Fourier-Mukai transforms for K3 and elliptic fibrations

Abstract

Given a non-singular variety with a K3 fibration f : X --> S we construct dual fibrations Y --> S by replacing each fibre X_s of f by a two-dimensional moduli space of stable sheaves on X_s. In certain cases we prove that the resulting scheme Y is a non-singular variety and construct an equivalence of derived categories of coherent sheaves \Phi : D(Y) --> D(X). Our methods also apply to elliptic and abelian surface fibrations. As an application we show how the equivalences \Phi identify certain moduli spaces of stable bundles on elliptic threefolds with Hilbert schemes of curves.