The period-index conjecture is false

Abstract

For any uncountable algebraically closed field $k$ of characteristic $0$ and any $d\ge 3$, we construct a variety over $k$ of dimension $d$ with a Brauer class which violates the period-index conjecture for Hodge-theoretic reasons. When $d=3$, our construction works even without the assumption that $k$ is uncountable; in particular, the period-index conjecture fails over $\overline{\mathbf{Q}}$.

17 pages