  Conditional on the Lefschetz standard conjecture in degree 2, we prove that
the index of a Brauer class on a smooth projective variety divides a fixed
power of its period, uniformly in smooth families. In the other direction, we
reinterpret in more classical terms recent work of Hotchkiss which gives
Hodge-theoretic lower bounds on the index of Brauer classes. We also prove
versions of our results over arbitrary algebraically closed base fields, and as
an application construct qualitatively new counterexamples to the integral Tate
conjecture.
