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
