Abstract
The cohomology groups Hk(X t ) of compact Kähler manifolds X t which vary in a smooth family over a complex base manifold S define a local system over S and the varying Hodge ags form the prototype of a variation of Hodge structure. These satisfy certain axioms which have been verified by Griffiths([Grif68]): the Hodge ags vary holomorphically and Griffith's transversality holds: the natural at connection shifts the index of the ags back by at most 1. The variations coming from families of compact Kähler manifolds are called geometric variations. In §10.4 we discuss these and show that the local system defined by the cohomology of the fibres of such a family indeed underlies a variation of Hodge structure.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
(2008). Variations of Hodge Structure. In: Mixed Hodge Structures. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol 52. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77017-6_11
Download citation
DOI: https://doi.org/10.1007/978-3-540-77017-6_11
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-77015-2
Online ISBN: 978-3-540-77017-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)