Gromov-Witten invariants : Crepant resolutions and simple flops

Citation

Cheong, Wan Keng (2010) Gromov-Witten invariants : Crepant resolutions and simple flops. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:05212010-060144793

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 [Sym^n(S)] of S and the equivariant extremal quantum cohomology of the Hilbert scheme Hilb^n(S) of n points in S. This proves a generalization of Ruan's Cohomological Crepant Resolution Conjecture for the case of Sym^n(S). Moreover, we determine the operators of small quantum multiplication by divisor classes on the orbifold quantum cohomology of [Sym^n(Ar)], where Ar is the minimal resolution of the cyclic quotient singularity C^2/Z_(r+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 [Sym^n(Ar)] and Hilb^n(Ar). More strikingly, this allows us to complete a tetrahedron of equivalences relating the Gromov-Witten theories of [Sym^n(Ar)]/Hilb^n(Ar) and the relative Gromov-Witten/Donaldson-Thomas theories of Ar x P^1. 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 P^r for positive integers r and d. The result generalizes the multiple cover formula for P^1 and reveals that any simple P^r 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.

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