2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/146741Let $π: {\mathbb C}_g \to {\mathbb M}_g$ 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.Geometric TopologyAlgebraic Geometry57R20 (Primary); 14H15, 32G15 (Secondary)text