International Congress of Mathematicians

Vector Bundles on a K3 Surface 497surfaces S$ = (2) n (2) n (2) c P 5 and Af s (2,O s (l),2) ^h P 2 in this exampleare both K3 surfaces. This is not an accident. In respect of moduli space, vectorbundles a K3 surface look like Picard varieties in the preceding section.Theorem 1 ([10],[11]) The moduli space M$(r, L, s) is smooth of dimension (L 2 ) —2rs + 2. Ms(r, L, s) is again a K3 surface if it is compact and and of dimension 2.A K3 surface S and a moduli K3 surface appearing as Ms(r,L,s) are notisomorphic in general 3 but their polarized Hodge structures, or periods, are isomorphicto each other over Q ([11]). The moduli is not always fine but there alwaysexists a universal P r_1 -bundle over the product S x Ms(r,L,s). Yet A be theassociated sheaf of Azumaya algebras, which is of rank r 2 and locally isomorphic tothe matrix algebra Mat r (OsxM)- A is isomorphic to End E if a universal family Eexists. The Hodge isometry between H 2 (S,Q) and H 2 (Ms(r,L,s),Q) is given byc 2 (-4)/2r G H 4 (S x Af,Q) ~ H 2 (S,Q) V ® H 2 (M,Q).Example 2 has also a K3 analogue. Let Si 2 C P 15 be a 10-dimensionalspinor variety SO(10)/U(5), that is, the orbit of a highest weight vector in theprojectivization of the 16-dimensional spinor representation. The anti-canonicalclass is 8 times the hyperplane section and a transversal linear section S = £12 nFi n • • • n H$ is a K3 surface (of degree 12). As is similar to G(2,5) the projectivedual Si2 C P 15 of Si2 is again a 10-dimensional spinor variety.Example 4 The moduli space Ms(2,Os(l),3)section S = £12 n {Hi,..., H$).is isomorphic to the dual linearIn this case, moduli is fine and the relation between S and the moduli K3 aredeeper. The universal vector bundle E on the product gives an equivalence betweenthe derived categories T)(S) and T)(S) of coherent sheaves, the duality S ~ S holds(cf. [17]) and moreover the Hilbert schemes HihV S and Hilb" S are isomorphic toeach other.Remark (1) Theorem 1 is generalized to the non-compact case by Abe [1].(2) If a universal family E exists, the derived functor with kernel E gives an equivalenceof derived categories of coherent sheaves on S and the moduli K3 (Bridgeland[4]). In even non-fine case the derived category D(S) is equivalent to that of themoduli K3 M twisted by a certain element a £ H 2 (M,Z/rZ) (Cäldäraru [5]).4. Shafarevieh conjectureLet S and T be algebraic K3 surfaces and / a Hodge isometry betweenH 2 (S, Z) and H 2 (T,Z). Then the associated cycle Z f £ H 4 (SxT,Z) ~H 2 (S,Z) V ®H 2 (T, Q) on the product S x T is a Hodge cycle. This is algebraic by virtue of theTorelli type theorem of Shafarevieh and Piatetskij-Shapiro. Shafarevieh conjecturedin [23] a generalization to Hodge isometries over Q. Our moduli theory is able toanswer this affirmatively.3 We take the complex number field C as ground field except for sections 2 and 7.