Fourier-Mukai transforms for K3 and elliptic fibrations

Authors:
Tom Bridgeland and Antony Maciocia

Journal:
J. Algebraic Geom. **11** (2002), 629-657

DOI:
https://doi.org/10.1090/S1056-3911-02-00317-X

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.

BK S. Barannikov, M. Kontsevich, Frobenius manifolds and formality of Lie algebras of polyvector fields, Internat. Math. Res. Notices **4** 201-215 (1998), also alg-geom 9710032.
BO A.I. Bondal, D.O. Orlov, Reconstruction of a variety from the derived category and groups of autoequivalences, Preprint (1997) alg-geom 9712029.
BB C. Bartocci, U. Bruzzo, D. Hernández Ruipérez, J.M. Muñoz Porras, Mirror symmetry on K3 surfaces via Fourier-Mukai transform, Comm. Math. Phys. **195** 1 79-93 (1998), also alg-geom 9704023.
Br1 T. Bridgeland, Fourier-Mukai transforms for elliptic surfaces, J. Reine Angew. Math. **498** 115-133 (1998), also alg-geom 9705002.
Br2 T. Bridgeland, Equivalences of triangulated categories and Fourier-Mukai transforms, Bull. London Math. Soc. **31** 1 25-34 (1999), also alg-geom 9809114.
Br3 T. Bridgeland, A. Maciocia, Complex surfaces with equivalent derived categories, Math. Zeit. **236** 4 677-697 (2001).
BKR T. Bridgeland, A. King, M. Reid, The McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc. **14** 3 535-554 (2001), also alg-geom 9908027.
Ei D. Eisenbud, Commutative algebra with a view toward algebraic geometry, G.T.M. **150**, Springer-Verlag, New York (1995).
Gr A. Grassi, Log contractions and equidimensional models of elliptic threefolds, J. Alg. Geom. **4** 255-276 (1995).
Syz P. Griffith, E.G. Evans, Syzygies, L.M.S. Lecture Notes in Mathematics **106**, C.U.P. (1985).
Ha R. Hartshorne, Residues and Duality, Lect. Notes Math. **20**, Springer-Verlag, Heidelberg (1966).
Muk1 S. Mukai, Duality between $\mathrm {D}(X)$ and $\mathrm {D}(\hat {X})$ with its application to Picard sheaves, Nagoya Math. J. **81** 153-175 (1981).
Muk2 S. Mukai, Symplectic structure of the moduli space of sheaves on an abelian or K3 surface, Invent. Math. **77** 101-116 (1984).
Muk3 S. Mukai, On the moduli space of bundles on K3 surfaces I, in: Vector Bundles on Algebraic Varieties, M.F. Atiyah et al., Oxford University Press (1987), 341-413.
O D.O. Orlov, Equivalences of derived categories and K3 surfaces, J. Math. Sci. (NY), **84** 5 1361-1381 (1997), also alg-geom 9606006.
OSS C. Okonek, M. Schneider, H. Spindler, Vector bundles on complex projective spaces, Progress in Mathematics **3**, Birkhauser, Boston, Mass. (1980).
Si C.T. Simpson, Moduli of representations of the fundamental group of a smooth projective variety I, Publ. Math. I.H.E.S. **79** 47-129 (1994).
Th R. Thomas, A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on K3 fibrations, J. Diff. Geom. **54** 367-438 (2000), also alg-geom 9806111.

BK S. Barannikov, M. Kontsevich, Frobenius manifolds and formality of Lie algebras of polyvector fields, Internat. Math. Res. Notices **4** 201-215 (1998), also alg-geom 9710032.
BO A.I. Bondal, D.O. Orlov, Reconstruction of a variety from the derived category and groups of autoequivalences, Preprint (1997) alg-geom 9712029.
BB C. Bartocci, U. Bruzzo, D. Hernández Ruipérez, J.M. Muñoz Porras, Mirror symmetry on K3 surfaces via Fourier-Mukai transform, Comm. Math. Phys. **195** 1 79-93 (1998), also alg-geom 9704023.
Br1 T. Bridgeland, Fourier-Mukai transforms for elliptic surfaces, J. Reine Angew. Math. **498** 115-133 (1998), also alg-geom 9705002.
Br2 T. Bridgeland, Equivalences of triangulated categories and Fourier-Mukai transforms, Bull. London Math. Soc. **31** 1 25-34 (1999), also alg-geom 9809114.
Br3 T. Bridgeland, A. Maciocia, Complex surfaces with equivalent derived categories, Math. Zeit. **236** 4 677-697 (2001).
BKR T. Bridgeland, A. King, M. Reid, The McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc. **14** 3 535-554 (2001), also alg-geom 9908027.
Ei D. Eisenbud, Commutative algebra with a view toward algebraic geometry, G.T.M. **150**, Springer-Verlag, New York (1995).
Gr A. Grassi, Log contractions and equidimensional models of elliptic threefolds, J. Alg. Geom. **4** 255-276 (1995).
Syz P. Griffith, E.G. Evans, Syzygies, L.M.S. Lecture Notes in Mathematics **106**, C.U.P. (1985).
Ha R. Hartshorne, Residues and Duality, Lect. Notes Math. **20**, Springer-Verlag, Heidelberg (1966).
Muk1 S. Mukai, Duality between $\mathrm {D}(X)$ and $\mathrm {D}(\hat {X})$ with its application to Picard sheaves, Nagoya Math. J. **81** 153-175 (1981).
Muk2 S. Mukai, Symplectic structure of the moduli space of sheaves on an abelian or K3 surface, Invent. Math. **77** 101-116 (1984).
Muk3 S. Mukai, On the moduli space of bundles on K3 surfaces I, in: Vector Bundles on Algebraic Varieties, M.F. Atiyah et al., Oxford University Press (1987), 341-413.
O D.O. Orlov, Equivalences of derived categories and K3 surfaces, J. Math. Sci. (NY), **84** 5 1361-1381 (1997), also alg-geom 9606006.
OSS C. Okonek, M. Schneider, H. Spindler, Vector bundles on complex projective spaces, Progress in Mathematics **3**, Birkhauser, Boston, Mass. (1980).
Si C.T. Simpson, Moduli of representations of the fundamental group of a smooth projective variety I, Publ. Math. I.H.E.S. **79** 47-129 (1994).
Th R. Thomas, A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on K3 fibrations, J. Diff. Geom. **54** 367-438 (2000), also alg-geom 9806111.

Additional Information

**Tom Bridgeland**

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

MR Author ID:
635821

ORCID:
0000-0001-5120-006X

Email:
tab@maths.ed.ac.uk

**Antony Maciocia**

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

Email:
A.Maciocia@ed.ac.uk

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