Abstract
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 α ∈ Br(k), compute the index of the class α k(X) ∈ Br(X) obtained from α 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.
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