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. doi:10.7907/6KZZ-MT72. https://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 [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(A_r)], where A_r is the minimal resolution of the cyclic quotient singularity C²/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 [Symⁿ(A_r)] and Hilbⁿ(A_r). More strikingly, this allows us to complete a tetrahedron of equivalences relating the Gromov-Witten theories of [Symⁿ(A_r)]/Hilbⁿ(A_r) 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 P^r for positive integers r and d. The result generalizes the multiple cover formula for P^r 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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