Hodge bundle
title: "Hodge bundle" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["moduli-theory", "invariant-theory", "algebraic-curves"] topic_path: "general/moduli-theory" source: "https://en.wikipedia.org/wiki/Hodge_bundle" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
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{{Citation | last=van der Geer | first=Gerard | author-link = Gerard van der Geer | contribution=Siegel modular forms and their applications | editor-last=Ranestad | editor-first=Kristian | title=The 1-2-3 of modular forms | pages=181–245 (at §13) | isbn=978-3-540-74117-6 | doi=10.1007/978-3-540-74119-0 | publisher=Springer-Verlag | location=Berlin | year=2008 | series=Universitext | mr=2409679 }} and string theory.{{Citation | last=Liu | first=Kefeng | author-link = Kefeng Liu | contribution=Localization and conjectures from string duality | editor1-last=Ge | editor1-first=Mo-Lin | editor2-last=Zhang | editor2-first=Weiping | title=Differential geometry and physics | isbn=978-981-270-377-4 | publisher=World Scientific | year=2006 | series=Nankai Tracts in Mathematics | volume=10 | pages=63–105 (at §5) | mr=2322389
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 bundleHere, "vector bundle" in the sense of quasi-coherent sheaf on an algebraic stack 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.,{{Citation | last1=Harris | first1=Joe | author1-link=Joe Harris (mathematician) | last2=Morrison | first2=Ian | title=Moduli of curves | year=1998 | series=Graduate Texts in Mathematics | volume=187 | publisher=Springer-Verlag | isbn=978-0-387-98429-2 | doi=10.1007/b98867 | mr=1631825 | page=155 :\Lambda_g=\pi*\omega_g.
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