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.
| Identifer | oai:union.ndltd.org:columbia.edu/oai:academiccommons.columbia.edu:10.7916/d8-g3da-xp58 |
| Date | January 2021 |
| Creators | Picciotto, Renata |
| Source Sets | Columbia University |
| Language | English |
| Detected Language | English |
| Type | Theses |
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