Hodge bundle

Let ${\displaystyle {\mathcal {M}}_{g}}$ be the moduli space of algebraic curves of genus g curves over some scheme. The Hodge bundle ${\displaystyle \Lambda _{g}}$ is a vector bundle^{[note 1]} on ${\displaystyle {\mathcal {M}}_{g}}$ whose fiber at a point C in ${\displaystyle {\mathcal {M}}_{g}}$ is the space of holomorphic differentials on the curve C. To define the Hodge bundle, let ${\displaystyle \pi \colon {\mathcal {C}}_{g}\rightarrow {\mathcal {M}}_{g}}$ be the universal algebraic curve of genus g and let ${\displaystyle \omega _{g}}$ be its relative dualizing sheaf. The Hodge bundle is the pushforward of this sheaf, i.e.,^[3]

${\displaystyle \Lambda _{g}=\pi _{*}\omega _{g}}$.

• ELSV formula