Opers with real monodromy and Eichler-Shimura isomorphism

2023/09/21 by Wakabayashi, Yasuhiro · 1 citation
#Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2309.12203

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 PSL₂-oper with real monodromy. The resulting decomposition of the (parabolic) de Rham cohomology group of its symmetric product defines a polarized real Hodge structure.

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