Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Fourier-Mukai transforms for K3 and elliptic fibrations

Authors: Tom Bridgeland and Antony Maciocia
Journal: J. Algebraic Geom. 11 (2002), 629-657
Published electronically: March 18, 2002
MathSciNet review: 1910263
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Abstract | References | Additional Information

Abstract: Given a non-singular variety with a K3 fibration $\pi\colon X\to S$we construct dual fibrations $\Hat{\pi}\colon Y\to S$ by replacing each fibre $X_s$ of $\pi$ 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 $\Phi\colon \operatorname{D}(Y)\to \operatorname{D}(X)$. Our methods also apply to elliptic and abelian surface fibrations. As an application we use the equivalences $\Phi$ to relate moduli spaces of stable bundles on elliptic threefolds to Hilbert schemes of curves.

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Additional Information

Tom Bridgeland
Affiliation: Department of Mathematics and Statistics, The University of Edinburgh, King’s Buildings, Mayfield Road, Edinburgh, EH9 3JZ, United Kingdom

Antony Maciocia
Affiliation: Department of Mathematics and Statistics, The University of Edinburgh, King’s Buildings, Mayfield Road, Edinburgh, EH9 3JZ, United Kingdom

Received by editor(s): May 23, 2000
Published electronically: March 18, 2002
Additional Notes: This research was carried out with the support of the Engineering and Physical Sciences Research Council of Great Britain