FourierMukai transforms for K3 and elliptic fibrations
Authors:
Tom Bridgeland and Antony Maciocia
Journal:
J. Algebraic Geom. 11 (2002), 629657
Published electronically:
March 18, 2002
MathSciNet review:
1910263
Fulltext PDF
Abstract 
References 
Additional Information
Abstract: Given a nonsingular variety with a K3 fibration we construct dual fibrations by replacing each fibre of by a twodimensional moduli space of stable sheaves on . In certain cases we prove that the resulting scheme is a nonsingular variety and construct an equivalence of derived categories of coherent sheaves . Our methods also apply to elliptic and abelian surface fibrations. As an application we use the equivalences to relate moduli spaces of stable bundles on elliptic threefolds to Hilbert schemes of curves.
 1.
S. Barannikov, M. Kontsevich, Frobenius manifolds and formality of Lie algebras of polyvector fields, Internat. Math. Res. Notices 4 201215 (1998), also alggeom 9710032.
 2.
A.I. Bondal, D.O. Orlov, Reconstruction of a variety from the derived category and groups of autoequivalences, Preprint (1997) alggeom 9712029.
 3.
C. Bartocci, U. Bruzzo, D. Hernández Ruipérez, J.M. Muñoz Porras, Mirror symmetry on K3 surfaces via FourierMukai transform, Comm. Math. Phys. 195 1 7993 (1998), also alggeom 9704023.
 4.
T. Bridgeland, FourierMukai transforms for elliptic surfaces, J. Reine Angew. Math. 498 115133 (1998), also alggeom 9705002.
 5.
T. Bridgeland, Equivalences of triangulated categories and FourierMukai transforms, Bull. London Math. Soc. 31 1 2534 (1999), also alggeom 9809114.
 6.
T. Bridgeland, A. Maciocia, Complex surfaces with equivalent derived categories, Math. Zeit. 236 4 677697 (2001).
 7.
T. Bridgeland, A. King, M. Reid, The McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc. 14 3 535554 (2001), also alggeom 9908027.
 8.
D. Eisenbud, Commutative algebra with a view toward algebraic geometry, G.T.M. 150, SpringerVerlag, New York (1995).
 9.
A. Grassi, Log contractions and equidimensional models of elliptic threefolds, J. Alg. Geom. 4 255276 (1995).
 10.
P. Griffith, E.G. Evans, Syzygies, L.M.S. Lecture Notes in Mathematics 106, C.U.P. (1985).
 11.
R. Hartshorne, Residues and Duality, Lect. Notes Math. 20, SpringerVerlag, Heidelberg (1966).
 12.
S. Mukai, Duality between and with its application to Picard sheaves, Nagoya Math. J. 81 153175 (1981).
 13.
S. Mukai, Symplectic structure of the moduli space of sheaves on an abelian or K3 surface, Invent. Math. 77 101116 (1984).
 14.
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), 341413.
 15.
D.O. Orlov, Equivalences of derived categories and K3 surfaces, J. Math. Sci. (NY), 84 5 13611381 (1997), also alggeom 9606006.
 16.
C. Okonek, M. Schneider, H. Spindler, Vector bundles on complex projective spaces, Progress in Mathematics 3, Birkhauser, Boston, Mass. (1980).
 17.
C.T. Simpson, Moduli of representations of the fundamental group of a smooth projective variety I, Publ. Math. I.H.E.S. 79 47129 (1994).
 18.
R. Thomas, A holomorphic Casson invariant for CalabiYau 3folds, and bundles on K3 fibrations, J. Diff. Geom. 54 367438 (2000), also alggeom 9806111.
 1.
 S. Barannikov, M. Kontsevich, Frobenius manifolds and formality of Lie algebras of polyvector fields, Internat. Math. Res. Notices 4 201215 (1998), also alggeom 9710032.
 2.
 A.I. Bondal, D.O. Orlov, Reconstruction of a variety from the derived category and groups of autoequivalences, Preprint (1997) alggeom 9712029.
 3.
 C. Bartocci, U. Bruzzo, D. Hernández Ruipérez, J.M. Muñoz Porras, Mirror symmetry on K3 surfaces via FourierMukai transform, Comm. Math. Phys. 195 1 7993 (1998), also alggeom 9704023.
 4.
 T. Bridgeland, FourierMukai transforms for elliptic surfaces, J. Reine Angew. Math. 498 115133 (1998), also alggeom 9705002.
 5.
 T. Bridgeland, Equivalences of triangulated categories and FourierMukai transforms, Bull. London Math. Soc. 31 1 2534 (1999), also alggeom 9809114.
 6.
 T. Bridgeland, A. Maciocia, Complex surfaces with equivalent derived categories, Math. Zeit. 236 4 677697 (2001).
 7.
 T. Bridgeland, A. King, M. Reid, The McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc. 14 3 535554 (2001), also alggeom 9908027.
 8.
 D. Eisenbud, Commutative algebra with a view toward algebraic geometry, G.T.M. 150, SpringerVerlag, New York (1995).
 9.
 A. Grassi, Log contractions and equidimensional models of elliptic threefolds, J. Alg. Geom. 4 255276 (1995).
 10.
 P. Griffith, E.G. Evans, Syzygies, L.M.S. Lecture Notes in Mathematics 106, C.U.P. (1985).
 11.
 R. Hartshorne, Residues and Duality, Lect. Notes Math. 20, SpringerVerlag, Heidelberg (1966).
 12.
 S. Mukai, Duality between and with its application to Picard sheaves, Nagoya Math. J. 81 153175 (1981).
 13.
 S. Mukai, Symplectic structure of the moduli space of sheaves on an abelian or K3 surface, Invent. Math. 77 101116 (1984).
 14.
 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), 341413.
 15.
 D.O. Orlov, Equivalences of derived categories and K3 surfaces, J. Math. Sci. (NY), 84 5 13611381 (1997), also alggeom 9606006.
 16.
 C. Okonek, M. Schneider, H. Spindler, Vector bundles on complex projective spaces, Progress in Mathematics 3, Birkhauser, Boston, Mass. (1980).
 17.
 C.T. Simpson, Moduli of representations of the fundamental group of a smooth projective variety I, Publ. Math. I.H.E.S. 79 47129 (1994).
 18.
 R. Thomas, A holomorphic Casson invariant for CalabiYau 3folds, and bundles on K3 fibrations, J. Diff. Geom. 54 367438 (2000), also alggeom 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
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
DOI:
http://dx.doi.org/10.1090/S105639110200317X
PII:
S 10563911(02)00317X
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
