Discoveries / algebra

The Period-Index Conjecture

For a Brauer class on a variety, the period-index conjecture bounds the index in terms of the period and the dimension. Disproved: for any uncountable algebraically closed field $k$ of characteristic $0$ and any $d \geq 3$ there is a $d$-dimensional variety over $k$ carrying a Brauer class that violates it, for Hodge-theoretic reasons. For $d = 3$ the construction needs no uncountability, so the conjecture fails already over $\overline{\mathbf{Q}}$.