Distinguished cycles on hyperKaehler varieties

The aim of this talk is to give an overview of various subrings of the Chow ring of hyperKaehler varieties, which like the Beauville-Voisin ring conjecturally inject in cohomology via the cycle class map. I will give examples of hyperKaehler varieties for which such injectivity statements are known to be true and I will explain how these various subrings relate to each other (conjecturally and in examples). I will also concomitantly draw the parallel picture for abelian varieties. This is based on works with Lie Fu, Robert Laterveer and Mingmin Shen.