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 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 is stable with respect to any polarization on . A similar result is proved for the Poincaré adjoint bundle over \(X \,{\times }\, M_G^{d, {\mathrm {rs}}}\), where \(M_G^{d, {\mathrm {rs}}}\) is the coarse moduli space of regularly stable principal G-bundles over X of fixed topological type d.