A Torelli theorem for moduli spaces of principal bundles over a curve

Biswas, Indranil; Hoffmann, Norbert

Annales de l'Institut Fourier, Tome 62 (2012) no. 1, pp. 87-106.

Soient ${\displaystyle X}$ et ${\displaystyle X^{\prime }}$ des surfaces de Riemann compactes de genre au moins 3, et ${\displaystyle G}$ et ${\displaystyle G^{\prime }}$ des groupes complexes réductifs non abéliens. Si une composante ${\displaystyle {ℳ}_G^d\left(X\right)}$ de l’espace des modules de $G$–fibrés principaux semi-stables sur $X$ est isomorphe à une composante ${\displaystyle {ℳ}_{G^{\prime }}^{d^{\prime }}\left(X^{\prime }\right)}$, alors $X$ est isomorphe à $X^{\prime }$.

Let $X$ and $X^{\prime }$ be compact Riemann surfaces of genus $\ge 3$, and let $G$ and $G^{\prime }$ be nonabelian reductive complex groups. If one component ${ℳ}_G^d\left(X\right)$ of the coarse moduli space for semistable principal $G$–bundles over $X$ is isomorphic to another component ${ℳ}_{G^{\prime }}^{d^{\prime }}\left(X^{\prime }\right)$, then $X$ is isomorphic to $X^{\prime }$.

Classification : 14D20, 14C34
Keywords: Principal bundle, moduli space, Torelli theorem
Mot clés : Fibré principal, espace des modules, théorème de Torelli

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