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An introduction to Hodge structures

journal contribution
posted on 08.10.2018, 12:40 by Sara Angela Filippini, Helge Ruddat, Alan ThompsonAlan Thompson
We begin by introducing the concept of a Hodge structure and give some of its basic properties, including the Hodge and Lefschetz decompositions. We then define the period map, which relates families of Kahler manifolds to the families of Hodge structures defined on their cohomology, and discuss its properties. This will lead us to the more general definition of a variation of Hodge structure and the Gauss-Manin connection. We then review the basics about mixed Hodge structures with a view towards degenerations of Hodge structures; including the canonical extension of a vector bundle with connection, Schmid's limiting mixed Hodge structure and Steenbrink's work in the geometric setting. Finally, we give an outlook about Hodge theory in the Gross-Siebert program.

Funding

A. Thompson was supported by a Fields-Ontario-PIMS postdoctoral fellowship with funding provided by NSERC, the Ontario Ministry of Training, Colleges and Universities, and an Alberta Advanced Education and Technology Grant.

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Fields Institute Monographs

Volume

34

Pages

83 - 130

Citation

FILIPPINI, S.A., RUDDAT, H. and THOMPSON, A., 2015. An Introduction to Hodge Structures. IN: Laza, R., Schutt, M. and Yui, N. (eds.) Calabi-Yau Varieties: Arithmetic, Geometry and Physics. Lecture Notes on Concentrated Graduate Courses. New York: Springer, pp. 83-130.

Publisher

© Springer

Version

AM (Accepted Manuscript)

Publisher statement

This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/

Publication date

2015

Notes

This book chapter is in closed access.

ISBN

9781493928293;9781493928309

ISSN

1069-5273

Book series

Fields Institute Monographs;34

Language

en

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