Johnson's homomorphisms and the Arakelov-Green function

Understand

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.