The Brauer group is a fundamental invariant of algebraic varieties and can be viewed as a higher analogue of the Picard group. Associated with any Brauer class are two invariants, the period and the index, whose relationship is the subject of the period-index problem. For hyper-Kähler varieties, whose geometry is governed by the second cohomology, one expects an even stronger form of this conjecture to hold. In this talk, I will present joint work with Daniel Huybrechts that provides new evidence for this expectation.

After a brief introduction to the problem, I will discuss a proof of a variant of the conjecture in which the classical index is replaced by a Hodge-theoretic analogue. I will then explain how to verify the conjecture for most Brauer classes on hyper-Kähler varieties of K3n-type and OG10-type. I will also discuss the impact of the recent AI assisted counterexamples on the conjecture, and possible adaptations both in the general case and in the hyper-Kähler case.