In this paper, we prove that Ruan's Cohomological Crepant Resolution Conjecture holds for the Hilbert-Chow morphisms. There are two main ideas in the proof. The first one is to use the representation theoretic approach proposed in [QW] which involves vertex operator techniques. The second is to prove certain universality structures about the 3-pointed genus-0 extremal Gromov-Witten invariants of the Hilbert schemes by using the indexing techniques from [LiJ], the product formula from [Beh2] and the co-section localization from [KL1, KL2, LL]. We then reduce Ruan's Conjecture from the case of an arbitrary surface to the case of smooth projective toric surfaces which has already been proved in [Che].
Hilbert schemes of points on surfacesLet X be a smooth projective complex surface with the canonical class K X and the Euler class e X , and X [n] be the Hilbert scheme of points in X. An element in X [n] is represented by a length-n 0-dimensional closed subscheme ξ of X. It is well known that X [n] is smooth. For a subset Y ⊂ X, define M n (Y ) = {ξ ∈ X [n] | Supp(ξ) = {x} for some x ∈ Y }.