Irreducibility of Moduli Spaces of Vector Bundles on K3 Surfaces

Let X be a projective K3 surface defined over C and H an ample divisor on X. For a coherent sheaf E on X, v(E) := ch(E) √ tdX ∈ H∗(X,Z) is the Mukai vector of E, where tdX is the Todd class of X. We denote the moduli space of stable sheaves E of v(E) = v by MH(v). If v is primitive and H is general (i.e. H does not lie on walls [Y3]), then MH(v) is a smooth projective scheme. In [Mu1], Mukai showed that MH(v) has a symplectic structure. In order to get more precise information, Mukai [Mu2] introduced a quite useful notion called Mukai lattice (H∗(X,Z), 〈 , 〉), where 〈 , 〉 is an integral primitive bilinear form on H∗(X,Z). By the language of this lattice, we can write down Riemann-Roch theorem in a simple form. In particular we get that dimMH(v) = 〈v2〉+ 2. If v is a primitive isotropic vector, then MH(v) is a surface with a symplectic structure. Mukai proved that MH(v) is a K3 surface and described the period in terms of Mukai lattice. If v is a primitive Mukai vector of 〈v2〉 > 0, then MH(v) is a higher dimensional symplectic manifold. If rk v = 1, then MH(v) is the Hilbert scheme of points on X. Beauville [B] proved that it is an example of higher dimensional irreducible symplectic manifold. For an irreducible symplectic manifold, Beauville [B] defined the period and proved local Torelli theorem. As an example, he also computed the period of the Hilbert scheme of points on X. For higher rank cases, Mukai [Mu3] (rank 2 case), O’Grady [O] (any rank but primitive c1 case) and the author [Y4] (asymptotic case) proved that MH(v) is an irreducible symplectic manifold and described the period of MH(v) in terms of Mukai lattice. For classification of MH(v), it is important to determine the period. Indeed, it is a birational invariant ([Mu3]), and affirmative solution of Torelli conjecture will imply that an irreducible symplectic manifold is determined by its period, up to birational equivalence. In this paper, by using [Y4] extensively, we prove the following theorem, which is expected by many people (for example, see [D], [Mu3], [O]).