Title: On Hilb/Sym Correspondence
Abstract: For a smooth surface S, the Hilbert scheme of n points on S, Hilb(S,n), is smooth of dimension 2n. The Hilbert-Chow morphism Hilb(S,n)\to S^n/S_n is a crepant resolution of the (singular) n-fold symmetric product variety S^n/S_n associated to S. The so-called crepant resolution conjecture in Gromov-Witten theory predicts in this case explicit equalities between generating functions of Gromov-Witten invariants of Hilb(S,n) and generating functions of Gromov-Witten invariants of the symmetric product stack Sym(S,n). In this talk, we discuss the formulation of this conjecture and proofs in known cases.