Quillen connection and the uniformization of Riemann surfaces

Abstract

The Quillen connection on \({\mathcal L}\, \longrightarrow \, {\mathcal M}_g\), where \({\mathcal L}^*\) is the Hodge line bundle over the moduli stack of smooth complex projective curves curves \({\mathcal M}_g\), \(g\, \ge \, 5\), is uniquely determined by the condition that its curvature is the Weil–Petersson form on \({\mathcal M}_g\). The bundle of holomorphic connections on \({\mathcal L}\) has a unique holomorphic isomorphism with the bundle on \({\mathcal M}_g\) given by the moduli stack of projective structures. This isomorphism takes the \(C^\infty\) section of the first bundle given by the Quillen connection on \({\mathcal L}\) to the \(C^\infty\) section of the second bundle given by the uniformization theorem. Therefore, any one of these two sections determines the other uniquely.