Hodge bundle
In mathematics, the Hodge bundle, named after W. V. D. Hodge, appears in the study of families of curves, where it provides an invariant in the moduli theory of algebraic curves. Furthermore, it has applications to the theory of modular forms on reductive algebraic groups and string theory.Definition
Let be the moduli space of algebraic curves of genus g curves over some scheme. The Hodge bundle is a vector bundle on whose fiber at a point C in is the space of holomorphic differentials on the curve C. To define the Hodge bundle, let be the universal algebraic curve of genus g and let be its relative dualizing sheaf. The Hodge bundle is the pushforward of this sheaf, i.e.,
