We prove stability of rank two tautological bundles on the Hilbert square of a surface (under a mild positivity condition) and compute their Chern classes. Let S be a smooth, complex projective surface, let Hilb(S) be the Hilbert scheme parametrizing subschemes of S of length 2. It is known by a classical theorem of Fogarty [Fo] that Hilb(S) is a smooth, projective variety of dimension 4. Let Z ⊂ S × Hilb(S) be the universal subscheme, denote by p : Z → S and by q : Z → Hilb(S) the projections. Given a line bundle L on S, the sheaf L[2] := q∗pL is a rank two vector bundle on Hilb(S), called the tautological vector bundle associated with L. In this note we prove the following Theorem. Assume that h0(S,L) ≥ 2. Then for N 0, the vector bundle L[2] is μHN -stable on Hilb(S). Here, HN is a polarization of the form Sym(NH) − E where H is an ample divisor on S and E ⊂ Hilb(S) denotes the exceptional divisor of the Hilbert–Chow morphism. The proof of the theorem relies upon the fundamental short exact sequence for tautological vector bundles on the blowup of S ×S and upon the corresponding result for curves which was proved by Mistretta [Mi]. Originally, our interest in this result came from the desire to produce vector bundles on Hilbert schemes of K3 surfaces with interesting metrics and with interesting Chern classes. For this reason we give a formula for the Chern classes of L[2] in terms of the symmetric product of c1(L), of [E] and of the characteristic classes of Hilb(S). Moduli spaces of stable sheaves on K3 surfaces have been studied extensively in the literature (cf. e.g. [Mu], [OG], [HL], [Ma]). These spaces are particularly interesting because they are among the few examples of compact Hyperkahler manifolds (cf. Huybrechts’ chapter in [GHJ]). In analogy it seems to be promising to study moduli spaces of stable sheaves on higher-dimensional Hyperkahler manifolds. In this note we present examples of stable sheaves on the second Hilbert scheme of a K3 surface which is one of the two prototypes of four-dimensional compact Hyperkahler manifolds. After introducing some notation in Section 1 we prove the theorem in Section 2. Finally we calculate the Chern classes of L[2] in Section 3. This work was supported by the SFB/TR 45 ‘Periods, Moduli Spaces and Arithmetic of Algebraic Varieties’ of the DFG (German Research Foundation) and by the Bonn International Graduate School in Mathematics (BIGS). 1 2 ULRICH SCHLICKEWEI Acknowledgements. This paper is a part of my Ph.D. thesis prepared at the University of Bonn. It is a great pleasure to thank my advisor Daniel Huybrechts for his constant support. I am also grateful to Luca Scala for helpful discussions on Chern characters of tautological vector bundles and to Ernesto Mistretta for explaining to me his results about stable vector bundles on symmetric products of curves. 1. Some notation Let ι∆ : ∆ ↪→ S × S be the diagonal. Denote by σ : S × S → S × S the blowup of S × S in ∆. The natural action of the symmetric group S2 on S ×S extends to a holomorphic action on S × S and Hilb(S) = S × S/S2. Let ιD : D ↪→ S × S be the exceptional divisor of σ. In the following diagram we summarize the situation and, at the same time, give names to the various natural maps.