Hodge bundle
Appearance
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 formsonreductivealgebraic groups[1]andstring theory.[2]
Definition
[edit]Letbe themoduli space of algebraic curvesofgenusgcurves over somescheme.TheHodge bundleis avector bundle[note 1]onwhosefiberat a pointCinis the space ofholomorphic differentialson the curveC.To define the Hodge bundle, letbe theuniversalalgebraic curve of genusgand letbe itsrelative dualizing sheaf.The Hodge bundle is thepushforwardof this sheaf, i.e.,[3]
- .
See also
[edit]Notes
[edit]- ^Here, "vector bundle" in the sense ofquasi-coherent sheaf on an algebraic stack
References
[edit]- ^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,ISBN978-3-540-74117-6,MR2409679
- ^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),ISBN978-981-270-377-4,MR2322389
- ^Harris, Joe;Morrison, Ian (1998),Moduli of curves,Graduate Texts in Mathematics,vol. 187,Springer-Verlag,p. 155,doi:10.1007/b98867,ISBN978-0-387-98429-2,MR1631825