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Moduli of stable maps with fields

Picciotto, Renata

Given a triple (𝑋,𝘌,𝘴) of a smooth projective variety, a rank 𝘳 vector bundle and a regular section, we construct a moduli of stable maps to 𝑋 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 𝘑 of 𝘴. 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 𝑋 and that of the moduli of stable maps to 𝘑 if the bundle 𝘌 is convex. We further generalize this result by considering (𝒳,Ι›,s) where 𝒳is a smooth Deligne--Mumford stack with projective coarse moduli space. In this setting, we can construct a moduli space of twisted stable maps to 𝒳with fields. This moduli space will have (possibly disconnected) components of constant virtual dimension indexed by 𝓃-tuples of components of the inertia stack of 𝒳. 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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More About This Work

Academic Units
Mathematics
Thesis Advisors
Liu, Chiu-Chu
Degree
Ph.D., Columbia University
Published Here
April 20, 2021