Abstract:
We will explain an axiomatic approach toward the cohomology rings of Hilbert schemes of points on an arbitrary projective surface and toward the Chen-Ruan orbifold cohomology rings of the symmetric products (For the latter, the Jucy-Murphy elements of the symmetric group play a key role.) A consequence of this is another proof of Ruan's conjecture for K3 surfaces saying that the cohomology rings of the Hilbert scheme and of the symmetric product are isomorphic. We then explain how to deal with a large class of quasi-projective surfaces. In particular for ALE spaces as well as the cotangent bundle of any algebraic curve, we varify again Ruan's conjecture. We will also explain a remarkable simple property of the cohomology rings ofHilbert schemes of points on a quasi-projective surface (which is not shared for projective surfaces). This is joint work with Wei-Ping Li (Hong Kong) and Zhenbo Qin (Missouri).
