Johnson's homomorphisms and the Arakelov-Green function

TL;DR

This paper introduces a new function on the moduli space of Riemann surfaces, computes its variations, and relates the Arakelov-Green function's Chern form to Johnson's homomorphisms via a flat connection.

Contribution

It establishes a novel connection between the Arakelov-Green function, Johnson's homomorphisms, and differential forms on the moduli space of Riemann surfaces.

Findings

Computed first and second variations of the introduced function.

Related the Chern form of the relative tangent bundle to Johnson's homomorphisms.

Connected the Arakelov-Green function with flat connections on the moduli space.

Abstract

Let $\pi: {\mathbb C}_g \to {\mathbb M}_g$ be the universal family of compact Riemann surfaces of genus $g \geq 1$. We introduce a real-valued function on the moduli space ${\mathbb M}_g$ and compute the first and the second variations of the function. As a consequence we relate the Chern form of the relative tangent bundle $T_{{\mathbb C}_g/{\mathbb M}_g}$ induced by the Arakelov-Green function with differential forms on ${\mathbb C}_g$ induced by a flat connection whose holonomy gives Johnson's homomorphisms on the mapping class group.