**Abstract:** A cubic hypersurface is a complex manifold realized as the zeros of a
cubic form in projective space. In 1969, Clemens and Griffiths applied Hodge theory to show that all cubic threefolds are irrational. On the other hand,
numerous examples of rational cubic fourfolds are known. In a series of recent papers, Voisin has revisited these questions using the technique of
decomposition of the diagonal.
We'll survey recent results applying these new ideas. |