Computation involving complex differential forms

Question

I am reading this lecture note on complex geometry and I'm stuck at one computation (seemingly basic) involving complex differential forms.

Suppose $X$ is a complex surface and $\omega$ is a holomorphic (1,0)-form, i.e. $\omega$ is killed by the operator $\overline{\partial}$. Let $\overline{\omega}$ be the corresponding (0,1) conjugate form. The author claims that

\begin{equation*}
d(\omega \wedge \overline{\omega}) = d\omega \wedge d \overline{\omega}
\end{equation*}

Now since $\partial \omega = \overline{\overline{\partial} \overline{\omega}}$, the right hand side is nothing but $\partial{\omega} \wedge \overline{\partial} \overline{\omega}$. But I cannot see how the left hand side can be written in the same expression (using the usual rule for exterior derivatives).

Any insight will be appreciated.

Answer 1

The LHS $d(\omega \wedge \overline\omega)$ is a three form while the RHS $d\omega \wedge d\overline\omega$ is a four form. They are not the same.

Looking at the note, they wrote

Now by Stokes Theorem $\int d\omega \wedge d\overline\omega = 0$ (because $ d(\omega \wedge \overline{\omega}) = d\omega \wedge d \overline{\omega} $).

I believe it is just a typo and they probably mean $$d(\omega \wedge d\overline{\omega}) = d\omega \wedge d \overline{\omega}.$$