Unramified Brauer group of the moduli spaces of $PGL_r({\mathbb C})$-bundles over curves

Abstract

Let $X$ be an irreducible smooth complex projective curve of genus at least two. Let $N$ be a connected component of the moduli space of semistable principal ${\rm PGL}_r({\mathbb C})$- bundles over $X$; it is a normal unirational complex projective variety. We prove that the Brauer group of a desingularization of $N$ is trivial.