# Hodge bundle

Let $\mathcal{M}_g$ be the moduli space of algebraic curves of genus g curves over some scheme. The Hodge bundle Λg is a vector bundle on $\mathcal{M}_g$ whose fiber at a point C in $\mathcal{M}_g$ is the space of holomorphic differentials on the curve C. To define the Hodge bundle, let $\pi:\mathcal{C}_g\rightarrow\mathcal{M}_g$ be the universal algebraic curve of genus g and let ωg be its relative dualizing sheaf. The Hodge bundle is the pushforward of this sheaf, i.e.[3]
$\Lambda_g=\pi_*\omega_g.\,$