Free, open-source online mathematics for students, teachers and workers

$\tilde{h}$, $g$ and $\omega$ may be $\mathbb{C}$-linearly extended to $TM_\mathbb{C}\times TM_\mathbb{C}\longrightarrow\mathbb{C}$:

$$ \begin{array}{rcl} \tilde{h} & = & h_{jk}\mathrm{d} x^j\otimes\mathrm{d} x^k-\mathrm{i} h_{jk}\mathrm{d} x^j\otimes\mathrm{d} y^k+\mathrm{i} h_{jk}\mathrm{d} y^j\otimes\mathrm{d} x^k+h_{jk}\mathrm{d} y^j\otimes\mathrm{d} y^k\\ & = & h_{jk}\left(\dfrac{\mathrm{d} z^j+\mathrm{d}\bar{z}^j}{2}\right)\otimes\left(\dfrac{\mathrm{d} z^k+\mathrm{d}\bar{z}^k}{2}\right) -\mathrm{i} h_{jk}\left(\dfrac{\mathrm{d} z^j+\mathrm{d}\bar{z}^j}{2}\right)\otimes\left(\dfrac{\mathrm{d} z^k-\mathrm{d}\bar{z}^k}{2\mathrm{i}}\right)\\ & + & \mathrm{i} h_{jk}\left(\dfrac{\mathrm{d} z^j-\mathrm{d}\bar{z}^j}{2\mathrm{i}}\right)\otimes\left(\dfrac{\mathrm{d} z^k+\mathrm{d}\bar{z}^k}{2}\right) +h_{jk}\left(\dfrac{\mathrm{d} z^j-\mathrm{d}\bar{z}^j}{2\mathrm{i}}\right)\otimes\left(\dfrac{\mathrm{d} z^k-\mathrm{d}\bar{z}^k}{2\mathrm{i}}\right)\\ & = &\dfrac{1}{4}[(h_{jk}-h_{jk}+h_{jk}-h_{jk})\mathrm{d} z^j\otimes\mathrm{d} z^k +(h_{jk}+h_{jk}+h_{jk}+h_{jk})\mathrm{d} z^j\otimes\mathrm{d} \bar{z}^k\\ & + & (h_{jk}-h_{jk}-h_{jk}+h_{jk})\mathrm{d} \bar{z}^j\otimes\mathrm{d} z^k +(h_{jk}+h_{jk}-h_{jk}-h_{jk})\mathrm{d} \bar{z}^j\otimes\mathrm{d} \bar{z}^k]\\ & = & h_{jk}\mathrm{d} z^j\otimes\mathrm{d} \bar{z}^k\\ &&\\ &&\\ g & = & \alpha_{jk}\mathrm{d} x^j\otimes\mathrm{d} x^k+\beta_{jk}\mathrm{d} x^j\otimes\mathrm{d} y^k-\beta_{jk}\mathrm{d} y^j\otimes\mathrm{d} x^k+\alpha_{jk}\mathrm{d} y^j\otimes\mathrm{d} y^k\\ & = & \alpha_{jk}\left(\dfrac{\mathrm{d} z^j+\mathrm{d}\bar{z}^j}{2}\right)\otimes\left(\dfrac{\mathrm{d} z^k+\mathrm{d}\bar{z}^k}{2}\right) +\beta_{jk}\left(\dfrac{\mathrm{d} z^j+\mathrm{d}\bar{z}^j}{2}\right)\otimes\left(\dfrac{\mathrm{d} z^k-\mathrm{d}\bar{z}^k}{2\mathrm{i}}\right)\\ & - & \beta_{jk}\left(\dfrac{\mathrm{d} z^j-\mathrm{d}\bar{z}^j}{2\mathrm{i}}\right)\otimes\left(\dfrac{\mathrm{d} z^k+\mathrm{d}\bar{z}^k}{2}\right) +\alpha_{jk}\left(\dfrac{\mathrm{d} z^j-\mathrm{d}\bar{z}^j}{2\mathrm{i}}\right)\otimes\left(\dfrac{\mathrm{d} z^k-\mathrm{d}\bar{z}^k}{2\mathrm{i}}\right)\\ & = & \dfrac{1}{4}[(\alpha_{jk}-\mathrm{i}\beta_{jk}+\mathrm{i}\beta_{jk}-\alpha_{jk})\mathrm{d} z^j\otimes\mathrm{d} z^k +(\alpha_{jk}+\mathrm{i}\beta_{jk}+\mathrm{i}\beta_{jk}+\alpha_{jk})\mathrm{d} z^j\otimes\mathrm{d}\bar{z}^k\\ & + & (\alpha_{jk}-\mathrm{i}\beta_{jk}-\mathrm{i}\beta_{jk}+\alpha_{jk})\mathrm{d}\bar{z}^j\otimes\mathrm{d} z^k +(\alpha_{jk}+\mathrm{i}\beta_{jk}-\mathrm{i}\beta_{jk}-\alpha_{jk})\mathrm{d}\bar{z}^j\otimes\mathrm{d}\bar{z}^k]\\ & = & \dfrac{h_{jk}}{2}\mathrm{d} z^j\otimes\mathrm{d}\bar{z}^k+\dfrac{\overline{h_{jk}}}{2}\mathrm{d}\bar{z}^j\otimes\mathrm{d} z^k = g_{j\bar{k}}\mathrm{d} z^j\otimes\mathrm{d}\bar{z}^k+g_{\bar{j}k}\mathrm{d}\bar{z}^j\otimes\mathrm{d} z^k\\ &&\\ &&\\ \omega & = & -\beta_{jk}\mathrm{d} x^j\otimes\mathrm{d} x^k-\alpha_{jk}\mathrm{d} x^j\otimes\mathrm{d} y^k\alpha_{jk}\mathrm{d} y^j\otimes\mathrm{d} x^k+\beta_{jk}\mathrm{d} y^j\otimes\mathrm{d} y^k\\ & = & -\beta_{jk}\left(\dfrac{\mathrm{d} z^j+\mathrm{d}\bar{z}^j}{2}\right)\otimes\left(\dfrac{\mathrm{d} z^k+\mathrm{d}\bar{z}^k}{2}\right) -\alpha_{jk}\left(\dfrac{\mathrm{d} z^j+\mathrm{d}\bar{z}^j}{2}\right)\otimes\left(\dfrac{\mathrm{d} z^k-\mathrm{d}\bar{z}^k}{2\mathrm{i}}\right)\\ & + & \alpha_{jk}\left(\dfrac{\mathrm{d} z^j-\mathrm{d}\bar{z}^j}{2\mathrm{i}}\right)\otimes\left(\dfrac{\mathrm{d} z^k+\mathrm{d}\bar{z}^k}{2}\right) +\beta_{jk}\left(\dfrac{\mathrm{d} z^j-\mathrm{d}\bar{z}^j}{2\mathrm{i}}\right)\otimes\left(\dfrac{\mathrm{d} z^k-\mathrm{d}\bar{z}^k}{2\mathrm{i}}\right)\\ & = & -\dfrac{1}{4}[(\beta_{jk}+\mathrm{i}\alpha_{jk}-\mathrm{i}\alpha_{jk}-\beta_{jk})\mathrm{d} z^j\otimes\mathrm{d} z^k +(\beta_{jk}-\mathrm{i}\alpha_{jk}-\mathrm{i}\alpha_{jk}+\beta_{jk})\mathrm{d} z^j\otimes\mathrm{d}\bar{z}^k\\ & + & (\beta_{jk}+\mathrm{i}\alpha_{jk}+\mathrm{i}\alpha_{jk}+\beta_{jk})\mathrm{d}\bar{z}^j\otimes\mathrm{d} z^k +(\beta_{jk}-\mathrm{i}\alpha_{jk}+\mathrm{i}\alpha_{jk}-\beta_{jk})\mathrm{d}\bar{z}^j\otimes\mathrm{d}\bar{z}^k]\\ & = & \dfrac{\mathrm{i} h_{jk}}{2}\mathrm{d} z^j\otimes\mathrm{d}\bar{z}^k-\dfrac{\mathrm{i}\overline{h_{jk}}}{2}\mathrm{d}\bar{z}^j\otimes\mathrm{d} z^k =\omega_{j\bar{k}}\mathrm{d} z^j\otimes\mathrm{d}\bar{z}^k+\omega_{\bar{j}k}\mathrm{d}\bar{z}^j\otimes\mathrm{d} z^k\\ \end{array} $$

and again $\tilde{h}=g-\mathrm{i}\omega$, yet now it makes no sense to say that $g=\text{Re}(\tilde{h})$, $\omega=-\text{Im}(\tilde{h})$ ($g$ and $\omega$ are complex-valued). Moreover, it must be remarked that seeing the complex extension of $\tilde{h}$ as $\tilde{h}:T^{1,0}M\times T^{0,1}M\longrightarrow\mathbb{C}$, then $\tilde{h}=h\circ(\text{id}\times c)$ (it's like adjusting $h$ for it to be $\mathbb{C}$-linear and also tensor).

We note also that

$$ \begin{array}{rcl} \omega & = & \dfrac{\mathrm{i} h_{jk}}{2}\mathrm{d} z^j\otimes\mathrm{d}\bar{z}^k-\dfrac{\mathrm{i}\overline{h_{jk}}}{2}\mathrm{d}\bar{z}^j\otimes\mathrm{d} z^k =\dfrac{\mathrm{i} h_{jk}}{2}\mathrm{d} z^j\otimes\mathrm{d}\bar{z}^k-\dfrac{\mathrm{i}\overline{h_{kj}}}{2}\mathrm{d}\bar{z}^k\otimes\mathrm{d} z^j\\ & = & \dfrac{\mathrm{i} h_{jk}}{2}(\mathrm{d} z^j\otimes\mathrm{d}\bar{z}^k-\mathrm{d}\bar{z}^k\otimes\mathrm{d} z^j)=\dfrac{\mathrm{i} h_{jk}}{2}\mathrm{d} z^j\wedge\mathrm{d}\bar{z}^k=\omega_{j\bar{k}}\mathrm{d} z^j\wedge\mathrm{d}\bar{z}^k\\ \end{array} $$

so $\omega$ is a (1,1)-form