Unipotent Schottky bundles on Riemann surfaces and complex tori

TL;DR

This paper establishes an equivalence between unipotent representations of Schottky groups and unipotent vector bundles on Riemann surfaces and complex tori, revealing deep connections between group representations and geometric structures.

Contribution

It proves that the Schottky functor induces an equivalence between unipotent representations and unipotent vector bundles, and characterizes flat bundles on complex tori as Schottky.

Findings

The Schottky functor creates an equivalence of categories.

Any flat bundle on a complex torus is Schottky.

Unipotent representations correspond to unipotent vector bundles.

Abstract

We study a natural map from representations of a free (resp. free abelian) group of rank g in GL_r(C), to holomorphic vector bundles of degree zero over a compact Riemann surface X of genus g (resp. complex torus X of dimension g). This map defines what is called a Schottky functor. Our main result is that this functor induces an equivalence between the category of unipotent representations of Schottky groups and the category of unipotent vector bundles on X. We also show that, over a complex torus, any vector or principal bundle with a flat holomorphic connection is Schottky.