Relative Quasimaps and Mirror Formulae. (22nd January 2020)

• Record Type: Journal Article
• Authors: Battistella, Luca; Nabijou, Navid
• Is Part Of: International mathematics research notices. Volume 2021:Number 10(2021)
• Page Start: 7885
• Page End: 7931
• Publication Date: 2020-01-22
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnz339
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: We construct and study the theory of relative quasimaps in genus zero, in the spirit of Gathmann. When $X$ is a smooth toric variety and $Y$ is a smooth very ample hypersurface in $X$, we produce a virtual class on the moduli space of relative quasimaps to $(X, Y)$, which we use to define relative quasimap invariants. We obtain a recursion formula which expresses each relative invariant in terms of invariants of lower tangency, and apply this formula to derive a quantum Lefschetz theorem for quasimaps, expressing the restricted quasimap invariants of $Y$ in terms of those of $X$ . Finally, we show that the relative $I$ -function of Fan–Tseng–You coincides with a natural generating function for relative quasimap invariants, providing mirror-symmetric motivation for the theory.