Hodge bundle

Last updated July 01, 2019

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]

Mathematics Field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure, space, and change. It has no generally accepted definition.

Sir William Vallance Douglas Hodge was a British mathematician, specifically a geometer.

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In mathematics, a curve is, generally speaking, an object similar to a line but that need not be straight. Thus, a curve is a generalization of a line, in that it may be curved.

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]

In mathematics, genus has a few different, but closely related, meanings. The most common concept, the genus of an (orientable) surface, is the number of "holes" it has, so that a sphere has genus 0 and a torus has genus 1. This is made more precise below.

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

.

Notes

↑ Here, "vector bundle" in the sense of quasi-coherent sheaf on an algebraic stack

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References

↑ 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, 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, 187, Springer-Verlag, p. 155, doi:10.1007/b98867, ISBN 978-0-387-98429-2, MR 1631825

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[3] Here, "vector bundle" in the sense of quasi-coherent sheaf on an algebraic stack

[1] 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

[2] Liu, Kefeng (2006), "Localization and conjectures from string duality", in Ge, Mo-Lin; Zhang, Weiping (eds.), Differential geometry and physics, Nankai Tracts in Mathematics, 10, World Scientific, pp. 63–105 (at §5), ISBN 978-981-270-377-4, MR 2322389

[4] Harris, Joe; Morrison, Ian (1998), Moduli of curves, Graduate Texts in Mathematics, 187, Springer-Verlag, p. 155, doi:10.1007/b98867, ISBN 978-0-387-98429-2, MR 1631825

Hodge bundle

Contents

Definition

See also

Notes

Related Research Articles

References