Topology of Algebraic Varieties

Being the natural generalization of K3 surfaces, hyperkaehler varieties, also known as irreducible holomorphic symplectic varieties, are one of the building blocks of smooth projective varieties with trivial canonical bundle. One of the guiding conjectures in the study of algebraic cycles of such varieties is Beauville's splitting principle. Concerning the weak form of the splitting principle, I want to report some progress on the closely related Beauville-Voisin conjecture. As a continuation of the recent work of Vial, I will formulate a motivic version of Ruan's hyperkaehler crepant resolution conjecture and explain a work in progress for the Hilbert schemes of K3 surfaces.