On the symplectic structure over the moduli space of projective structures on a surface. (English) Zbl 1539.14054
Given a closed oriented surface \(\Sigma\) of genus \(g\), denote by \(\mathrm{Proj}(\Sigma)\) the space of smooth projective structures on \(\Sigma\), hence the structures defined by an atlas on \(\Sigma\), taking values in \(\mathbb{C}\mathbb{P}^1\) and having Mobius transformations as coordinates transformations. Each such structure yields, in particular, a flat \(\mathrm{PSL}(2,\mathbb{C})\)-bundle on \(\Sigma\) which, in turn, defines a representation \(\pi_1\Sigma\to\mathrm{PSL}(2,\mathbb{C})\), by monodromy, and therefore determines a class in the character variety \(\chi(\Sigma,\mathrm{PSL}(2,\mathbb{C}))=\mathrm{Hom}(\pi_1\Sigma,\mathrm{PSL}(2,\mathbb{C}))/\hspace{-.1cm}/\mathrm{PSL}(2,\mathbb{C})\). This correspondence gives rise to a map \[f:\mathcal{P}(\Sigma)=\mathrm{Proj}(\Sigma)/\mathrm{Diff}^0(\Sigma)\to \chi(\Sigma,\mathrm{PSL}(2,\mathbb{C})),\] with \(\mathrm{Diff}^0(\Sigma)\) given by the connected component, containing the identity element, of the group of orientation preserving diffeomorphisms of \(\Sigma\).
The character variety \(\chi(\Sigma,\mathrm{PSL}(2,\mathbb{C}))\) is endowed with the so-called Atiyah-Bott-Goldman symplectic form, which induces a symplectic form on \(\mathcal{P}(\Sigma)\). Moreover, if \(\mathrm{Mod}(\Sigma)\) is the mapping-class group of \(\Sigma\), then the Atiyah-Bott-Goldman symplectic form if \(\mathrm{Mod}(\Sigma)\)-invariant and the map \(f\) is \(\mathrm{Mod}(\Sigma)\)-equivariant. So in the end one gets a holomorphic symplectic form \(\Theta_g\) on \(\mathcal{P}(\Sigma)_g=\mathcal{P}(\Sigma)/\mathrm{Mod}(\Sigma)\).
The aim of this paper is to produce a new construction of the symplectic form \(\Theta_g\). It turns out that \(\mathcal{P}(\Sigma)_g\) is in fact an algebraic orbifold, and the construction given by the author proves that actually the symplectic form \(\Theta_g\) is algebraic (which is something which does not directly follow from the previous construction because the map \(f\) above is not algebraic).
The character variety \(\chi(\Sigma,\mathrm{PSL}(2,\mathbb{C}))\) is endowed with the so-called Atiyah-Bott-Goldman symplectic form, which induces a symplectic form on \(\mathcal{P}(\Sigma)\). Moreover, if \(\mathrm{Mod}(\Sigma)\) is the mapping-class group of \(\Sigma\), then the Atiyah-Bott-Goldman symplectic form if \(\mathrm{Mod}(\Sigma)\)-invariant and the map \(f\) is \(\mathrm{Mod}(\Sigma)\)-equivariant. So in the end one gets a holomorphic symplectic form \(\Theta_g\) on \(\mathcal{P}(\Sigma)_g=\mathcal{P}(\Sigma)/\mathrm{Mod}(\Sigma)\).
The aim of this paper is to produce a new construction of the symplectic form \(\Theta_g\). It turns out that \(\mathcal{P}(\Sigma)_g\) is in fact an algebraic orbifold, and the construction given by the author proves that actually the symplectic form \(\Theta_g\) is algebraic (which is something which does not directly follow from the previous construction because the map \(f\) above is not algebraic).
Reviewer: André Oliveira (Porto)
MSC:
| 14H15 | Families, moduli of curves (analytic) |
| 32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |
| 14H60 | Vector bundles on curves and their moduli |
References:
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| [3] | Biswas, I.; Looijenga, E., Projective structures on Riemann surfaces. Pure Appl. Math. Q. (2023), in press |
| [4] | Gallo, D.; Kapovich, M.; Marden, A., The monodromy groups of Schwarzian equations on closed Riemann surfaces. Ann. Math., 625-704 (2000) · Zbl 0977.30028 |
| [5] | Goldman, W. M., The symplectic nature of fundamental groups of surfaces. Adv. Math., 200-225 (1984) · Zbl 0574.32032 |
| [6] | Gunning, R. C., Lectures on Riemann Surfaces. Mathematical Notes (1966), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0175.36801 |
| [7] | Hejhal, D. A., Monodromy groups and linearly polymorphic functions. Acta Math., 1-55 (1975) · Zbl 0333.34002 |
| [8] | Hubbard, J. H., The monodromy of projective structures, Riemann surfaces and related topics, 257-275 |
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