The pre-symplectic geometry of opers and the holonomy map

Andrew Sanders

Published 2018-04-12Version 1

Given a connected complex semi-simple Lie group $G$ and a Riemann surface $X,$ a $G$-oper on $X$ is a higher rank generalization of a complex projective structure on $X.$ These objects play an important role in integrable systems and geometric representation theory, a status that was cemented by the seminal work of Beilinson-Drinfeld \cite{BD91}. For $G$ a connected complex simple Lie group of adjoint type, we study the global deformation theory of $G$-opers on a connected, closed, oriented smooth surface $\Sigma$ of genus at least two. We exhibit the deformation space of $G$-opers on $\Sigma$ as a holomorphic fiber bundle over Teichm\"{u}ller space, and elucidate the relationship with the deformation space of complex projective structures. Then, we show that there is a family of identifications of the deformation space of $G$-opers with a holomorphic vector bundle $\mathcal{B}_{G}(\Sigma)$ over Teichm\"{u}ller space whose typical fiber over a Riemann surface $X$ is a sum of spaces of pluri-canonical sections. Finally, we show that the holonomy map from the deformation space of $G$-opers to the deformation space of flat $G$-bundles on $\Sigma$ is a holomorphic immersion. As a consequence of this result, we show that the deformation space of $G$-opers carries a (pre-symplectic) closed holomorphic differential $2$-form of constant rank, and we prove that a sub-family of the identifications of $\mathcal{B}_{G}(\Sigma)$ with the deformation space of $G$-opers is a holomorphic pre-symplectic map for a natural holomorphic pre-symplectic form on $\mathcal{B}_{G}(\Sigma).$ These results generalize the fundamental features of the deformation space of complex projective structures on $\Sigma$ to the setting of $G$-opers.