Given $$G^{\mu\nu} = -2 \frac{\partial L}{\partial F_{\mu\nu}}$$ it is written in http://arxiv.org/abs/hep-th/9506035 that the field equations are (eq 1.8) $$\partial_{[\alpha} \star G_{\beta \gamma]}= 0$$
How is that? Why the dual and not the $G_{\mu\nu}$ itself? Here L is general form of Lagrangian which is a function of $F_{\mu\nu}$ and $g_{\mu\nu}$ and $\partial_{[\alpha} \star G_{\beta \gamma]}= 0$ is the anti-symmetrized way of writing the equation of motion. Please note that here we are considering duality.