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 formula_1 be the moduli space of algebraic curves of genus "g" curves over some scheme. The Hodge bundle formula_2 is a vector bundle on formula_1 whose fiber at a point "C" in formula_1 is the space of holomorphic differentials on the curve "C". To define the Hodge bundle, let formula_5 be the universal algebraic curve of genus "g" and let formula_6 be its relative dualizing sheaf. The Hodge bundle is the pushforward of this sheaf, i.e.,