On this page, the author says:
We now define a Hermitian manifold as a complex manifold where there is a preferred class of coordinate systems in which unmixed components of metric tensor vanish ($g_{\alpha\beta}=g_{\bar{\alpha}\bar{\beta}}=0$).
My question is why do those vanish if the metric is Hermitian?
Let $g$ be a Riemannian metric which is hermitian with respect to an almost complex structure $J$, i.e. $g(JX, JY) = g(X, Y)$ for all vector fields $X$ and $Y$. Then we have
$$g_{\alpha\beta} = g(\partial_{\alpha}, \partial_{\beta}) = g(J\partial_{\alpha}, J\partial_{\beta}) = g(i\partial_{\alpha}, i\partial_{\beta}) = -g(\partial_{\alpha}, \partial_{\beta})= - g_{\alpha\beta} \implies g_{\alpha\beta} = 0,$$
and
$$g_{\bar{\alpha}\bar{\beta}} = g(\bar{\partial}_a, \bar{\partial}_b) = g(J\bar{\partial}_a, J\bar{\partial}_b) = g(-i\bar{\partial}_a, -i\bar{\partial}_b) = -g(\bar{\partial}_a, \bar{\partial}_b) = - g_{\bar{\alpha}\bar{\beta}} \implies g_{\bar{\alpha}\bar{\beta}} = 0.$$