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 Fan-Tseng-You coincides with a natural generating function for relative quasimap invariants, providing mirror-symmetric motivation for the theory.

32 pages, 1 figure; comments welcome. v2: added a stronger version of the quantum Lefschetz theorem. v3: additional section exploring applications to relative mirror symmetry, new title and introduction. v4: final version published in IMRN

References in corpus (2)

• Lectures on K-theoretic computations in enumerative geometry
• Quasimap Wall-crossings and Mirror Symmetry

Cited by in corpus (4)

• Curve counting in genus one: elliptic singularities & relative geometry
• The local-orbifold correspondence for simple normal crossings pairs
• The local/logarithmic correspondence and the degeneration formula for quasimaps
• Quantum Lefschetz without curves