Pseudo-Kähler Geometry of Properly Convex Projective Structures on the torus

In this paper we prove the existence of a pseudo-Kähler structure on the deformation space B₀(T²) of properly convex ℝℙ²-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 B₀(T²) with the complement of the zero section of the total space of the bundle of cubic holomorphic differentials over the Teichmüller space. We show that the S¹-action on B₀(T²), 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 SL(2,ℝ)-action over B₀(T²).