Stability of the Poincaré bundle

Abstract

Let X be an irreducible smooth projective curve, of genus at least two, over an algebraically closed field k. Let ${\mathcal{M}}_G^d$ denote the moduli stack of principal G-bundles over X of fixed topological type $d\in \ddot{I}{€}_1(G)$, where G is any almost simple affine algebraic group over k. We prove that the universal bundle over $X\times {\mathcal{M}}_G^d$ is stable with respect to any polarization on $X\times {\mathcal{M}}_G^d$. A similar result is proved for the Poincaré 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.

7 pages