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^[1] and string theory.^[2]

Definition

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

\Lambda _{g}=\pi _{*}\omega _{g}

.

^ van der Geer, Gerard (2008), "Siegel modular forms and their applications", in Ranestad, Kristian (ed.), The 1-2-3 of modular forms, Universitext, Berlin: Springer-Verlag, pp. 181–245 (at §13), doi:10.1007/978-3-540-74119-0, ISBN 978-3-540-74117-6, MR 2409679
^ Liu, Kefeng (2006), "Localization and conjectures from string duality", in Ge, Mo-Lin; Zhang, Weiping (eds.), Differential geometry and physics, Nankai Tracts in Mathematics, vol. 10, World Scientific, pp. 63–105 (at §5), ISBN 978-981-270-377-4, MR 2322389
^ Harris, Joe; Morrison, Ian (1998), Moduli of curves, Graduate Texts in Mathematics, vol. 187, Springer-Verlag, p. 155, doi:10.1007/b98867, ISBN 978-0-387-98429-2, MR 1631825