Relative quasimaps and mirror formulae

TL;DR

This paper develops a theory of relative quasimaps for genus zero, establishing a virtual class, recursion formulas, and a mirror symmetry connection, extending quasimap invariants to relative settings involving hypersurfaces in toric varieties.

Contribution

It introduces a new framework for relative quasimaps in genus zero, including virtual classes, recursion relations, and a mirror symmetry interpretation for hypersurfaces in toric varieties.

Findings

Established a virtual class for relative quasimaps to (X,Y).

Derived a recursion formula relating invariants of different tangencies.

Connected the relative I-function with mirror symmetry and quasimap invariants.

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.