Pseudo-K\"ahler geometry of properly convex projective structures on the torus

In this paper we prove the existence of a pseudo-K"ahler structure on the deformation space ${\mathcal{B}}_0(T^2)$ of properly convex $\mathbb R\mathbb P^2$-structures over the torus. In particular, the pseudo-Riemannian metric and the symplectic form are compatible with the complex structure inherited from the identification of ${\mathcal{B}}_0(T^2)$ with the complement of the zero section of the total space of the bundle of cubic holomorphic differentials over the Teichm"uller space. We show that the $S^1$-action on ${\mathcal{B}}_0(T^2)$, given by rotation of the fibers, is Hamiltonian and it preserves both the metric and the symplectic form. Finally, we prove the existence of a moment map for the $\mathrm{S}\mathrm{L}(2,\mathbb{R})$-action over ${\mathcal{B}}_0(T^2)$.