Counterexamples to the integral Hodge conjecture

The Hodge conjecture predicts which rational cohomology classes on a smooth complex projective variety can be represented by linear combinations of complex subvarieties. The integral Hodge conjecture, the analogous conjecture for integral homology classes, is known to be false in general (the first counterexamples were given in dimension $7$ by Atiyah and Hirzebruch). I’ll survey some of the known results on this conjecture, and then present some new counterexamples. This is joint work with Olivier Benoist.