The period-index conjecture is false for motivic reasons

Abstract

For any integer $d \geq 3$ and any algebraically closed field $k$ of characteristic $0$ and transcendence degree at least $d-3$, or of characteristic $p>2$ and transcendence degree at least $d-2$, we construct a $d$-dimensional variety over $k$ with a Brauer class of period $2$ and index $2^d$, violating the period-index conjecture; in particular, the period-index conjecture fails in dimension $3$ over $\overline{\mathbf{Q}}$ and $\overline{\mathbf{F}_p(t)}$, and in all dimensions $d \geq 3$ over $\mathbf{C}$ and $\overline{\mathbf{F}_p((t))}$. To bound the index of Brauer classes from below, we employ an obstruction of motivic nature, which requires proving the nonexistence of integral Hodge or Tate classes satisfying a certain equation modulo $2$. Motivated by these counterexamples, we propose a period-index conjecture with corrections at small primes and, as evidence, prove its Hodge-theoretic counterpart.