2021 Theses Doctoral
Moduli of stable maps with fields
Given a triple (π,π,π΄) of a smooth projective variety, a rank π³ vector bundle and a regular section, we construct a moduli of stable maps to π with fields together with a cosection localized virtual class. We show the class coincides up to a sign with the virtual fundamental class on the moduli space of stable maps to the vanishing locus π‘ of π΄. We show that this gives a generalization of the Quantum Lefschetz hyperplane principle, which relates the virtual classes of the moduli of stable maps to π and that of the moduli of stable maps to π‘ if the bundle π is convex. We further generalize this result by considering (π³,Ι,s) where π³is a smooth Deligne--Mumford stack with projective coarse moduli space. In this setting, we can construct a moduli space of twisted stable maps to π³with fields. This moduli space will have (possibly disconnected) components of constant virtual dimension indexed by π-tuples of components of the inertia stack of π³. We show that its cosection localized virtual class on each component agrees up to a sign with the virtual fundamental class of a corresponding component of the moduli of twisted stable maps to ΖΆ=s=0. This generalizes similar comparison results of Chang--Li, Kim--Oh and Chang--Li and presents a different approach from Chen--Janda--Webb.
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Picciotto_columbia_0054D_16446.pdf application/pdf 477 KB Download File
More About This Work
- Academic Units
- Mathematics
- Thesis Advisors
- Liu, Chiu-Chu
- Degree
- Ph.D., Columbia University
- Published Here
- April 20, 2021