Gromov-Witten Invariants: Crepant Resolutions and Simple Flops

Abstract

Let S be any smooth toric surface. We establish a ring isomorphism between the equivariant extended Chen-Ruan cohomology of the n-fold symmetric product stack [Symn(S)] of S and the equivariant extremal quantum cohomology of the Hilbert scheme Hilbn(S) of n points in S. This proves a generalization of Ruan's Cohomological Crepant Resolution Conjecture for the case of Symn(S). Moreover, we determine the operators of small quantum multiplication by divisor classes on the orbifold quantum cohomology of [Symn(Ar)], where Ar is the minimal resolution of the cyclic quotient singularity C2/Zr+1. Under the assumption of the nonderogatory conjecture, these operators completely determine the quantum ring structure, which gives an affirmative answer to Bryan-Graber's Crepant Resolution Conjecture on [Symn(Ar)] and Hilbn(Ar). More strikingly, this allows us to complete a tetrahedron of equivalences relating the Gromov-Witten theories of [Symn(Ar)]/Hilbn(Ar) and the relative Gromov-Witten/Donaldson-Thomas theories of Ar x P1. Finally, we prove a closed formula for an excess integral over the moduli space of degree d stable maps from unmarked curves of genus one to the projective space Pr for positive integers r and d. The result generalizes the multiple cover formula for Pr and reveals that any simple Pr flop of smooth projective varieties preserves the theory of extremal Gromov-Witten invariants of arbitrary genus. It also provides examples for which Ruan's Minimal Model Conjecture holds.