Opers with real monodromy and Eichler-Shimura isomorphism
Yasuhiro Wakabayashi

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 -opers on pointed Riemann surfaces (for a simple algebraic group of adjoint type) and their monodromy maps. In the first part, we review some general facts on -opers, or more generally, principal -bundles with holomorphic connection having simple poles along marked points, including the correspondence with -representations of the fundamental group. One of the main results, proved in the second part, asserts that the space of certain -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 -oper with real monodromy. The resulting decomposition of the (parabolic) de Rham cohomology group of its symmetric product defines a polarized…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Algebra and Geometry · Algebraic Geometry and Number Theory · Geometry and complex manifolds
