Citation
Cheong, Wan Keng (2010) GromovWitten invariants : Crepant resolutions and simple flops. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:05212010060144793
Abstract
Let S be any smooth toric surface. We establish a ring isomorphism between the equivariant extended ChenRuan cohomology of the nfold 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 BryanGraber'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 GromovWitten theories of [Sym^n(Ar)]/Hilb^n(Ar) and the relative GromovWitten/DonaldsonThomas 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 GromovWitten invariants of arbitrary genus. It also provides examples for which Ruan's Minimal Model Conjecture holds.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  symmetric product; Hilbert scheme; orbifold; crepant resolution; simple flop; GromovWitten invariant 
Degree Grantor:  California Institute of Technology 
Division:  Physics, Mathematics and Astronomy 
Major Option:  Mathematics 
Awards:  Scott Russell Johnson Graduate Dissertation Prize in Mathematics, 2010 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  29 April 2010 
Record Number:  CaltechTHESIS:05212010060144793 
Persistent URL:  http://resolver.caltech.edu/CaltechTHESIS:05212010060144793 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  5820 
Collection:  CaltechTHESIS 
Deposited By:  Wan Keng Cheong 
Deposited On:  25 May 2010 16:11 
Last Modified:  22 Aug 2016 21:19 
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