Index reduction for Brauer classes via stable sheaves (with an appendix by Bhargav Bhatt)

We use twisted sheaves to study the problem of index reduction for Brauer classes. In general terms, this problem may be phrased as follows: given a field $k$, a $k$-variety $X$, and a class $\alpha \in \Br(k)$, compute the index of the class $\alpha_{k(X)} \in \Br(X)$ obtained from $\alpha$ by extension of scalars to $k(X)$. We give a general method for computing index reduction which refines classical results of Schofield and van den Bergh. When $X$ is a curve of genus 1, we use Atiyah's theorem on the structure of stable vector bundles with integral slope to show that our formula simplifies dramatically, giving a complete solution to the index reduction problem in this case. Using the twisted Fourier-Mukai transform, we show that a similarly simple formula describes homogeneous index reduction on torsors under higher-dimensional abelian varieties.