Principal Schottky Bundles over Riemann surfaces

TL;DR

This paper introduces and studies Schottky G-bundles over Riemann surfaces, establishing their relation to branes, trivial topological type, and the surjectivity of the Schottky map in specific cases, advancing understanding of their moduli space.

Contribution

It generalizes the Schottky moduli map to principal G-bundles, proves its local surjectivity, and establishes surjectivity onto flat bundles in special cases.

Findings

All Schottky G-bundles have trivial topological type.

The Schottky map is locally surjective at the good and unitary locus.

The Schottky map is surjective onto flat bundles for abelian G and G over elliptic curves.

Abstract

We introduce and study (strict) Schottky G-bundles over a compact Riemann surface X, where G is a connected reductive algebraic group. Strict Schottky representations are shown to be related to branes in the moduli space of G-Higgs bundles over X, and we prove that all Schottky $G$-bundles have trivial topological type. Generalizing the Schottky moduli map introduced in Florentino to the setting of principal bundles, we prove its local surjectivity at the good and unitary locus. Finally, we prove that the Schottky map is surjective onto the space of flat bundles for two special classes: when G is an abelian group over an arbitrary X, and the case of a general G-bundle over an elliptic curve.