International Congress of Mathematicians

484 L. GöttscheIt is usually best to look at the Hilbert schemes S^ for all n at the sametime, and to study their invariants in terms of generating functions, because newstructures emerge this way. For Euler numbers, Betti numbers and conjecturallyfor the elliptic genus these generating functions will be modular forms and Jacobiforms. This fits into general conjectures from physics about invariants of modulispaces. Also the cohomology rings of the S^ for different n are closely tied together.The direct sum over n of all the cohomologies is a representation for the Heisenbergalgebra modeled on the cohomology of S, and the cohomology rings of the S^ canbe described in terms of vertex operators. In the case that the canonical divisor ofthe surface S is trivial, this leads to an elementary description of the cohomologyrings of the S^n\which coincides with the orbifold cohomology ring of the symmetricpower, giving a nontrivial check of a conjecture relating the cohomology ring of anice resolution of an orbifold to the recently defined orbifold cohomology ring.The Hilbert schemes S^ are closely related to other moduli spaces of objectson S, including moduli of vector bundles and moduli of curves e.g. via the Serrecorrespondence and the Mukai Fourier transform. This leads to applications to thegeometry and topology of these moduli spaces, to Donaldson invariants, and alsoto formulas in the enumerative geometry of curves on surfaces and Gromov-Witteninvariants. We want to explain some of these results and connections. We will notattempt to give a complete overview, but rather give a glimpse of some of the morestriking results.1. The Hilbert scheme of pointsIn this article S will usually be a smooth projective surface over the complexnumbers. We will study the Hilbert scheme S^ = Hilb n (S) of subschemes of lengthn on S. The points of S^ correspond to finite subschemes W C S of length n,in particular a general point corresponds just to a set of n distinct points on S.S^ is projective and comes with a universal family Z n (S) C S^ x S, consisting ofthe (W, x) with x £ W. An important role in applications of S^ is played by thetautological vector bundles F^ := n*q*(L) of rank n on S^. Here n : Z n (S) —^ S^and q : Z n (S) —¥ S are the projections and F is a line bundle on S.Closely related to S^ is the symmetric power S^ = S n /G n , the quotientof S n by the action of the symmetric group G n . The points of S^ correspond toeffective 0-cycles ^ n»[xj], where the xi are distinct points of S and the sum of therii is n. The forgetful mapp : SW -• S {n) ,W ^ Y, len (°w,x) Mis a morphism. The symmetric power S^ is singular, as for instance the fix-locus ofany transposition in G n has codimension 2. On the other hand by [22] S^ is smoothand connected of dimension 2n and p : S^ —t S^ is a resolution of singularities.In fact this is a particularly nice resolution: If Y is a Gorenstein variety, i.e. thedualizing sheaf is a line bundle Ky, a resolution / : X —t Y of singularities is calledcrêpant if it preserves the canonical divisor, that is f*Ky = Kx- It is easy to see