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.