Abstract
We construct stable sheaves over K3 fibrations using a relative Fourier‐Mukai transform which describes the sheaves in terms of spectral data. This procedure is similar to the construction for elliptic fibrations, which we also describe. On K3 fibered Calabi‐Yau threefolds, we show that the Fourier‐Mukai transform induces an embedding of the relative Jacobian of spectral line bundles on spectral covers into the moduli space of sheaves of given invariants. This makes the moduli space of spectral sheaves to a generic torus fibration over the moduli space of curves of given arithmetic genus on the Calabi‐Yau manifold.
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