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

TL;DR

This paper establishes a pseudo-K"ahler structure on the deformation space of convex projective structures on the torus, linking complex, symplectic, and metric structures with group actions.

Contribution

It introduces a pseudo-K"ahler structure on the deformation space of convex projective structures on the torus, with compatible metric and symplectic form, and analyzes associated group actions.

Findings

Existence of a pseudo-K"ahler structure on the deformation space.

The $S^1$-action is Hamiltonian and preserves the structures.

A moment map for the $ ext{SL}(2,b R)$-action exists.

Abstract

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 $\mathrm{SL}(2,\mathbb R)$-action over $\mathcal{B}_0(T^2)$.