When is a $(p,q)$-form real?

Question

Let $M$ be a complex manifold, and let $\omega$ be a $(p,q)$-form. Then, in holomorphic coordinates $(z^1,\dots , z^n)$, $\omega$ can be expressed as $\omega = f_{U,V} dz^U \wedge d\bar{z}^V$, where $dz^U = dz^{u_1}\wedge\cdots\wedge dz^{u_p}$ and the same for $dz^V$. Are there any necessary and sufficient conditions on $f_{U,V}$ for $\omega$ to be real?

Answer 1

A complex form $\omega$ is real if $\overline{\omega} = \omega$. If $\omega$ is a $(p, q)$-form, then $\overline{\omega}$ is a $(q, p)$-form, so a necessary condition for a $(p, q)$-form to be real is that $p = q$. This condition is not sufficient however (consider the constant $(0, 0)$-form $i$).

Assuming $\omega$ is a $(p, p)$-form, and writing $\omega = f_{UV}dz^U\wedge d\bar{z}^V$, then

$$\overline{\omega} = \overline{f_{UV}} d\bar{z}^U\wedge dz^V = (-1)^{p^2}\overline{f_{UV}}dz^V\wedge d\bar{z}^U = (-1)^p\overline{f_{UV}}dz^V\wedge d\bar{z}^U.$$

Therefore, the form $\omega$ is real if and only if $f_{VU} = (-1)^p\overline{f_{UV}}$.