Pseudo-Kähler geometry of properly convex projective structures on the torus
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866929618685001728 |
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| author | Rungi, Nicholas Tamburelli, Andrea |
| author_facet | Rungi, Nicholas Tamburelli, Andrea |
| contents | In this paper we prove the existence of a pseudo-Kähler 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 Teichmüller 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 $\mathcal{B}_0(T^2)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2112_08979 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Pseudo-Kähler geometry of properly convex projective structures on the torus Rungi, Nicholas Tamburelli, Andrea Differential Geometry Representation Theory Symplectic Geometry In this paper we prove the existence of a pseudo-Kähler 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 Teichmüller 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 $\mathcal{B}_0(T^2)$. |
| title | Pseudo-Kähler geometry of properly convex projective structures on the torus |
| topic | Differential Geometry Representation Theory Symplectic Geometry |
| url | https://arxiv.org/abs/2112.08979 |