\Phi' = f \circ \Phi$. (Compare the situation for vector fields on the real line.)

In our world, by contrast, we're irresistably drawn to write a complex number $z$ in the form $x + iy$ with $x$ and $y$ real; to form a new independent complex variable $\bar{z} = x - iy$; to consider real vector fields such as $\dd_{x}$, $\dd_{y}$, and $$ X = X_{1}(x, y)\, \dd_{x} + X_{2}(x, y)\, \dd_{y}, $$ and interpret their flows as real paths; etc., etc. How are we supposed to make sense of holomorphic coordinates and vector fields in this real setting?

The standard answers boil down to:

1. We can pass from a holomorphic manifold of complex dimension $m$ to a real manifold of real dimension $n = 2m$ equipped with an extra complex structure $J$ that effects multiplication by $i$ at each point.
2. Conversely, given complex manifold—a smooth manifold $N$ of real dimension $n = 2m$ and a complex structure $J$—the real tangent bundle $TN$ acquires the structure of a smooth complex vector bundle of complex rank $m$. We extend $J$ to the complexified tangent bundle $TN \otimes \Cpx$ by $J(v \otimes \alpha) = J(v) \otimes \alpha$ for all tangent vectors $v$ and all complex $\alpha$. The complexified bundle then decomposes into complementary real subbundles, often denoted $T^{1,0}N$ and $T^{0,1}N$, on which the extension of $J$ acts as multiplication by $i$ and by $-i$. These are the well-known formulas \begin{align*} v^{1,0} &= \tfrac{1}{2}(v \otimes 1 - J(v) \otimes i), \\ v^{0,1} &= \tfrac{1}{2}(v \otimes 1 + J(v) \otimes i). \end{align*} (If, further, $J$ satisfies a differential equation—vanishing of the Nijenhuis tensor, we can assemble the $2m$ real coordinates into $m$ complex coordinates so that changes of coordinates are holomorphic with holomorphic inverse, and our complex manifold becomes a holomorphic manifold.)

Particularly, the real vector bundles $(T^{1,0}N, i)$ and $(TN, J)$ are isomorphic as complex vector bundles. This, at last, tells us "what a holomorphic vector (field) is geometrically" and explains the (technically sloppy) way people speak of "flow lines" of holomorphic vector fields: A "holomorphic vector" $v^{1,0}$, which from the real perspective is not a vector at all, corresponds to (or "is" if we're being sloppy) the honest real vector $v = v^{1,0} + v^{0,1}$.

Specifically, a holomorphic vector field $X$ "is" (ahem) twice its real part, $X + \bar{X}$. For example, \begin{align*} z\, \dd_{z} &= (x + iy)\, \tfrac{1}{2}(\dd_{x} - i\, \dd_{y}) \\ &= \tfrac{1}{2}\bigl[(x\, \dd_{x} + y\, \dd_{y}) - i(-y\, \dd_{x} + x\, \dd_{y})\bigr] \end{align*} is the Euler field $x\, \dd_{x} + y\, \dd_{y}$, while the rotation field $-y\, \dd_{x} + x\, \dd_{y}$ is $iz\, \dd_{z}$.