Symplectic coordinates on the deformation spaces of convex projective structures on 2-orbifolds

TL;DR

This paper demonstrates that the deformation space of convex projective structures on certain 2-orbifolds admits a global Darboux coordinate system, revealing a detailed symplectic geometric structure and decomposing it into smaller symplectic components.

Contribution

It establishes the existence of a global Darboux coordinate system on the deformation space, extending the understanding of symplectic structures on convex projective structures on 2-orbifolds.

Findings

Decomposition of the deformation space into smaller symplectic spaces

Construction of symplectic form for orbifolds with boundary

Existence of global Darboux coordinates on the deformation space

Abstract

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