Stable Sheaves Over K3 Fibrations

TL;DR

This paper develops a method to construct and analyze stable sheaves over K3 fibrations using a relative Fourier-Mukai transform, revealing their structure and moduli space properties.

Contribution

It introduces a spectral data approach for stable sheaves on K3 fibrations and shows how the Fourier-Mukai transform embeds the Jacobian into the sheaf moduli space.

Findings

Spectral sheaves form a torus fibration over the moduli space of curves.

The Fourier-Mukai transform provides an embedding of the Jacobian into the moduli space.

The moduli space of spectral sheaves is described as a generic torus fibration.

Abstract

We construct stable sheaves over K3 fibrations using a relative Fourier-Mukai transform which describes the sheaves in terms of spectral data similar to the construction for elliptic fibrations. On K3 fibered Calabi-Yau threefolds we show that the Fourier-Mukai transform induces an embedding ion 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.