Relative Quasimaps and Mirror Formulae

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.