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An Introduction to the Hodge Conjecture for Abelian Varieties

Part of the Lecture Notes in Mathematics book series (LNMCIME,volume 1594)

Abstract

In this lecture we give a brief introduction to the Hodge conjecture for abelian varieties. We describe in some detail the abelian varieties of Weil-type. These are examples due to A. Weil of abelian varieties for which the Hodge conjecture is still open in general.

The Mumford-Tate groups are a very usefull tool for finding the Hodge classes in the cohomology of an abelian variety. We recall their main properties and illustrate it with an example.

Finally we discuss recent results on the Hodge conjecture for abelian fourfolds. Most of this material is well known, and we just hope to provide an easy going introduction.

Mathematics Subject Classification (1991):

  • 14C25
  • 14C30
  • 14D07
  • 19E15
  • 32I25

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© 1994 Springer-Verlag Berlin/Heidelberg

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van Geemen, B. (1994). An Introduction to the Hodge Conjecture for Abelian Varieties. In: Bardelli, F., Albano, A. (eds) Algebraic Cycles and Hodge Theory. Lecture Notes in Mathematics, vol 1594. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-49046-3_5

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  • DOI: https://doi.org/10.1007/978-3-540-49046-3_5

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-58692-0

  • Online ISBN: 978-3-540-49046-3

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