Abstract
Historically, one of the main motivations for the development of Hodge theory was the study of cycles. This certainly was one of the principal preoccupations of Sir William Hodge who stated his famous conjecture that algebraic cycles can be detected in cohomology by looking at the integral classes having pure Hodge type. In this chapter we shall explain this as well as Grothendieck's generalization. To state the latter requires certain subtle properties implied by the existence of functorial mixed Hodge structures on possibly singular and non-compact algebraic varieties derived in the previous chapters. This can be found in § 7.1. Intermediate Jacobians find their natural place in this section.
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). Applications to Algebraic Cycles and to Singularities. 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_8
Download citation
DOI: https://doi.org/10.1007/978-3-540-77017-6_8
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)