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From the Ricci tensor we define the Ricci form:

$$\rho(u,v)=R(Ju,v)$$

that is an anti-symmetric 2-form locally described by

$$\rho_{j\bar{k}}=\rho(\partial_{j},\partial_{\overline{k}})=R(\mathrm{i}\partial_{j},\partial_{\overline{k}})=\mathrm{i} R_{j\bar{k}}$$

$$\rho_{\bar{j}k}=\rho(\partial_{\overline{j}},\partial_{k})=R(-\mathrm{i}\partial_{\overline{j}},\partial_{k})=-\mathrm{i} R_{\bar{j}k}$$

$$\rho_{jk}=\rho_{\bar{j}\bar{k}}=0$$

and also

$$\rho=\rho_{j\bar{k}}\mathrm{d} z^j\otimes\mathrm{d}\bar{z}^k+\rho_{\bar{j}k}\mathrm{d}\bar{z}^j\otimes\mathrm{d} z^k=-\mathrm{i}\dfrac{\partial ^2}{\partial z^j \partial \bar{z}^k}(\log \det g)(\mathrm{d} z^j\otimes\mathrm{d}\bar{z}^k-\mathrm{d}\bar{z}^k\otimes\mathrm{d} z^j)$$

$$=-\mathrm{i}\dfrac{\partial ^2}{\partial z^j \partial \bar{z}^k}(\log \det g)\mathrm{d} z^j\wedge\mathrm{d}\bar{z}^k=-\mathrm{i}\partial\bar{\partial}\log \det g$$