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.
Highlights
Relative Gromov–Witten theory for smooth pairs (X, Y) occupies a central place in modern enumerative geometry, owing both to its intrinsic interest and to the role it plays in the degeneration formula [17, 24, 25, 31, 34]
There are several possible approaches here, depending on one’s viewpoint on mirror symmetry, but the end goal of each should be to obtain pleasant closed formulae for generating functions of relative Gromov– Witten invariants
1.2.1 Construction and recursion We begin by constructing moduli spaces of relative stable quasimaps in the spirit of Gathmann, that is, as substacks of moduli spaces of quasimaps: Q0,α(X|Y, β) → Q0,n(X, β)
Summary
Relative Gromov–Witten theory for smooth pairs (X, Y) occupies a central place in modern enumerative geometry, owing both to its intrinsic interest and to the role it plays in the degeneration formula [17, 24, 25, 31, 34]. There are several possible approaches here, depending on one’s viewpoint on mirror symmetry, but the end goal of each should be to obtain pleasant closed formulae for generating functions of relative Gromov– Witten invariants. Recent work of Fan–Tseng–You [14] uses the correspondence between relative invariants and the Gromov–Witten invariants of orbifolds [1] in order to derive a mirror theorem for certain restricted generating functions of relative invariants. Our motivation comes from the theory of stable quasimaps; this theory dates back to Givental’s earliest work on mirror symmetry [21], and since has been systematised and extended in order to prove a large class of mirror theorems in the absolute setting [5, 6, 9, 10]. The belief is that a fully f ledged theory of relative quasimaps should lead to an powerful collection of mirror theorems in the relative setting
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