Opers with real monodromy and Eichler-Shimura isomorphism

TL;DR

This paper studies $G$-opers on Riemann surfaces with real monodromy, proving their space is discrete and establishing an Eichler-Shimura isomorphism that links these structures to polarized real Hodge structures.

Contribution

It proves the discreteness of $G$-opers with real monodromy and establishes the Eichler-Shimura isomorphism for $ ext{PSL}_2$-opers, extending known results.

Findings

The space of certain $G$-opers with real monodromy is discrete.

Established Eichler-Shimura isomorphism for $ ext{PSL}_2$-opers.

Decomposition of cohomology yields polarized real Hodge structures.

Abstract

The purpose of the present paper is to investigate $G$-opers on pointed Riemann surfaces (for a simple algebraic group $G$ of adjoint type) and their monodromy maps. In the first part, we review some general facts on $G$-opers, or more generally, principal $G$-bundles with holomorphic connection having simple poles along marked points, including the correspondence with $G$-representations of the fundamental group. One of the main results, proved in the second part, asserts that the space of certain $G$-opers with real monodromy forms a discrete set. This fact generalizes the discreteness theorem for real projective structures, already proved by G. Faltings. As an application, we establish the Eichler-Shimura isomorphism for each $\mathrm{PSL}_2$-oper with real monodromy. The resulting decomposition of the (parabolic) de Rham cohomology group of its symmetric product defines a polarized real Hodge structure.