Quillen connection and the uniformization of Riemann surfaces

TL;DR

This paper explores the relationship between the Quillen connection, the Weil--Petersson form, and the uniformization of Riemann surfaces, establishing a unique correspondence between holomorphic connections and projective structures.

Contribution

It demonstrates a unique isomorphism linking the Quillen connection on the Hodge line bundle to the moduli stack of projective structures, connecting differential geometry and complex analysis.

Findings

The Quillen connection's curvature equals the Weil--Petersson form.

A unique holomorphic isomorphism exists between holomorphic connections and projective structures.

The sections determined by the Quillen connection and uniformization theorem are equivalent.

Abstract

The Quillen connection on ${\mathcal L} \rightarrow {\mathcal M}_g$, where ${\mathcal L}^*$ is the Hodge line bundle over the moduli stack of smooth complex projective curves curves ${\mathcal M}_g$, $g \geq 5$, 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.