# Stability of the Poincar\'e bundle

## TL;DR

This paper proves the stability of the universal and Poincaré bundles over moduli spaces of principal G-bundles on algebraic curves, ensuring their robustness under various polarizations.

## Contribution

It establishes the stability of the universal bundle and Poincaré adjoint bundle over moduli spaces of principal G-bundles on curves, a significant step in understanding their geometric properties.

## Key findings

- Universal bundle over X × M_G^d is stable under any polarization.
- Poincaré adjoint bundle over X × M_G^{d, rs} is stable.
- Results hold for almost simple algebraic groups over algebraically closed fields.

## Abstract

Let X be an irreducible smooth projective curve, of genus at least two, over an algebraically closed field k. Let $\mathcal{M}^d_G$ denote the moduli stack of principal G-bundles over X of fixed topological type $d \in \pi_1(G)$, where G is any almost simple affine algebraic group over k. We prove that the universal bundle over $X \times \mathcal{M}^d_G$ is stable with respect to any polarization on $X \times \mathcal{M}^d_G$. A similar result is proved for the Poincar\'e adjoint bundle over $X \times M_G^{d, rs}$, where $M_G^{d, rs}$ is the coarse moduli space of regularly stable principal G-bundles over X of fixed topological type d.