We prove that every Brauer class over a field splits over a torsor under an abelian variety. If the index of the class is not congruent to 2 modulo 4, we show that the Albanese variety of any smooth curve of positive genus that splits the class also splits the class. This can fail when the index is congruent to 2 modulo 4, but adding a single genus 1 factor to the Albanese suffices to split the class. This is joint work with Max Lieblich.