Contractibility of the space of rational maps

TL;DR

This paper introduces an algebraic model for the space of rational maps from a smooth curve to an algebraic group and proves its homological contractibility, impacting the understanding of G-bundle moduli spaces.

Contribution

It provides a new algebro-geometric framework for the space of rational maps and establishes its homological contractibility, linking it to the uniformization of G-bundles.

Findings

The space of rational maps is homologically contractible.

The moduli space Bun(G) is uniformized by a rational affine Grassmannian.

The uniformizing map has contractible fibers.

Abstract

We define an algebro-geometric model for the space of rational maps from a smooth curve X to an algebraic group G, and show that this space is homologically contractible. As a consequence, we deduce that the moduli space Bun(G) of G-bundles on X is uniformized by the appropriate rational version of the affine Grassmannian, where the uniformizing map has contractible fibers.