Derived categories
Let X
be a complex algebraic variety and let D(X) be the bounded derived
category of coherent sheaves on X. Two varieties X,Y for which
D(X)=D(Y) are called Fourier-Mukai partners.
Non-isomorphic Fourier-Mukai partners must have the same dimension and
the same divisibility of the canonical class. For surfaces it can be
shown that this phenomenon only happens for algebraic tori, K3-surfaces
and some elliptic surfaces.
For abelian varieties (of any dimension) Mukai, showed in [M1] that the
Fourier-transform (which connects an abelian variety and its dual) yields
Fourier-Mukai parters - whence the terminology. Mukai [Mu2] also showed
that certain moduli spaces Y of stable sheaves on K3-surfaces X are
Fourier-Mukai partners. Orlov [O] showed the converse: if X and Y are
Fourier-Mukai partners and X is either an abelian surface or a K3-surface,
then Y is isomorphic to a fine, two-dimensional moduli space of stable
sheaves on X. See the survey article [H-VdB].
Huybrechts, in a series
of articles [H1-2,H-M-S], links the Fourier-Mukai partners for
K3-surfaces with constructions from mathematical physics
(mirror-symmetry).
[H-VdB] Lutz Hille and Michel Van den Bergh: Fourier-Mukai Transforms
http://fr.arxiv.org/abs/math/0402043v2
[H1] Daniel Huybrechts: Generalized Calabi-Yau structures, K3 surfaces, and B-fields
http://fr.arxiv.org/abs/math/0306162
[H2]Daniel Huybrechts: Derived and abelian equivalence of K3 surfaces, http://fr.arxiv.org/abs/math/0604150
[H-M-S] Daniel Huybrechts, Emanuele Macri, Paolo Stellari: Derived equivalences of K3 surfaces and orientation http://fr.arxiv.org/abs/0710.1645
[O] D. O. Orlov: Equivalences of derived categories and K3 surfaces, J. Math.
Sci. (New York) 84 (1997), no. 5, 1361–1381, Algebraic geometry, 7.
[Mu1] S. Mukai: Duality between D(X) and D( ˆX) with its application to Picard
sheaves, Nagoya Math. J. 81 (1981), 153–175.
[Mu2] S. Mukai: On the moduli space of bundles on K3 surfaces. I, Vector bundles
on algebraic varieties (Bombay, 1984), Tata Inst. Fund. Res. Stud. Math.,
vol. 11, Tata Inst. Fund. Res., Bombay, 1987, pp. 341–413.