Relative quasimaps and mirror formulae
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 FanTsengYou coincides with a natural generating function for relative quasimap invariants, providing mirrorsymmetric motivation for the theory.
 Publication:

arXiv eprints
 Pub Date:
 October 2017
 DOI:
 10.48550/arXiv.1710.11158
 arXiv:
 arXiv:1710.11158
 Bibcode:
 2017arXiv171011158B
 Keywords:

 Mathematics  Algebraic Geometry;
 14N35
 EPrint:
 32 pages, 1 figure