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In [[mathematics]], the '''Hodge bundle''', named after [[W. V. D. Hodge]], appears in the study of families of [[curves]], where it provides an [[invariant (mathematics)|invariant]] in the [[moduli theory]] of [[algebraic curve]]s. Furthermore, it has applications to the theory of [[modular form]]s on [[reductive group|reductive]] [[algebraic groups]]<ref>{{harvnb|van der Geer|2008|loc=§13}}</ref> and [[string theory]].<ref>{{harvnb|Liu|2006|loc=§5}}</ref>

Let <math>\mathcal{M}_g</math> be the [[moduli space of algebraic curves]] of [[genus (mathematics)|genus]] ''g'' curves over some [[scheme (algebraic geometry)|scheme]]. The '''Hodge bundle''' Λ<sub>''g''</sub> is a [[vector bundle]] on <math>\mathcal{M}_g</math> whose [[vector bundle|fiber]] at a point ''C'' in <math>\mathcal{M}_g</math> is the space of [[holomorphic differential]]s on the curve ''C''. To define the Hodge bundle, let <math>\pi:\mathcal{C}_g\rightarrow\mathcal{M}_g</math> be the [[universal object|universal]] algebraic curve of genus ''g'' and let ω<sub>''g''</sub> be its [[relative dualizing sheaf]]. The Hodge bundle is the [[pushforward]] of this sheaf, i.e.<ref>{{harvnb|Harris|Morrison|1998|p=155}}</ref>
:<math>\Lambda_g=\pi_*\omega_g.\,</math>

==See also==
*[[ELSV formula]]

==Notes==
{{reflist}}

==References==
*{{Citation
| last=van der Geer
| first=Gerard
| contribution=Siegel modular forms and their applications
| editor-last=Ranestad
| editor-first=Kristian
| title=The 1-2-3 of modular forms
| pages=181–245
| 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
}}
*{{Citation
| last=Harris
| first=Joe
| 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
| mr=1631825
}}
*{{Citation
| last=Liu
| first=Kefeng
| contribution=Localization and conjectures from string duality
| editor-last=Ge
| editor-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
| mr=2322389
}}

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