International Congress of Mathematicians

Vector Bundles on a K3 Surface 501action. In the above construction, we take the quotient of Alt^2, which is nothingbut the affine variety of Gieseker matrices of suitable rank 2 vector bundles, by ageneral linear group GF(4).The Jacobian, or the Picard variety, of a curve is more fundamental. Weil [27]constructed Pic s G as an algebraic variety using the symmetric product Sym s Gand showed its projectivity by Lefschetz' 30 theorem. Later Seshadri and Oda[24] constructed Pic^ G for arbitray d (over the same ground field as G) by alsotaking the GIT quotient of Quot schemes. The above constructions eliminate Quotschemes and the concept of linearization from those of Gieseker, Seshadri and Oda.References[i[2[io;[n[12;[13;[14;K. Abe: A remark on the 2-dimensional moduli spaces of vector bundles onK3 surfaces, Math. Res. Letters, 7(2000), 463^470.E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris : Geometry of algebraiccurves, I, Springer-Verlag, 1985.M. F. Atiyah : Vector bundles over an elliptic curve, Proc. London Math.Soc, 7(1957), 414-452.T. Bridgeland: Equivalences of triangulated categories and Fourier-Mukaitransformations, Bull. London Math. Soc, 31(1999), 25-34.A. Cäldäraru: Non-fine moduli spaces of sheaves on K3 surfaces, preprint.D. Gieseker: On the moduli of vector bundles on an algebraic surface, Ann.Math. 106(1977), 45-60.A. Grothendieck : Techniques de construction et theorem d'existence engéométrie algébrique, iV: Les schémas de Hilbert, Sem. Bourbaki, t. 13,1960/61, n° 221.V.A. Iskovskih : Fano 3-folds, II, Izv. Akad. Nauk SSSR,42(1978) : Englishtranslation, Math. USSR Izv. 12(1978), 469^505.S. Mori and S. Mukai: The uniruledness of the moduli space of curves of genus11, in 'Algebraic Geometry, Proceedings, Tokyo/Kyoto 1982', Series: LectureNotes in Mathematics, vol. 1016, (M, Raynaud and T. Shioda eds.), SpringerVerlag, 1983, 334^353.S. Mukai: Symplectic structure of the moduli space of sheaves on an abelianor K3 surface, Invent. Math., 77(1984), 10H16.—: On the moduli space of bundles on K3 surfaces, I, in 'Vector Bundles onAlgebraic Varieties ', Tata Institute of Fundamental Research, Bombay, 1987,341-413.— : Curves, K3 surfaces and Fano manifolds of genus < 10, in 'AlgebraicGeometry and Commutative Algebra in honor of Masayoshi NAGATA', (H.Hijikata and H. Hironaka et al eds.), Kinokuniya, Tokyo, 1988, 367^377.— : Biregular classification of Fano threefolds and Fano manifolds of coindex3, Proc. Nat. Acad. Sci., USA, 86 (1989), 3000^3002.— : New developments in the theory of Fano 3-folds: Vector bundle methodand moduli problem, Sugaku, 47(1995), 125^144.: English translation, SugakuExpositions, to appear.