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An Introduction to Hodge Structures

  • Sara Angela Filippini
  • Helge RuddatEmail author
  • Alan ThompsonEmail author
Part of the Fields Institute Monographs book series (FIM, volume 34)

Abstract

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 Kähler 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.

Notes

Acknowledgements

A part of these notes was written while the authors were in residence at the Fields Institute Thematic Program on Calabi-Yau Varieties: Arithmetic, Geometry and Physics; we would like to thank the Fields Institute for their support and hospitality. 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.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Institut für MathematikUniversität ZürichZürichSwitzerland
  2. 2.Mathematisches InstitutUniversität MainzMainzGermany
  3. 3.Department of Pure MathematicsUniversity of WaterlooWaterlooCanada

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