Johnson's homomorphisms and the Arakelov-Green function

Let $\pi :{\mathbb{C}}_g\to {\mathbb{M}}_g$ be the universal family of compact Riemann surfaces of genus $g\ge 1$. 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 $T_{{\mathbb{C}}_g\mathrm{/}{\mathbb{M}}_g}$ 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.