On Nori's Obstruction to Universal Bundles

TL;DR

This paper proves that for certain classical groups, no universal principal bundle exists over any Zariski open subset of the moduli space of semistable bundles on a curve.

Contribution

It establishes the non-existence of universal bundles over Zariski open subsets for moduli spaces associated with $SL_n$, $Sp(2n)$, and $SO(2n)$ groups.

Findings

No universal bundle exists on any Zariski open subset of the moduli space.

The result applies to classical groups $SL_n$, $Sp(2n)$, and $SO(2n)$.

The proof involves geometric and algebraic techniques related to the structure of the moduli space.

Abstract

Let $G$ be $Sl_n, Sp(2n)$ or SO(2n). We consider the moduli space $M$ of semistable principal $G$-bundles over a curve $X$. Our main result is that if $U$ is a Zariski open subset of $M$ then there is no universal bundle on $U\times X$.