Holomorphic 1-forms are closed

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Theorem

Holomorphic 1-forms are closed.

Proof

Let $\omega$ be a holomorphic 1-form.

By wedge product of differential forms is antisymmetric

$\rd z \wedge\rd z=0$

By Cauchy-Riemann Equations

$\frac{\partial f}{\partial \bar{z}}=0$

Then:

$

\rd \omega=\rd(f(z)\rd z)=\frac{\partial f}{\partial z}\rd z \wedge\rd z+\frac{\partial f}{\partial \bar{z}}\rd \bar{z} \wedge\rd z=0+0=0 $ so $\omega$ is closed.

$\blacksquare$

Also see

• Results about $1$-forms can be found here.

Source

• Ben Dribus: Abel’s Theorem () 2. Basic Notions