Hodge bundle

Inmathematics,theHodge bundle,named afterW. V. D. Hodge,appears in the study of families ofcurves,where it provides aninvariantin themoduli theoryofalgebraic curves.Furthermore, it has applications to the theory ofmodular formsonreductive algebraic groups^[1]andstring theory.^[2]

Definition

Let ${\mathcal {M}}_{g}$ be themoduli space of algebraic curvesofgenusgcurves over somescheme.TheHodge bundle $\Lambda _{g}$ is avector bundle^{[note 1]}on ${\mathcal {M}}_{g}$ whosefiberat a pointCin ${\mathcal {M}}_{g}$ is the space ofholomorphic differentialson the curveC.To define the Hodge bundle, let $\pi \colon {\mathcal {C}}_{g}\rightarrow {\mathcal {M}}_{g}$ be theuniversalalgebraic curve of genusgand let $\omega _{g}$ be itsrelative dualizing sheaf.The Hodge bundle is thepushforwardof 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