International Congress of Mathematicians

496 Shigeru MukaiIn higher rank case it is natural to consider the moduli problem of F underthe restriction that det F is unchanged. In view of the above diagram, vectorbundles and their moduli reflect the geometry of the morphism X —y P*H°(L)via Grassmannians and Pliicker relations. In this article we consider the case whereX is a K3 surface, which is one of two 2-dimensional analogues of an elliptic curveand seems an ideal place to see such reflection.2. Curves of genus oneThe moduli space of line bundles on an algebraic variety is called the Picardvariety. The Picard variety Pic C of an algebraic curve C is decomposed into thedisjoint union IJ d€Z Picd C by the degree d of line bundles. Here we consider thecase of genus 1. All components Pic^G are isomorphic to G if the ground field1is algebraically closed. But this is no more true otherwise. For example theJacobian Pico G has always a rational point but G itself does not. 2 We give otherexamples:Example 1 Let G4 be an intersection of two quadrics qi (x) = q 2 (x) = 0 in theprojective space P 3 and F the pencil of defining quadrics. Then the Picard varietyPÌC2 G4 is the double cover of F ~ P 1 and the branch locus consists of 4 singularquadrics in P. Precisely speaking, its equation is given by r 2 = disc (Xiqi + X2q2).Let G(2,5) C P 9 be the 6-dimensional Grassmann variety embedded into P 9by the Pliicker coordinate. Its projective dual is the dual Grassmannian G(5,2) cP 9 , where G(2,5) parameterizes 2-dimensional subspaces and G(5,2) quotient spaces.Example 2 A transversal linear section G = G(2,5) n Hi n • • • n H 5 is a curve genus1 and of degree 5. Its Picard variety PÌC2 G is isomorphic to the dual linear sectionG = G(5,2) n {Hi,... ,H 5 ), the intersection with the linear subspace spanned by 5points Hi,... ,H 5 £ P 9 .3. Moduli K3 surfacesA compact complex 2-dimensional manifold S is a K3 surface if the canonicalbundle is trivial and the irregularity vanishes, that is, Kg = F 1 (ös) = 0. Asmooth quartic surface S4 C P 3 is the most familiar example. Let us first look atthe 2-dimensional generalization of Example 1:Example 3 Let Ss be an intersection of three general quadrics in P 5 and N thenet of defining quadrics. Then the moduli space Ms(2,ös(l),2) is a double coverof N ~ P 2 and the branch locus, which is of degree 6, consists of singular quadricsin N.Here Ms(r, L,s), L being a line bundle, is the moduli space of stable sheavesF on a K3 surface S with rank r, det F ~ F and x(E) = r + s. Surprisingly two1 More precisely, this holds true if C has a rational point.2 Two components Pico C and Pic 9 _i C deserve the name Jacobian. They coincide in our case9=1-