Kuga-Satake varieties and the Hodge conjecture

Details Kuga-Satake varieties and the Hodge conjecture Kuga-Satake varieties and the Hodge conjecture

1999-03-24

arxiv.org/abs/math/9903146v1

Mathematics

Algebraic Geometry

23 pages LaTeX. To appear in: The Arithmetic and Geometry of Algebraic Cycles. Eds. J. Lewis et al. (Proceedings of the NATO A

Scientific paper

This paper gives an introduction to Kuga-Satake varieties and discusses some aspects of the Hodge conjecture related to them. Kuga-Satake varieties are abelian varieties associated to certain weight two Hodge structures, for example the second cohomology group of a K3 surface. We give a detailed account of the construction of Kuga-Satake varieties and of their decomposition in simple subvarieties. We recall the Hodge conjecture and we point out a connection between the Hodge conjecture for abelian fourfolds and Kuga-Satake varieties. We discuss the implications of the Hodge conjecture on the geometry of surfaces whose second cohomology group has a Kuga-Satake variety. We conclude with some recent results on Kuga-Satake varieties of Hodge structures on which an imaginary quadratic field acts.

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