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.
Let be the moduli space of algebraic curves of genus g curves over some scheme. The Hodge bundle Λg 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 ωg be its relative dualizing sheaf. The Hodge bundle is the pushforward of this sheaf, i.e.
- van der Geer, Gerard (2008), "Siegel modular forms and their applications", in Ranestad, Kristian, The 1-2-3 of modular forms, Universitext, Berlin: Springer-Verlag, pp. 181–245, doi:10.1007/978-3-540-74119-0, ISBN 978-3-540-74117-6, MR 2409679
- Harris, Joe; Morrison, Ian (1998), Moduli of curves, Graduate Texts in Mathematics, 187, Springer-Verlag, ISBN 978-0-387-98429-2, MR 1631825
- Liu, Kefeng (2006), "Localization and conjectures from string duality", in Ge, Mo-Lin; Zhang, Weiping, Differential geometry and physics, Nankai Tracts in Mathematics, 10, World Scientific, pp. 63–105, ISBN 978-981-270-377-4, MR 2322389