Complex Differential Forms and Notation

Question

If I have a complex form $\Omega$, what does the notation $\operatorname{Re}(\Omega)$ and $\operatorname{Im}(\Omega)$ mean? How does this relate to the decomposition of the space of $(p,q)$-forms and how does it relate to the notation $\Omega\wedge\bar{\Omega}$?

Answer 1

If $V$ is a real vector space, the complexification $V\otimes_{\mathbb{R}}\mathbb{C}$ is a complex vector space which has an induced complex conjugation map given by $\overline{v\otimes z} = v\otimes\overline{z}$.

If $w \in V\otimes_{\mathbb{R}}\mathbb{C}$, then $w = v_1\otimes 1 + v_2\otimes i$ where $v_1, v_2 \in V$. Then $\operatorname{Re}(w) = v_1\otimes 1$ and $\operatorname{Im}(w) = v_2\otimes 1$ so that $w = \operatorname{Re}(w) + i\operatorname{Im}(w)$. Note that $\operatorname{Re}(w) = \frac{1}{2}(w + \overline{w})$ and $\operatorname{Im}(w) = -\frac{i}{2}(w - \overline{w})$.

If $V = \mathcal{E}(X)$ denotes the vector space of differential forms, then its complexification $V\otimes_{\mathbb{R}}\mathbb{C}$ is the vector space of complex differential forms. So $\operatorname{Re}(\Omega) = \frac{1}{2}(\Omega + \overline{\Omega})$ and $\operatorname{Im}(\Omega) = -\frac{i}{2}(\Omega - \overline{\Omega})$.

If $\Omega$ is a $k$-form, then $\Omega = \Omega^{k,0} + \Omega^{k-1,1} + \dots + \Omega^{1,k-1} + \Omega^{k,0}$ where $\Omega^{p,q}$ is a $(p, q)$-form. Then

\begin{align*} \operatorname{Re}(\Omega) &= \frac{1}{2}\left[\left(\Omega^{k,0} + \overline{\Omega^{0,k}}\right) + \left(\Omega^{k-1,1} + \overline{\Omega^{1,k-1}}\right) + \dots + \left(\Omega^{1,k-1} + \overline{\Omega^{k-1,1}}\right) + \left(\Omega^{0,k} + \overline{\Omega^{k,0}}\right) \right]\\ \operatorname{Im}(\Omega) &= -\frac{i}{2}\left[\left(\Omega^{k,0} - \overline{\Omega^{0,k}}\right) + \left(\Omega^{k-1,1} - \overline{\Omega^{1,k-1}}\right) + \dots + \left(\Omega^{1,k-1} - \overline{\Omega^{k-1,1}}\right) + \left(\Omega^{0,k} - \overline{\Omega^{k,0}}\right) \right]. \end{align*}

As $\overline{\Omega\wedge\overline{\Omega}} = \overline{\Omega}\wedge\Omega = (-1)^{k(n-k)}\Omega\wedge\overline{\Omega}$. It follows that if $k$ is odd and $n$ is even, then $\Omega\wedge\overline{\Omega}$ is purely imaginary. In all other cases, $\Omega\wedge\overline{\Omega}$ is real.