Symplectic Coordinates on the Deformation Spaces of Convex Projective Structures on 2-Orbifolds

Abstract

Let 𝒪 be a closed orientable 2-orbifold of negative Euler characteristic. Huebschmann constructed the Atiyah-Bott-Goldman type symplectic form ω on the deformation space C(𝒪) of convex projective structures on 𝒪. We show that the deformation space C(𝒪) of convex projective structures on 𝒪 admits a global Darboux coordinate system with respect to ω. To this end, we show that C(𝒪) can be decomposed into smaller symplectic spaces. In the course of the proof, we also study the deformation space C(𝒪) for an orbifold 𝒪 with boundary and construct the symplectic form on the deformation space of convex projective structures on 𝒪 with fixed boundary holonomy.