An example of a function not invariant under complex conjugation

Question

Let $X$ be a complex manifold. The complex conjugation induces naturally an anti-holomorphic function $\sigma: X\to X$. A $C^{\infty}$-function $f:U\to \mathbb C $ is said invariant under conjugation if $f=f\circ\sigma$.

what is an example of function $f$ which is not invariant under conjugation?

This is a rather basic question, but I have a conceptual problem: $\sigma$ as function between sets is just the identity, we just change the charts by composing with the conjugation. So I don't understand how the equality $f=f\circ\sigma$ can fail.

Answer 1

The most obvious example: Let $X=\mathbb{C}$, then $\sigma$ is just the usual complex conjugation. The identity function $\mathbb{C}\to\mathbb{C}$ is not invariant under conjugation, because $\operatorname{id}\circ\sigma=\sigma\neq\operatorname{id}$. Similarly the "imaginary part" is not invariant (it gains a minus sign).

On the other hand, the "real part" is invariant under $\sigma$.