Kuga-satake Varieties and the Hodge Conjecture

Kuga-Satake varieties are abelian varieties associated to certain weight two Hodge structures, for example the second cohomology group of a K3 surface. We start with an introduction to Hodge structures and we give a detailed account of the construction of Kuga-Satake varieties. The Hodge conjecture is discussed in section 2. An excellent survey of the Hodge conjecture for abelian varieties is [G]. We point out a connection between the Hodge conjecture for abelian varieties and KugaSatake varieties in section 9. In section 10 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, inspired by an example of C. Voisin, on Kuga-Satake varieties of Hodge structures on which an imaginary quadratic field acts. I’m indebted to E. Izadi, G. Lombardo and M. Nori for helpful discussions.